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Quaternions and spatial rotation

Unit quaternions, known as versors, provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation about an arbitrary axis. Rotation and orientation quaternions have applications in computer graphics, computer vision, robotics, navigation, molecular dynamics, flight dynamics, orbital mechanics of satellites, and crystallographic texture analysis.

When used to represent rotation, unit quaternions are also called rotation quaternions as they represent the 3D rotation group. When used to represent an orientation (rotation relative to a reference coordinate system), they are called orientation quaternions or attitude quaternions. A spatial rotation around a fixed point of $$\theta$$ radians about a unit axis $$(X,Y,Z)$$ that denotes the Euler axis is given by the quaternion $$(C, X \, S, Y \, S, Z \, S)$$, where $$C = \cos(\theta/2)$$ and $$S=\sin(\theta/2)$$.

Compared to rotation matrices, quaternions are more compact, efficient, and numerically stable. Compared to Euler angles, they are simpler to compose. However, they are not as intuitive and easy to understand and, due to the periodic nature of sine and cosine, rotation angles differing precisely by the natural period will be encoded into identical quaternions and recovered angles in radians will be limited to $$[0,2\pi]$$.

Using quaternions as rotations
In 3-dimensional space, according to Euler's rotation theorem, any rotation or sequence of rotations of a rigid body or coordinate system about a fixed point is equivalent to a single rotation by a given angle $$\theta$$ about a fixed axis (called the Euler axis) that runs through the fixed point. The Euler axis is typically represented by a unit vector $$\vec{u}$$ ($$\hat{e}$$ in the picture). Therefore, any rotation in three dimensions can be represented as via a vector $$\vec{u}$$ and an angle $$\theta$$.

Quaternions give a simple way to encode this axis–angle representation using four real numbers, and can be used to apply (calculate) the corresponding rotation to a position vector $(x,y,z)$, representing a point relative to the origin in R3.

Euclidean vectors such as $(2, 3, 4)$ or $(a_{x}, a_{y}, a_{z})$ can be rewritten as $2 i + 3 j + 4 k$ or $a_{x} i + a_{y} j + a_{z} k$, where $i$, $j$, $k$ are unit vectors representing the three Cartesian axes (traditionally $x$, $y$, $z$), and also obey the multiplication rules of the fundamental quaternion units by interpreting the Euclidean vector $(a_{x}, a_{y}, a_{z})$ as the vector part of the pure quaternion $(0, a_{x}, a_{y}, a_{z})$.

A rotation of angle $$\theta$$ around the axis defined by the unit vector
 * $$\mathbf{u} = (u_x, u_y, u_z) = u_x\mathbf{i} + u_y\mathbf{j} + u_z\mathbf{k}$$

can be represented by conjugation by a unit quaternion $q$. Since the quaternion product $$\ (0 + u_x\mathbf{i} + u_y\mathbf{j} + u_z\mathbf{k})(0 - u_x\mathbf{i} - u_y\mathbf{j} - u_z\mathbf{k})$$ gives 1, using the Taylor series of the exponential function, the extension of Euler's formula results:

 * $$ \mathbf{q} = e^{\frac{\theta}{2}{(u_x\mathbf{i} + u_y\mathbf{j} + u_z\mathbf{k})}} = \cos \frac{\theta}{2} + (u_x\mathbf{i} + u_y\mathbf{j} + u_z\mathbf{k}) \sin \frac{\theta}{2} = \cos \frac{\theta}{2} + \mathbf u \sin \frac{\theta}{2} $$

It can be shown that the desired rotation can be applied to an ordinary vector $$\mathbf{p} = (p_x, p_y, p_z) = p_x\mathbf{i} + p_y\mathbf{j} + p_z\mathbf{k}$$ in 3-dimensional space, considered as the vector part of the pure quaternion $$\mathbf{p'}$$, by evaluating the conjugation of $p'$ by $q$, given by:
 * $$L(\mathbf{p'}) := \mathbf{q} \mathbf{p'} \mathbf{q}^{-1} = (0, \mathbf r),$$
 * $$\mathbf r = (\cos^2\frac{\theta}{2}-\sin^2\frac{\theta}{2}||\mathbf u||^2)\mathbf p + 2\sin^2\frac{\theta}{2} (\mathbf u . \mathbf p)\mathbf u + 2\cos\frac{\theta}{2}\sin\frac{\theta}{2}(\mathbf u \times \mathbf p), $$

using the Hamilton product, where the vector part of the pure quaternion $L(p') = (0, r_{x}, r_{y}, r_{z})$ is the new position vector of the point after the rotation. In a programmatic implementation, the conjugation is achieved by constructing a pure quaternion whose vector part is $p$, and then performing the quaternion conjugation. The vector part of the resulting pure quaternion is the desired vector $r$. Clearly, $$L$$ provides a linear transformation of the quaternion space to itself; also, since $$\mathbf q $$ is unitary, the transformation is an isometry. Also, $$L(\mathbf q) = \mathbf q $$ and so $$L$$ leaves vectors parallel to $$\mathbf q$$ invariant. So, by decomposing $$\mathbf p $$ as a vector parallel to the vector part $$(u_x, u_y, u_z)\sin \frac{\theta}{2}$$ of $$\mathbf q $$ and a vector normal to the vector part of $$\mathbf q $$  and showing that the application of $$L$$ to the normal component of $$\mathbf p$$ rotates it, the claim is shown. So let $$\mathbf n$$ be the component of $$\mathbf p$$ orthogonal to the vector part of $$\mathbf q$$ and let $$\mathbf n_T = \mathbf n \times \mathbf u $$. It turns out that the vector part of $$L(0, \mathbf n)$$ is given by
 * $$\left(\cos^2\frac{\theta}{2} -\sin^2\frac{\theta}{2}\right) \mathbf{n} + 2 \left(\cos\frac{\theta}{2}\sin\frac{\theta}{2} \right)\mathbf{n}_T = \cos\theta \mathbf{n} + \sin\theta \mathbf{n}_T$$.

The conjugation of $p'$ by $q$ can be expressed with fewer arithmetic operations as:
 * $$\mathbf{r} = \mathbf{p} + 2\cos\frac{\theta}{2}\sin\frac{\theta}{2}(\mathbf u \times \mathbf p) + 2 \sin^2\frac{\theta}{2} \mathbf u \times (\mathbf u \times \mathbf p) .$$

A geometric fact independent of quaternions is the existence of a two-to-one mapping from physical rotations to rotational transformation matrices. If 0 ⩽ $$\theta$$ ⩽ $$2\pi$$, a physical rotation about $$\vec{u}$$ by $$\theta$$ and a physical rotation about $$-\vec{u}$$ by $$2\pi-\theta$$ both achieve the same final orientation by disjoint paths through intermediate orientations. By inserting those vectors and angles into the formula for $q$ above, one finds that if $q$ represents the first rotation, $−q$ represents the second rotation. This is a geometric proof that conjugation by $q$ and by $−q$ must produce the same rotational transformation matrix. That fact is confirmed algebraically by noting that the conjugation is quadratic in $q$, so the sign of $q$ cancels, and does not affect the result. (See 2:1 mapping of SU(2) to SO(3)) If both rotations are a half-turn $$(\theta=\pi)$$, both $q$ and $−q$ will have a real coordinate equal to zero. Otherwise, one will have a positive real part, representing a rotation by an angle less than $$\pi$$, and the other will have a negative real part, representing a rotation by an angle greater than $$\pi$$.

Mathematically, this operation carries the set of all "pure" quaternions $p$ (those with real part equal to zero)—which constitute a 3-dimensional space among the quaternions—into itself, by the desired rotation about the axis u, by the angle θ. (Each real quaternion is carried into itself by this operation. But for the purpose of rotations in 3-dimensional space, we ignore the real quaternions.)

The rotation is clockwise if our line of sight points in the same direction as $$\vec{u}$$.

In this (which?) instance, $q$ is a unit quaternion and
 * $$ \mathbf{q}^{-1} = e^{-\frac{\theta}{2}{(u_x\mathbf{i} + u_y\mathbf{j} + u_z\mathbf{k})}} = \cos \frac{\theta}{2} - (u_x\mathbf{i} + u_y\mathbf{j} + u_z\mathbf{k}) \sin \frac{\theta}{2} .$$

It follows that conjugation by the product of two quaternions is the composition of conjugations by these quaternions: If $p$ and $q$ are unit quaternions, then rotation (conjugation) by $pq$ is
 * $$\mathbf{p q} \vec{v} (\mathbf{p q})^{-1} = \mathbf{p q} \vec{v} \mathbf{q}^{-1} \mathbf{p}^{-1} = \mathbf{p} (\mathbf{q} \vec{v} \mathbf{q}^{-1}) \mathbf{p}^{-1}$$,

which is the same as rotating (conjugating) by $q$ and then by $p$. The scalar component of the result is necessarily zero.

The quaternion inverse of a rotation is the opposite rotation, since $$\mathbf{q}^{-1} (\mathbf{q} \vec{v} \mathbf{q}^{-1}) \mathbf{q} = \vec{v}$$. The square of a quaternion rotation is a rotation by twice the angle around the same axis. More generally $q^{n}$ is a rotation by $n$ times the angle around the same axis as $q$. This can be extended to arbitrary real $n$, allowing for smooth interpolation between spatial orientations; see Slerp.

Two rotation quaternions can be combined into one equivalent quaternion by the relation:

 * $$\mathbf{q}' = \mathbf{q}_2 \mathbf{q}_1 $$

in which $q'$ corresponds to the rotation $q_{1}$ followed by the rotation $q_{2}$. Thus, an arbitrary number of rotations can be composed together and then applied as a single rotation. (Note that quaternion multiplication is not commutative.)

Example conjugation operation

Conjugating $i$ by $j$ refers to the operation $k$.

Consider the rotation $f$ around the axis $$\vec{v} = \mathbf{i} + \mathbf{j} + \mathbf{k}$$, with a rotation angle of 120°, or $2\pi⁄3$ radians.

 * $$\alpha = \frac{2 \pi}{3}$$

The length of $$\vec{v}$$ is √3, the half angle is $π⁄3$ (60°) with cosine $1⁄2$, ($p$) and sine $√3⁄2$, ($q$). We are therefore dealing with a conjugation by the unit quaternion

 * $$\begin{align}

u &= \cos\frac{\alpha}{2} + \sin\frac{\alpha}{2}\cdot \frac{1}{\| \vec{v} \| }\vec{v}\\ &= \cos \frac{\pi}{3} + \sin \frac{\pi}{3}\cdot \frac{1}{\sqrt{3}}\vec{v}\\ &= \frac{1}{2} + \frac{\sqrt{3}}{2}\cdot \frac{1}{\sqrt{3}}\vec{v}\\ &= \frac{1}{2} + \frac{\sqrt{3}}{2}\cdot \frac{\mathbf{i}+\mathbf{j}+\mathbf{k}}{\sqrt{3}}\\ &= \frac{1 + \mathbf{i} + \mathbf{j} + \mathbf{k}}{2} \end{align}$$

If $f$ is the rotation function,
 * $$f(a\mathbf{i} + b\mathbf{j} + c\mathbf{k}) = u (a\mathbf{i} + b\mathbf{j} + c\mathbf{k}) u^{-1}$$

It can be proven that the inverse of a unit quaternion is obtained simply by changing the sign of its imaginary components. As a consequence,
 * $$u^{-1} = \dfrac{1- \mathbf{i} - \mathbf{j} - \mathbf{k}}{2}$$

and
 * $$f(a\mathbf{i} + b\mathbf{j} + c\mathbf{k}) = \dfrac{1 + \mathbf{i} + \mathbf{j} + \mathbf{k}}{2}(a\mathbf{i} + b\mathbf{j} + c\mathbf{k}) \dfrac{1 - \mathbf{i} - \mathbf{j} - \mathbf{k}}{2}$$

This can be simplified, using the ordinary rules for quaternion arithmetic, to
 * $$f(a\mathbf{i} + b\mathbf{j} + c\mathbf{k}) = c\mathbf{i} + a\mathbf{j} + b\mathbf{k}$$

As expected, the rotation corresponds to keeping a cube held fixed at one point, and rotating it 120° about the long diagonal through the fixed point (observe how the three axes are permuted cyclically).

Quaternion-derived rotation matrix
A quaternion rotation $$\mathbf{p'} = \mathbf{q} \mathbf{p} \mathbf{q}^{-1}$$ (with $$\mathbf{q} = q_r + q_i \mathbf{i} + q_j \mathbf{j} + q_k \mathbf{k}$$) can be algebraically manipulated into a matrix rotation $$\mathbf{p'} = \mathbf{R p}$$, where $$\mathbf{R}$$ is the rotation matrix given by:

 * $$\mathbf{R} = \begin{bmatrix}

1 - 2s (q_j^2 + q_k^2) & 2s (q_i q_j - q_k q_r) & 2s (q_i q_k + q_j q_r) \\ 2s (q_i q_j + q_k q_r) & 1 - 2s (q_i^2 + q_k^2) & 2s (q_j q_k - q_i q_r) \\ 2s (q_i q_k - q_j q_r) & 2s (q_j q_k + q_i q_r) & 1 - 2s (q_i^2 + q_j^2) \end{bmatrix} $$

Here $$s = \|q\|^{-2}$$ and if $q$ is a unit quaternion, $$s = 1^{-2} = 1$$.

This can be obtained by using vector calculus and linear algebra if we express $$\mathbf p$$ and $$\mathbf q$$ as scalar and vector parts and use the formula for the multiplication operation in the equation $$\mathbf{p'} = \mathbf{q} \mathbf{p} \mathbf{q}^{-1}$$. If we write $$\mathbf p$$ as $$\left(0,\ \mathbf p\right)$$, $$\mathbf p'$$ as $$\left(0,\ \mathbf p'\right)$$ and $$\mathbf q$$ as $$\left(q_r,\ \mathbf v\right)$$, where $$\mathbf v = \left(q_i, q_j, q_k\right)$$, our equation turns into $$\left(0,\ \mathbf p'\right) = \left(q_r,\ \mathbf v\right) \left(0,\ \mathbf p\right) s\left(q_r,\ -\mathbf v\right)$$. By using the formula for multiplication of two quaternions that are expressed as scalar and vector parts,

 * $$\left(r_1,\ \vec{v}_1\right) \left(r_2,\ \vec{v}_2\right) = \left(r_1 r_2 - \vec{v}_1\cdot\vec{v}_2,\ r_1\vec{v}_2 + r_2\vec{v}_1 + \vec{v}_1\times\vec{v}_2\right),$$

this equation can be rewritten as

 * $$\begin{align}

(0,\ \mathbf p') = &((q_r,\ \mathbf v) (0,\ \mathbf p)) s(q_r,\ -\mathbf v) \\ = &(q_r 0 - \mathbf v \cdot \mathbf p,\ q_r\mathbf p + 0\mathbf v + \mathbf v \times \mathbf p)s(q_r,\ -\mathbf v) \\ = &s(-\mathbf v \cdot \mathbf p,\ q_r\mathbf p + \mathbf v \times \mathbf p)(q_r,\ -\mathbf v) \\ = &s(-\mathbf v \cdot \mathbf p q_r - (q_r\mathbf p + \mathbf v \times \mathbf p)\cdot(-\mathbf v),\ (-\mathbf v \cdot \mathbf p)(-\mathbf v) + q_r(q_r\mathbf p + \mathbf v \times \mathbf p) + (q_r\mathbf p + \mathbf v \times \mathbf p)\times(-\mathbf v)) \\ = &s\left(-\mathbf v \cdot \mathbf p q_r + q_r\mathbf v \cdot \mathbf p,\ \mathbf v\left(\mathbf v \cdot \mathbf p\right) + q_r^2\mathbf p + q_r\mathbf v \times \mathbf p + \mathbf v \times \left(q_r \mathbf p + \mathbf v \times \mathbf p\right)\right) \\ = &\left(0,\ s\left(\mathbf v \otimes \mathbf v + q_r^2\mathbf{I} + 2 q_r[\mathbf v]_{\times} + [\mathbf v]_{\times}^2\right) \mathbf p\right), \end{align}$$

where $$\otimes$$ denotes the outer product, $$\mathbf{I}$$ is the identity matrix and $$[\mathbf v]_{\times}$$ is the transformation matrix that when multiplied from the right with a vector $$\mathbf u$$ gives the cross product $$\mathbf v \times \mathbf u$$.

Since $$\mathbf p' = \mathbf{R}\mathbf p$$, we can identify $$\mathbf{R}$$ as $$s\left(\mathbf v \otimes \mathbf v + q_r^2\mathbf{I} + 2 q_r[\mathbf v]_{\times} + [\mathbf v]_{\times}^2\right)$$, which upon expansion should result in the expression written in matrix form above.

Recovering the axis-angle representation
The expression $$\mathbf{q} \mathbf{p} \mathbf{q}^{-1}$$ rotates any vector quaternion $$\mathbf{p}$$ around an axis given by the vector $$\mathbf{a}$$ by the angle $$\theta$$, where $$\mathbf{a}$$ and $$\theta$$ depends on the quaternion $$\mathbf{q} = q_r + q_i \mathbf{i} + q_j \mathbf{j} + q_k \mathbf{k}$$.

$$\mathbf{a}$$ and $$\theta$$ can be found from the following equations:

 * $$\begin{align}

(a_x, a_y, a_z) ={} &\frac{(q_i, q_j, q_k)}{\sqrt{q_i^2 + q_j^2 + q_k^2}} \\[2pt] \theta = 2 \operatorname{atan2} &\left(\sqrt{q_i^2 + q_j^2 + q_k^2},\, q_r \right), \end{align}$$

where $$\operatorname{atan2}$$ is the two-argument arctangent. While $$\theta = 2 \operatorname{acos} (q_r)$$ works, it is numerically unstable (inaccurate) near $$q_r = \pm1$$ for numbers with finite precision.

Care should be taken when the quaternion approaches a scalar, since due to degeneracy the axis of an identity rotation is not well-defined.

The composition of spatial rotations
A benefit of the quaternion formulation of the composition of two rotations RB and RA is that it yields directly the rotation axis and angle of the composite rotation RC = RBRA.

Let the quaternion associated with a spatial rotation R be constructed from its rotation axis S with the rotation angle $$\varphi$$ around this axis. The associated quaternion is given by $$ S = \cos\frac{\varphi}{2} + \mathbf{S}\sin\frac{\varphi}{2}. $$ Then the composition of the rotation RB with RA is the rotation RC = RBRA with rotation axis and angle defined by the product of the quaternions $$ A = \cos\frac{\alpha}{2} + \mathbf{A}\sin\frac{\alpha}{2} \quad \text{and} \quad B = \cos\frac{\beta}{2} + \mathbf{B}\sin\frac{\beta}{2}, $$ that is $$ C = \cos\frac{\gamma}{2} + \mathbf{C}\sin\frac{\gamma}{2} = \left(\cos\frac{\beta}{2} + \mathbf{B} \sin\frac{\beta}{2}\right) \left(\cos\frac{\alpha}{2} + \mathbf{A}\sin\frac{\alpha}{2}\right). $$

Expand this product to obtain $$ \cos\frac{\gamma}{2} + \mathbf{C}\sin\frac{\gamma}{2} = \left(    \cos\frac{\beta}{2} \cos\frac{\alpha}{2} -    \mathbf{B} \cdot \mathbf{A} \sin\frac{\beta}{2} \sin\frac{\alpha}{2}\right) + \left(\mathbf{B} \sin\frac{\beta}{2} \cos\frac{\alpha}{2} +   \mathbf{A} \sin\frac{\alpha}{2} \cos\frac{\beta}{2} +    \mathbf{B} \times \mathbf{A} \sin\frac{\beta}{2} \sin\frac{\alpha}{2}  \right). $$

Divide both sides of this equation by the identity, which is the law of cosines on a sphere, $$\cos\frac{\gamma}{2} = \cos\frac{\beta}{2} \cos\frac{\alpha}{2} - \mathbf{B} \cdot \mathbf{A} \sin\frac{\beta}{2} \sin\frac{\alpha}{2},$$ and compute $$ \mathbf{C} \tan\frac{\gamma}{2} = \frac{ \mathbf{B} \tan\frac{\beta}{2} + \mathbf{A} \tan\frac{\alpha}{2} + \mathbf{B} \times \mathbf{A} \tan\frac{\beta}{2}\tan\frac{\alpha}{2} }{1 - \mathbf{B} \cdot \mathbf{A} \tan\frac{\beta}{2}\tan\frac{\alpha}{2}}. $$

This is Rodrigues' formula for the axis of a composite rotation defined in terms of the axes of the two rotations. He derived this formula in 1840 (see page 408).

The three rotation axes A, B, and C form a spherical triangle and the dihedral angles between the planes formed by the sides of this triangle are defined by the rotation angles. Hamilton presented the component form of these equations showing that the quaternion product computes the third vertex of a spherical triangle from two given vertices and their associated arc-lengths, which is also defines an algebra for points in Elliptic geometry.

Axis–angle composition
The normalized rotation axis, removing the $\cos\frac{\gamma}{2}$ from the expanded product, leaves the vector which is the rotation axis, times some constant. Care should be taken normalizing the axis vector when $$\gamma$$ is $$0$$ or $$k 2\pi$$ where the vector is near $$0$$; which is identity, or 0 rotation around any axis.

$$\begin{align} \gamma &= 2\cos^{-1}\left(\cos\frac{\beta}{2} \cos\frac{\alpha}{2} - \mathbf{B} \cdot \mathbf{A} \sin\frac{\beta}{2} \sin\frac{\alpha}{2}\right) \\ \mathbf{D} &= \mathbf{B} \sin\frac{\beta}{2} \cos\frac{\alpha}{2} + \mathbf{A} \sin\frac{\alpha}{2} \cos\frac{\beta}{2} + \mathbf{B} \times \mathbf{A} \sin\frac{\beta}{2} \sin\frac{\alpha}{2} \end{align}$$

Or with angle addition trigonometric substitutions... $$\begin{align} \gamma &= 2 \cos^{-1} \left(\left(1 - \mathbf{A} \cdot \mathbf{B}\right) \cos\frac{\beta - \alpha}{2} + \left(1 + \mathbf{A}\cdot \mathbf{B}\right) \cos\frac{\beta + \alpha}{2}\right) \\ \mathbf{D} &= \left(\sin\frac{\beta + \alpha}{2} + \sin\frac{\beta - \alpha}{2}\right) \mathbf{A} + \left(\sin\frac{\beta + \alpha}{2} - \sin\frac{\beta - \alpha}{2}\right) \mathbf{B} + \left(\cos\frac{\beta - \alpha}{2} - \cos\frac{\beta + \alpha}{2}\right) \mathbf{B} \times \mathbf{A} \end{align}$$

finally normalizing the rotation axis: $\frac{\mathbf{D}}{2 \sin\frac{1}{2}\gamma}$ or $\frac{\mathbf{D}}{\|\mathbf{D}\|}$.

Differentiation with respect to the rotation quaternion
The rotated quaternion $p ↦ qpq^{−1}$ needs to be differentiated with respect to the rotating quaternion $p ↦ q p$, when the rotation is estimated from numerical optimization. The estimation of rotation angle is an essential procedure in 3D object registration or camera calibration. For unitary $q = 1 + i + j + k⁄2$ and pure imaginary $cos  60° = 0.5$, that is for a rotation in 3D space, the derivatives of the rotated quaternion can be represented using matrix calculus notation as
 * $$\begin{align}

\frac{\partial \mathbf{p'}}{\partial \mathbf{q}} \equiv \left[\frac{\partial \mathbf{p'}}{\partial q_0},\frac{\partial \mathbf{p'}}{\partial q_x},\frac{\partial \mathbf{p'}}{\partial q_y},\frac{\partial \mathbf{p'}}{\partial q_z}  \right] = \left[ \mathbf{pq}-(\mathbf{pq})^*,(\mathbf{pqi})^*-\mathbf{pqi},(\mathbf{pqj})^*-\mathbf{pqj},(\mathbf{pqk})^*-\mathbf{pqk} \right]. \end{align}$$

A derivation can be found in.

Quaternions
The complex numbers can be defined by introducing an abstract symbol $sin  60° ≈ 0.866$ which satisfies the usual rules of algebra and additionally the rule $p' = q p q^{−1}$. This is sufficient to reproduce all of the rules of complex number arithmetic: for example:
 * $$(a+b\mathbf{i})(c+d\mathbf{i}) = ac + ad\mathbf{i} + b\mathbf{i}c + b\mathbf{i}d\mathbf{i} = ac + ad\mathbf{i} + bc\mathbf{i} + bd\mathbf{i}^2 = (ac - bd) + (bc + ad) \mathbf{i}.$$

In the same way the quaternions can be defined by introducing abstract symbols $q$, $q$, $p$ which satisfy the rules $i$ and the usual algebraic rules except the commutative law of multiplication (a familiar example of such a noncommutative multiplication is matrix multiplication). From this all of the rules of quaternion arithmetic follow, such as the rules on multiplication of quaternion basis elements. Using these rules, one can show that:
 * $$\begin{align}

&(a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k}) (e + f\mathbf{i} + g\mathbf{j} + h\mathbf{k}) = \\ &(ae - bf - cg - dh) + (af + be + ch - dg) \mathbf{i} + (ag - bh + ce + df) \mathbf{j} + (ah + bg - cf + de) \mathbf{k}. \end{align}$$

The imaginary part $$b\mathbf{i} + c\mathbf{j} + d\mathbf{k}$$ of a quaternion behaves like a vector $$\vec{v} = (b,c,d)$$ in three-dimensional vector space, and the real part $a$ behaves like a scalar in $i^{2} = −1$. When quaternions are used in geometry, it is more convenient to define them as a scalar plus a vector:

 * $$a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k} = a + \vec{v}.$$

Some might find it strange to add a number to a vector, as they are objects of very different natures, or to multiply two vectors together, as this operation is usually undefined. However, if one remembers that it is a mere notation for the real and imaginary parts of a quaternion, it becomes more legitimate. In other words, the correct reasoning is the addition of two quaternions, one with zero vector/imaginary part, and another one with zero scalar/real part:
 * $$q_1 = s + \vec{v} = \left(s, \vec{0}\right) + \left(0, \vec{v}\right).$$

We can express quaternion multiplication in the modern language of vector cross and dot products (which were actually inspired by the quaternions in the first place ). When multiplying the vector/imaginary parts, in place of the rules $i$ we have the quaternion multiplication rule:
 * $$\vec{v} \vec{w} = - \vec{v} \cdot \vec{w} + \vec{v} \times \vec{w},$$

where:
 * $$\vec{v} \vec{w}$$ is the resulting quaternion,
 * $$\vec{v} \times \vec{w}$$ is vector cross product (a vector),
 * $$\vec{v} \cdot \vec{w}$$ is vector scalar product (a scalar).

Quaternion multiplication is noncommutative (because of the cross product, which anti-commutes), while scalar–scalar and scalar–vector multiplications commute. From these rules it follows immediately that (see ):

 * $$q_1 q_2 = \left(s + \vec{v}\right) \left(t + \vec{w}\right) = \left(s t - \vec{v} \cdot \vec{w}\right) + \left(s \vec{w} + t \vec{v} + \vec{v} \times \vec{w}\right).$$

The (left and right) multiplicative inverse or reciprocal of a nonzero quaternion is given by the conjugate-to-norm ratio (see details):

 * $$q_1^{-1} = \left(s + \vec{v}\right)^{-1} = \frac{\left(s + \vec{v}\right)^*}{\lVert s + \vec{v} \rVert^2} = \frac{s - \vec{v}}{s^2 + \lVert \vec{v} \rVert^2},$$

as can be verified by direct calculation (note the similarity to the multiplicative inverse of complex numbers).

Rotation identity
Let $$\vec{u}$$ be a unit vector (the rotation axis) and let $$q = \cos \frac{\alpha}{2} + \vec{u} \sin \frac{\alpha}{2}$$. Our goal is to show that
 * $$\vec{v}' = q \vec{v} q^{-1} = \left( \cos \frac{\alpha}{2} + \vec{u} \sin \frac{\alpha}{2} \right) \, \vec{v} \, \left( \cos \frac{\alpha}{2} - \vec{u} \sin \frac{\alpha}{2} \right)$$

yields the vector $$\vec{v}$$ rotated by an angle $$\alpha$$ around the axis $$\vec{u}$$. Expanding out (and bearing in mind that $$\vec{u}\vec{v} = \vec{u} \times \vec{v} - \vec{u} \cdot \vec{v}$$), we have

 * $$\begin{align}

\vec{v}' &= \vec{v} \cos^2 \frac{\alpha}{2} + \left(\vec{u}\vec{v} - \vec{v}\vec{u}\right) \sin \frac{\alpha}{2} \cos \frac{\alpha}{2} - \vec{u}\vec{v}\vec{u} \sin^2 \frac{\alpha}{2} \\[6pt] &= \vec{v} \cos^2 \frac{\alpha}{2} + 2 \left(\vec{u} \times \vec{v}\right) \sin \frac{\alpha}{2} \cos \frac{\alpha}{2} - \left(\left(\vec{u} \times \vec{v}\right) - \left(\vec{u} \cdot \vec{v}\right)\right)\vec{u} \sin^2 \frac{\alpha}{2} \\[6pt] &= \vec{v} \cos^2 \frac{\alpha}{2} + 2 \left(\vec{u} \times \vec{v}\right) \sin \frac{\alpha}{2} \cos \frac{\alpha}{2} - \left(\left(\vec{u} \times \vec{v}\right)\vec{u} - \left(\vec{u} \cdot \vec{v}\right)\vec{u}\right) \sin^2 \frac{\alpha}{2} \\[6pt] &= \vec{v} \cos^2 \frac{\alpha}{2} + 2 \left(\vec{u} \times \vec{v}\right) \sin \frac{\alpha}{2} \cos \frac{\alpha}{2} - \left(\left(\left(\vec{u} \times \vec{v}\right) \times \vec{u} - \left(\vec{u} \times \vec{v}\right) \cdot \vec{u}\right) - \left(\vec{u} \cdot \vec{v}\right)\vec{u}\right) \sin^2 \frac{\alpha}{2} \\[6pt] &= \vec{v} \cos^2 \frac{\alpha}{2} + 2 \left(\vec{u} \times \vec{v}\right) \sin \frac{\alpha}{2} \cos \frac{\alpha}{2} - \left(\left(\vec{v} - \left(\vec{u} \cdot \vec{v}\right)\vec{u}\right) - 0 - \left(\vec{u} \cdot \vec{v}\right)\vec{u}\right) \sin^2 \frac{\alpha}{2} \\[6pt] &= \vec{v} \cos^2 \frac{\alpha}{2} + 2 \left(\vec{u} \times \vec{v}\right) \sin \frac{\alpha}{2} \cos \frac{\alpha}{2} - \left(\vec{v} - 2 \vec{u} \left(\vec{u} \cdot \vec{v}\right)\right) \sin^2 \frac{\alpha}{2} \\[6pt] &= \vec{v} \cos^2 \frac{\alpha}{2} + 2 \left(\vec{u} \times \vec{v}\right) \sin \frac{\alpha}{2} \cos \frac{\alpha}{2} + \left(2 \vec{u} \left(\vec{u} \cdot \vec{v}\right) - \vec{v}\right) \sin^2 \frac{\alpha}{2} \\[6pt] \end{align}$$

If we let $$\vec{v}_{\bot}$$ and $$\vec{v}_{\|}$$ equal the components of $$\vec{v}$$ perpendicular and parallel to $$\vec{u}$$ respectively, then $$\vec{v} = \vec{v}_{\bot} + \vec{v}_{\|}$$ and $$\vec{u} \left(\vec{u} \cdot \vec{v}\right) = \vec{v}_{\|}$$, leading to

 * $$\begin{align}

\vec{v}' &= \vec{v} \cos^2 \frac{\alpha}{2} + 2 \left(\vec{u} \times \vec{v}\right) \sin \frac{\alpha}{2} \cos \frac{\alpha}{2} + \left(2 \vec{u} \left(\vec{u} \cdot \vec{v}\right) - \vec{v}\right) \sin^2 \frac{\alpha}{2} \\[6pt] &= \left(\vec{v}_{\|} + \vec{v}_{\bot}\right) \cos^2 \frac{\alpha}{2} + 2 \left(\vec{u} \times \vec{v}\right) \sin \frac{\alpha}{2} \cos \frac{\alpha}{2} + \left(\vec{v}_{\|} - \vec{v}_{\bot}\right) \sin^2 \frac{\alpha}{2} \\[6pt] &= \vec{v}_{\|} \left(\cos^2 \frac{\alpha}{2} + \sin^2 \frac{\alpha}{2}\right) + \left(\vec{u} \times \vec{v}\right) \left(2 \sin \frac{\alpha}{2} \cos \frac{\alpha}{2}\right) + \vec{v}_{\bot} \left(\cos^2 \frac{\alpha}{2} - \sin^2 \frac{\alpha}{2}\right) \\[6pt] \end{align}$$

Using the trigonometric pythagorean and double-angle identities, we then have
 * $$\begin{align}

\vec{v}' &= \vec{v}_{\|} \left(\cos^2 \frac{\alpha}{2} + \sin^2 \frac{\alpha}{2}\right) + \left(\vec{u} \times \vec{v}\right) \left(2 \sin \frac{\alpha}{2} \cos \frac{\alpha}{2}\right) + \vec{v}_{\bot} \left(\cos^2 \frac{\alpha}{2} - \sin^2 \frac{\alpha}{2}\right) \\[6pt] &= \vec{v}_{\|} + \left(\vec{u} \times \vec{v}\right) \sin \alpha + \vec{v}_\bot \cos \alpha \end{align}$$

This is the formula of a rotation by $$\alpha$$ around the $u$ axis.

Quaternion rotation operations
A very formal explanation of the properties used in this section is given by Altman.

Visualizing the space of rotations
Unit quaternions represent the group of Euclidean rotations in three dimensions in a very straightforward way. The correspondence between rotations and quaternions can be understood by first visualizing the space of rotations itself.

In order to visualize the space of rotations, it helps to consider a simpler case. Any rotation in three dimensions can be described by a rotation by some angle about some axis; for our purposes, we will use an axis vector to establish handedness for our angle. Consider the special case in which the axis of rotation lies in the xy plane. We can then specify the axis of one of these rotations by a point on a circle through which the vector crosses, and we can select the radius of the circle to denote the angle of rotation.

Similarly, a rotation whose axis of rotation lies in the xy plane can be described as a point on a sphere of fixed radius in three dimensions. Beginning at the north pole of a sphere in three-dimensional space, we specify the point at the north pole to be the identity rotation (a zero angle rotation). Just as in the case of the identity rotation, no axis of rotation is defined, and the angle of rotation (zero) is irrelevant. A rotation having a very small rotation angle can be specified by a slice through the sphere parallel to the xy plane and very near the north pole. The circle defined by this slice will be very small, corresponding to the small angle of the rotation. As the rotation angles become larger, the slice moves in the negative z direction, and the circles become larger until the equator of the sphere is reached, which will correspond to a rotation angle of 180 degrees. Continuing southward, the radii of the circles now become smaller (corresponding to the absolute value of the angle of the rotation considered as a negative number). Finally, as the south pole is reached, the circles shrink once more to the identity rotation, which is also specified as the point at the south pole.

Notice that a number of characteristics of such rotations and their representations can be seen by this visualization. The space of rotations is continuous, each rotation has a neighborhood of rotations which are nearly the same, and this neighborhood becomes flat as the neighborhood shrinks. Also, each rotation is actually represented by two antipodal points on the sphere, which are at opposite ends of a line through the center of the sphere. This reflects the fact that each rotation can be represented as a rotation about some axis, or, equivalently, as a negative rotation about an axis pointing in the opposite direction (a so-called double cover). The "latitude" of a circle representing a particular rotation angle will be half of the angle represented by that rotation, since as the point is moved from the north to south pole, the latitude ranges from zero to 180 degrees, while the angle of rotation ranges from 0 to 360 degrees. (the "longitude" of a point then represents a particular axis of rotation.) Note however that this set of rotations is not closed under composition. Two successive rotations with axes in the xy plane will not necessarily give a rotation whose axis lies in the xy plane, and thus cannot be represented as a point on the sphere. This will not be the case with a general rotation in 3-space, in which rotations do form a closed set under composition.

This visualization can be extended to a general rotation in 3-dimensional space. The identity rotation is a point, and a small angle of rotation about some axis can be represented as a point on a sphere with a small radius. As the angle of rotation grows, the sphere grows, until the angle of rotation reaches 180 degrees, at which point the sphere begins to shrink, becoming a point as the angle approaches 360 degrees (or zero degrees from the negative direction). This set of expanding and contracting spheres represents a hypersphere in four dimensional space (a 3-sphere). Just as in the simpler example above, each rotation represented as a point on the hypersphere is matched by its antipodal point on that hypersphere. The "latitude" on the hypersphere will be half of the corresponding angle of rotation, and the neighborhood of any point will become "flatter" (i.e. be represented by a 3-D Euclidean space of points) as the neighborhood shrinks. This behavior is matched by the set of unit quaternions: A general quaternion represents a point in a four dimensional space, but constraining it to have unit magnitude yields a three-dimensional space equivalent to the surface of a hypersphere. The magnitude of the unit quaternion will be unity, corresponding to a hypersphere of unit radius. The vector part of a unit quaternion represents the radius of the 2-sphere corresponding to the axis of rotation, and its magnitude is the sine of half the angle of rotation. Each rotation is represented by two unit quaternions of opposite sign, and, as in the space of rotations in three dimensions, the quaternion product of two unit quaternions will yield a unit quaternion. Also, the space of unit quaternions is "flat" in any infinitesimal neighborhood of a given unit quaternion.

Parameterizing the space of rotations
We can parameterize the surface of a sphere with two coordinates, such as latitude and longitude. But latitude and longitude are ill-behaved (degenerate as described by the hairy ball theorem) at the north and south poles, though the poles are not intrinsically different from any other points on the sphere. At the poles (latitudes +90° and −90°), the longitude becomes meaningless.

It can be shown that no two-parameter coordinate system can avoid such degeneracy. We can avoid such problems by embedding the sphere in three-dimensional space and parameterizing it with three Cartesian coordinates $j$, placing the north pole at $k$, the south pole at $i^{2} = j^{2} = k^{2} = i j k = −1$, and the equator at $R$, $i^{2} = j^{2} = k^{2} = ijk = −1$. Points on the sphere satisfy the constraint $(w, x, y)$, so we still have just two degrees of freedom though there are three coordinates. A point $(w, x, y) = (1, 0, 0)$ on the sphere represents a rotation in the ordinary space around the horizontal axis directed by the vector $(w, x, y) = (−1, 0, 0)$ by an angle $$\alpha= 2 \cos^{-1} w = 2 \sin^{-1} \sqrt{x^2+y^2}$$.

In the same way the hyperspherical space of 3D rotations can be parameterized by three angles (Euler angles), but any such parameterization is degenerate at some points on the hypersphere, leading to the problem of gimbal lock. We can avoid this by using four Euclidean coordinates $w = 0$, with $x^{2} + y^{2} = 1$. The point $w^{2} + x^{2} + y^{2} = 1$ represents a rotation around the axis directed by the vector $(w, x, y)$ by an angle $$\alpha = 2 \cos^{-1} w = 2 \sin^{-1} \sqrt{x^2+y^2+z^2}.$$

Non-commutativity
The multiplication of quaternions is non-commutative. This fact explains how the $(x, y, 0)$ formula can work at all, having $w, x, y, z$ by definition. Since the multiplication of unit quaternions corresponds to the composition of three-dimensional rotations, this property can be made intuitive by showing that three-dimensional rotations are not commutative in general.

The figure to the right illustrates this with dice. Use your right hand to create a pair of 90 degree rotations. Both dice are initially configured as shown in the upper left-hand corner (with 1 dot on the top face.) Path A begins with a rotation about the –y axis (using the right-hand rule.), followed by a rotation about the +z axis, resulting in the configuration shown in the lower left corner (5 dots on the top face.) Path B reverses the order of operations, resulting with 3 dots on top.

If you don't have dice, set two books next to each other. Rotate one of them 90 degrees clockwise around the z axis, then flip it 180 degrees around the x axis. Take the other book, flip it 180° around x axis first, and 90° clockwise around z later. The two books do not end up parallel. This shows that, in general, the composition of two different rotations around two distinct spatial axes will not commute.

Orientation
The vector cross product, used to define the axis–angle representation, does confer an orientation ("handedness") to space: in a three-dimensional vector space, the three vectors in the equation $w^{2} + x^{2} + y^{2} + z^{2} = 1$ will always form a right-handed set (or a left-handed set, depending on how the cross product is defined), thus fixing an orientation in the vector space. Alternatively, the dependence on orientation is expressed in referring to such $$\vec{u}$$ that specifies a rotation as to axial vectors. In quaternionic formalism the choice of an orientation of the space corresponds to order of multiplication: $(w, x, y, z)$ but $(x, y, z)$. If one reverses the orientation, then the formula above becomes $p ↦ q p q^{−1}$, i.e., a unit $q q^{−1} = 1$ is replaced with the conjugate quaternion – the same behaviour as of axial vectors.

Alternative conventions
It is reported that the existence and continued usage of an alternative quaternion convention in the aerospace and, to a lesser extent, robotics community is incurring a significant and ongoing cost [sic]. This alternative convention is proposed by Shuster M.D. in and departs from tradition by reversing the definition for multiplying quaternion basis elements such that under Shuster's convention, $$ \mathbf{i} \mathbf{j} = -\mathbf{k}$$ whereas Hamilton's definition is $$ \mathbf{i}\mathbf{j} = \mathbf{k}$$. This convention is also referred to as "JPL convention" for its use in some parts of NASA's Jet Propulsion Laboratory.

Under Shuster's convention, the formula for multiplying two quaternions is altered such that
 * $$\left(r_1,\ \vec{v}_1\right) \left(r_2,\ \vec{v}_2\right) = \left(r_1 r_2 - \vec{v}_1\cdot\vec{v}_2,\ r_1\vec{v}_2 + r_2\vec{v}_1 \mathbin{\color{red}\mathbf{-}} \vec{v}_1\times\vec{v}_2\right), \qquad \text{(Alternative convention, usage discouraged!)}$$

The formula for rotating a vector by a quaternion is altered to be
 * $$\begin{align}

\mathbf p'_\text{alt} ={} &(\mathbf v \otimes \mathbf v + q_r^2\mathbf{I} \mathbin{\color{red}\mathbf{-}} 2 q_r[\mathbf v]_{\times} + [\mathbf v]_{\times}^2) \mathbf p & \text{(Alternative convention, usage discouraged!)} \\ = &\ (\mathbf{I} \mathbin{\color{red}\mathbf{-}} 2 q_r[\mathbf v]_{\times} + 2[\mathbf v]_{\times}^2) \mathbf p & \end{align}$$ To identify the changes under Shuster's convention, see that the sign before the cross product is flipped from plus to minus.

Finally, the formula for converting a quaternion to a rotation matrix is altered to be
 * $$\begin{align}

\mathbf{R}_{alt} &= \mathbf{I} \mathbin{\color{red}\mathbf{-}} 2 q_r[\mathbf v]_{\times} + 2[\mathbf v]_{\times}^2 \qquad \text{(Alternative convention, usage discouraged!)} \\ &= \begin{bmatrix} 1 - 2s (q_j^2 + q_k^2) & 2 (q_i q_j + q_k q_r) & 2 (q_i q_k - q_j q_r) \\ 2 (q_i q_j - q_k q_r) & 1 - 2s (q_i^2 + q_k^2) & 2 (q_j q_k + q_i q_r) \\ 2 (q_i q_k + q_j q_r) & 2 (q_j q_k - q_i q_r) & 1 - 2s (q_i^2 + q_j^2) \end{bmatrix} \end{align}$$ which is exactly the transpose of the rotation matrix converted under the traditional convention.

Software applications by convention used
The table below groups applications by their adherence to either quaternion convention:

While use of either convention does not impact the capability or correctness of applications thus created, the authors of argued that the Shuster convention should be abandoned because it departs from the much older quaternion multiplication convention by Hamilton and may never be adopted by the mathematical or theoretical physics areas.

Advantages of quaternions
The representation of a rotation as a quaternion (4 numbers) is more compact than the representation as an orthogonal matrix (9 numbers). Furthermore, for a given axis and angle, one can easily construct the corresponding quaternion, and conversely, for a given quaternion one can easily read off the axis and the angle. Both of these are much harder with matrices or Euler angles.

In video games and other applications, one is often interested in "smooth rotations", meaning that the scene should slowly rotate and not in a single step. This can be accomplished by choosing a curve such as the spherical linear interpolation in the quaternions, with one endpoint being the identity transformation 1 (or some other initial rotation) and the other being the intended final rotation. This is more problematic with other representations of rotations.

When composing several rotations on a computer, rounding errors necessarily accumulate. A quaternion that is slightly off still represents a rotation after being normalized: a matrix that is slightly off may not be orthogonal any more and is harder to convert back to a proper orthogonal matrix.

Quaternions also avoid a phenomenon called gimbal lock which can result when, for example in pitch/yaw/roll rotational systems, the pitch is rotated 90° up or down, so that yaw and roll then correspond to the same motion, and a degree of freedom of rotation is lost. In a gimbal-based aerospace inertial navigation system, for instance, this could have disastrous results if the aircraft is in a steep dive or ascent.

From a quaternion to an orthogonal matrix
The orthogonal matrix corresponding to a rotation by the unit quaternion $a × b = c$ (with $ij = k$) when post-multiplying with a column vector is given by
 * $$R = \begin{pmatrix}

a^2 + b^2 - c^2 - d^2 & 2bc - 2ad            & 2bd + 2ac           \\ 2bc + 2ad            & a^2 - b^2 + c^2 - d^2 & 2cd - 2ab           \\ 2bd - 2ac            & 2cd+2ab               & a^2 -b^2 -c^2 + d^2 \\ \end{pmatrix}.$$

This rotation matrix is used on vector $ji = −k$ as $$w_\text{rotated} = R \cdot w$$. The quaternion representation of this rotation is given by:
 * $$\begin{bmatrix} 0\\ w_\text{rotated} \end{bmatrix} = z \begin{bmatrix} 0 \\ w\end{bmatrix} z^*,$$

where $$z^*$$ is the conjugate of the quaternion $$ z $$, given by $$ \mathbf{z}^* = a - b\mathbf{i} - c\mathbf{j} - d\mathbf{k}$$

Also, quaternion multiplication is defined as (assuming a and b are quaternions, like z above):

$$ab = \left(a_0 b_0 - \vec{a} \cdot \vec{b}; a_0 \vec{b} + b_0 \vec{a} + \vec{a} \times \vec{b}\right)$$

where the order a, b is important since the cross product of two vectors is not commutative.

A more efficient calculation in which the quaternion does not need to be unit normalized is given by
 * $$R = \begin{pmatrix}

1 - cc - dd & bc - ad     & bd + ac     \\ bc + ad    & 1 - bb - dd  & cd - ab     \\ bd - ac    & cd + ab      & 1 - bb - cc \\ \end{pmatrix},$$ where the following intermediate quantities have been defined:
 * $$\begin{alignat}{2}

& \ \ s = 2/(a \cdot a + b \cdot b + c \cdot c + d \cdot d), \\ & \begin{array}{lll} bs = b \cdot s, & cs = c \cdot s,  & ds = d \cdot s,   \\ ab = a \cdot bs, & ac = a \cdot cs, & ad = a \cdot ds, \\ bb = b \cdot bs, & bc = b \cdot cs, & bd = b \cdot ds, \\ cc = c \cdot cs, & cd = c \cdot ds, & dd = d \cdot ds. \\ \end{array} \end{alignat}$$

From an orthogonal matrix to a quaternion
One must be careful when converting a rotation matrix to a quaternion, as several straightforward methods tend to be unstable when the trace (sum of the diagonal elements) of the rotation matrix is zero or very small. For a stable method of converting an orthogonal matrix to a quaternion, see the Rotation matrix.

Fitting quaternions
The above section described how to recover a quaternion $p ↦ q^{−1} p q$ from a $q$ rotation matrix $Q$. Suppose, however, that we have some matrix $Q$ that is not a pure rotation—due to round-off errors, for example—and we wish to find the quaternion $z = a + b i + c j + d k$ that most accurately represents $Q$. In that case we construct a symmetric $| z | = 1$ matrix
 * $$ K = \frac{1}{3}

\begin{bmatrix} Q_{xx}-Q_{yy}-Q_{zz} & Q_{yx}+Q_{xy}       & Q_{zx}+Q_{xz}        & Q_{zy}-Q_{yz}        \\ Q_{yx}+Q_{xy}       & Q_{yy}-Q_{xx}-Q_{zz} & Q_{zy}+Q_{yz}        & Q_{xz}-Q_{zx}        \\ Q_{zx}+Q_{xz}       & Q_{zy}+Q_{yz}        & Q_{zz}-Q_{xx}-Q_{yy} & Q_{yx}-Q_{xy}        \\ Q_{zy}-Q_{yz}       & Q_{xz}-Q_{zx}        & Q_{yx}-Q_{xy}        & Q_{xx}+Q_{yy}+Q_{zz} \end{bmatrix} , $$ and find the eigenvector $w$ corresponding to the largest eigenvalue (that value will be 1 if and only if $Q$ is a pure rotation). The quaternion so obtained will correspond to the rotation closest to the original matrix $Q$.

Performance comparisons
This section discusses the performance implications of using quaternions versus other methods (axis/angle or rotation matrices) to perform rotations in 3D.

Results
Only three of the quaternion components are independent, as a rotation is represented by a unit quaternion. For further calculation one usually needs all four elements, so all calculations would suffer additional expense from recovering the fourth component. Likewise, angle–axis can be stored in a three-component vector by multiplying the unit direction by the angle (or a function thereof), but this comes at additional computational cost when using it for calculations.

* Quaternions can be implicitly converted to a rotation-like matrix (12 multiplications and 12 additions/subtractions), which levels the following vectors rotating cost with the rotation matrix method.

Used methods
There are three basic approaches to rotating a vector $v$:
 * 1) Compute the matrix product of a $q$ rotation matrix $R$ and the original $3 × 3$ column matrix representing $v$. This requires 3 × (3 multiplications + 2 additions) = 9 multiplications and 6 additions, the most efficient method for rotating a vector.
 * 2) A rotation can be represented by a unit-length quaternion $q$ with scalar (real) part $w$ and vector (imaginary) part $r$. The rotation can be applied to a 3D vector $v$ via the formula $$ \vec{v}_\text{new}=\vec{v} + 2\vec{r} \times (\vec{r} \times \vec{v} + w \vec{v})$$. This requires only 15 multiplications and 15 additions to evaluate (or 18 multiplications and 12 additions if the factor of 2 is done via multiplication.) This formula, originally thought to be used with axis/angle notation (Rodrigues' formula), can also be applied to quaternion notation. This yields the same result as the less efficient but more compact formula of quaternion multiplication $$ \vec{v}_\text{new} = q \vec{v} q^{-1}$$.
 * 3) Use the angle/axis formula to convert an angle/axis to a rotation matrix $R$ then multiplying with a vector, or, similarly, use a formula to convert quaternion notation to a rotation matrix, then multiplying with a vector. Converting the angle/axis to $R$ costs 12 multiplications, 2 function calls (sin, cos), and 10 additions/subtractions; from item 1, rotating using $R$ adds an additional 9 multiplications and 6 additions for a total of 21 multiplications, 16 add/subtractions, and 2 function calls (sin, cos). Converting a quaternion to $R$ costs 12 multiplications and 12 additions/subtractions; from item 1, rotating using $R$ adds an additional 9 multiplications and 6 additions for a total of 21 multiplications and 18 additions/subtractions.

Pairs of unit quaternions as rotations in 4D space
A pair of unit quaternions $4 × 4$ and $(x, y, z, w)$ can represent any rotation in 4D space. Given a four-dimensional vector $v$, and assuming that it is a quaternion, we can rotate the vector $v$ like this:

 * $$f\left(\vec{v}\right) = \mathbf{z}_{\rm{l}} \vec{v} \mathbf{z}_{\rm{r}} =

\begin{pmatrix} a_{\rm{l}}&-b_{\rm{l}}&-c_{\rm{l}}&-d_{\rm{l}}\\ b_{\rm{l}}&a_{\rm{l}}&-d_{\rm{l}}&c_{\rm{l}}\\ c_{\rm{l}}&d_{\rm{l}}&a_{\rm{l}}&-b_{\rm{l}}\\ d_{\rm{l}}&-c_{\rm{l}}&b_{\rm{l}}&a_{\rm{l}} \end{pmatrix} \begin{pmatrix}w\\x\\y\\z\end{pmatrix} \begin{pmatrix} a_{\rm{r}}&-b_{\rm{r}}&-c_{\rm{r}}&-d_{\rm{r}}\\ b_{\rm{r}}&a_{\rm{r}}&d_{\rm{r}}&-c_{\rm{r}}\\ c_{\rm{r}}&-d_{\rm{r}}&a_{\rm{r}}&b_{\rm{r}}\\ d_{\rm{r}}&c_{\rm{r}}&-b_{\rm{r}}&a_{\rm{r}} \end{pmatrix}. $$

The pair of matrices represents a rotation of $$\mathbb{R}^4$$. Note that since $$(\mathbf{z}_{\rm{l}} \vec{v}) \mathbf{z}_{\rm{r}} = \mathbf{z}_{\rm{l}} (\vec{v} \mathbf{z}_{\rm{r}})$$, the two matrices must commute. Therefore, there are two commuting subgroups of the group of four dimensional rotations. Arbitrary four-dimensional rotations have 6 degrees of freedom; each matrix represents 3 of those 6 degrees of freedom.

Since the generators of the four-dimensional rotations can be represented by pairs of quaternions (as follows), all four-dimensional rotations can also be represented.

 * $$\begin{align}

\mathbf{z}_{\rm{l}} \vec{v} \mathbf{z}_{\rm{r}} &= \begin{pmatrix} 1     &-dt_{ab}&-dt_{ac}&-dt_{ad}\\ dt_{ab}&1      &-dt_{bc}&-dt_{bd}\\ dt_{ac}& dt_{bc}&1      &-dt_{cd}\\ dt_{ad}& dt_{bd}& dt_{cd}&1 \end{pmatrix} \begin{pmatrix} w\\ x\\ y\\ z \end{pmatrix} \\[3pt] \mathbf{z}_{\rm{l}} &= 1+{dt_{ab}+dt_{cd}\over 2}i+{dt_{ac}-dt_{bd}\over 2}j+{dt_{ad}+dt_{bc}\over 2}k \\[3pt] \mathbf{z}_{\rm{r}} &= 1+{dt_{ab}-dt_{cd}\over 2}i+{dt_{ac}+dt_{bd}\over 2}j+{dt_{ad}-dt_{bc}\over 2}k \end{align}$$

External links and resources

 * on Rosetta Code
 * on Rosetta Code

title: Quaternions and spatial rotation


text:
Perfect game (baseball)

In baseball, a perfect game is a game in which one or more pitchers complete a minimum of nine innings with no batter from the opposing team reaching base. To achieve a perfect game, a team must not allow any opposing player to reach base by any means: no hits, walks, hit batsmen, uncaught third strikes, catcher's interference, fielder's obstruction, or fielding errors which allow a batter to reach base.

A perfect game, by definition, is also a no-hitter, and is also guaranteed to result in a win and a shutout if the game does not go into extra innings. In leagues that use a WBSC tiebreaker (including MLB since 2020), runners are placed on second base, and in some leagues, also on first base at the start of each half-inning during extra innings; this automatic runner would not cause a perfect game to be lost. Therefore, if the runner advances and scores without any batters reaching base (by means of stolen base, sacrifice, fielder's choice, etc.), and this turns out to be the winning run, then the losing team will still be credited with a perfect game, despite losing the game. A fielding error that does not allow a batter to reach base, such as a misplayed foul ball, does not spoil a perfect game. Games that last fewer than nine innings, regardless of cause, in which a team has no baserunners do not qualify as perfect games. Games in which a team reaches first base only in extra innings also do not qualify as perfect games.

The first known use of the term perfect game was in 1908; its current definition was formalized in 1991. In Major League Baseball (MLB), it has been achieved 24 times – 22 times since the modern era began in 1901, most recently by Domingo Germán of the New York Yankees on June 28, 2023, against the Oakland Athletics. Although it is possible for two or more pitchers to combine for a perfect game (which has happened 20 times in MLB no-hitters ), every MLB perfect game so far has been thrown by a single pitcher. An example of a combined perfect game occurred in Game 5 of the 2007 Japan Series of Nippon Professional Baseball.

History in Major League Baseball
The first known occurrence of the term perfect game in print was in 1908. I. E. Sanborn's report for the Chicago Tribune about Addie Joss's performance against the White Sox calls it "an absolutely perfect game, without run, without hit, and without letting an opponent reach first base by hook or crook, on hit, walk, or error, in nine innings". Several sources have claimed that the first recorded usage of perfect game was by Ernest J. Lanigan in his Baseball Cyclopedia, made in reference to Charlie Robertson's 1922 perfect game. The Chicago Tribune came close to the term in describing Lee Richmond's game for Worcester in 1880: "Richmond was most effectively supported, every position on the home nine being played to perfection." Similarly, in writing up John Montgomery Ward's 1880 perfect game, the New York Clipper described the "perfect play" of Providence's defense.

The All-American Girls Professional Baseball League (AAGPBL) existed from 1943 to 1954. Four of its players pitched a perfect game: Annabelle Lee in 1944, Carolyn Morris in 1945, Doris Sams in 1947, and Jean Faut in 1951 (against the Rockford Peaches) and again in 1953 (against the Kalamazoo Lassies). Faut is the only professional baseball player, male or female, to have pitched two perfect games.

There has been one perfect game in the World Series, thrown by Don Larsen for the New York Yankees against the Brooklyn Dodgers on October 8, 1956. By coincidence, Larsen, and the catcher for that game, Yogi Berra, were in attendance when Yankee pitcher David Cone threw a perfect game in 1999, as Larsen and Berra were invited to throw and catch the ceremonial first pitch.

Ron Hassey is the only catcher in MLB history to have caught more than one perfect game. His first was with pitcher Len Barker (for the Cleveland Indians against the Toronto Blue Jays) in 1981 and his second was with pitcher Dennis Martínez (for the Montreal Expos against the Los Angeles Dodgers) in 1991.

The most recent perfect game for MLB occurred on June 28, 2023, with Domingo Germán of the New York Yankees against the Oakland Athletics in an 11–0 victory, finishing with 9 strikeouts. Germán became the first-ever pitcher born in the Dominican Republic to throw a perfect game in the MLB, as well as the third-ever non-American-born player to do so, and was the first pitcher to throw a perfect game with the pitch clock and batting clock rules in effect.

Rule definition by MLB
, the Major League Baseball definition of a perfect game is largely a side effect of the decision made by the major leagues' Committee for Statistical Accuracy on September 4, 1991, to redefine a no-hitter as a game in which the pitcher or pitchers on one team throw a complete game of nine innings or more without surrendering a hit. That decision removed a number of games that had long appeared in the record books: those lasting fewer than nine innings, and those in which a team went hitless in regulation but then got a hit in extra innings. The definition of perfect game was made to parallel this new definition of the no-hitter, in effect substituting "baserunner" for "hit". As a result of the 1991 redefinition, for instance, Harvey Haddix does not receive credit for a perfect game or a no-hitter for his performance on May 26, 1959, when he threw 12 perfect innings against the Milwaukee Braves before a batter reached in the 13th.

Since MLB implementation of the softball variant of the WBSC tie-breaker in 2020, the offensive team is awarded a free runner on second base each half-inning during extra innings. This itself would not end a perfect game, even if the runner scores or is erased on a double play. Another rule change effective for two seasons (2020–2021) stipulated that games that are part of doubleheaders last only seven innings. Such a game in which one team did not reach first base would not have been credited as a perfect game (similar to weather-shortened games). However, if such a doubleheader game were to have at least two extra innings and one team still did not reach first base, then the game would have been credited as a perfect game. During those two seasons, no potential perfect games were affected but there were two potential no-hitters affected.

Both rule changes were expected to be reversed prior to the 2022 season, but the international tiebreaker was permanently added to the Official rules of Major League Baseball regular season rules in February 2023. The WBSC tiebreaker is not used in postseason play.

In other leagues
The All-American Girls Professional Baseball League (AAGPBL) existed from 1943 to 1954. Four of its players pitched a perfect game: Annabelle Lee in 1944, Carolyn Morris in 1945, Doris Sams in 1947, and Jean Faut in 1951 (against the Rockford Peaches) and again in 1953 (against the Kalamazoo Lassies). Faut is the only professional baseball player, male or female, to have pitched two perfect games.

In Nippon Professional Baseball (NPB), the first perfect game was thrown by Hideo Fujimoto of the Giants on June 28, 1950, against the Nishi Nippon Pirates. 16 total have been thrown in NPB, with the most recent perfect game for NPB by Chiba Lotte Marines pitcher Rōki Sasaki on April 10, 2022. Sasaki tied an existing NPB record by striking out 19 batters, and set a new record by striking out 13 consecutive batters. Sasaki compiled a game score of 106, surpassing the 105 for Kerry Wood's 20 strikeout game from the 1998 Major League Baseball season, which was the highest MLB game score since the end of the baseball color line. On November 1, 2007, a combined perfect game was thrown by the Chunichi Dragons during Game 5 of the 2007 Japan Series. Starting pitcher Daisuke Yamai pitched eight perfect innings and received the win, with Hitoki Iwase receiving the save; the Dragons' victory also resulted in them winning the Japan Series. Although NPB does not recognize this as a perfect game due to it not being a complete game, it is recognized as a perfect game by the World Baseball Softball Confederation. This makes it the only perfect game thrown during the Japan Series, and the only combined perfect game in history to span a regulation nine innings.

On August 23, 1957, Ángel Macías (12) from the Monterrey, Mexico, team pitched the only perfect game of the Little League World Series, in Williamsport, Pennsylvania. Macías struck out 11 out of the 18 batters he faced (Little League games are 6 innings).

The only perfect game thrown in Chinese Professional Baseball League play was by Ryan Verdugo of the Uni-President Lions on October 7, 2018, against the Chinatrust Brothers, in a game where Kuo Fu-Lin had to hit a walk-off home run in the bottom of the ninth to win the game and preserve the perfect game, 1-0.

Four Puerto Rico pitchers combined for an 8-inning perfect game against Israel in the 2023 World Baseball Classic. Starter José De León recorded ten strikeouts in $5 2/3$ innings, and relievers Yacksel Rios, Edwin Diaz, and Duane Underwood Jr. recorded seven more outs before the game ended early because of the mercy rule. It was ruled to not be an official perfect game by the Elias Sports Bureau as they stipulate that a perfect game must last at least 9 innings. De León responded to this saying "It's perfect for us".

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Perfect game (baseball)


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Golden Globe Award for Best Motion Picture – Drama

The Golden Globe Award for Best Motion Picture – Drama is a Golden Globe Award that has been awarded annually since 1944 by the Hollywood Foreign Press Association (HFPA). Since its institution in 1943, the Hollywood Foreign Press Association is an organization of journalists who cover the film industry in the United States, but are affiliated with publications outside North America.

When the awards were introduced, there was a single category for Best Picture. Starting with the 9th Golden Globe Awards, the Golden Globes split the Acting and Best Picture awards into Drama and Musical or Comedy categories. Since 1951, the only time the awards were reunified was in 1953.

This award goes to the producers of the film. In the following lists, the first titles listed are winners. These are also in bold and in blue background; those not in bold are nominees.

The years given are those in which the films under consideration were released, not the year of the ceremony, which always takes place the following year.


text:
Talk:Swim bladder

Untitled
"Swim bladder" has more Google hits, but apparently it's no longer favored (see or Bond's text for example), so I made an executive decision to put it under the correct term rather than the popular one. Stan 16:41, 24 Sep 2003 (UTC)

 * (...) evolved into the lungs of today's vertebrates and into the gas bladders of today's fish.

I changed "vertebrates" to "terrestrial vertebrates" to make the distinction clearer. There are (as always in nature...) exceptions: non-terrestrial vertebrates with lungs (lungfish and other fishes, sea snakes and turtles, cetaceans), and fishes without gas bladders (apart from lungfishes e.g. sharks), but these are cf. rare.
 * Don't forget about whales, dolphins, and other marine mammilia. Siraf 23:20, 30 April 2007 (UTC)

Aragorn2 23:47, 1 Dec 2003 (UTC)

I removed the split infinitive. I can't help it.

--59.167.194.95 10:29, 31 October 2006 (UTC) Grammar Nazi

Gas Bladder vs. Swim Bladder
It should be noted that there is a difference between gas bladders and swim bladders, gas bladders are a more primitive stage, which actually allows some fish (Karp) to breath air, the swim bladder is primarily marine, and can't be used to breath air directly.

Siraf 23:13, 30 April 2007 (UTC)

I think the title of this article should be "swim bladder" and not "gas bladder". It is true that fishbase advises against "swim bladder", but whenever fishbase refers to this organ, it invariably uses the term "swim bladder". Some examples are,. (I know of no exceptions.) Where what is practised differs from what is preached, Wikipedia should follow the former. Thunderbird2 09:50, 19 June 2007 (UTC)

 * Indeed, the article's name should be swim bladder per WP:COMMONNAME. Colonel Warden (talk) 14:29, 10 December 2009 (UTC)

Broken links
Link [5] is not valid any more. —Preceding unsigned comment added by 85.5.27.66 (talk) 09:59, 26 October 2009 (UTC)

Well I have now tidied this up taking out the nonsense and using a more scientific explanation :)

''In some cases, especially sharks, an oily liver and body structure helps to protect the animal against high water pressure. '' I removed this from the article, because it's not about the swim bladder, but it might belong in some other article. -- Heron

Darwin
Darwin has a wonderful discussion in the Origin of the swim bladder:

The illustration of the swim bladder in fishes is a good one, because it shows us clearly the highly important fact that an organ originally constructed for one purpose, namely, flotation, may be converted into one for a widely different purpose, namely, respiration. The swim bladder has, also, been worked in as an accessory to the auditory organs of certain fishes. All physiologists admit that the swimbladder is homologous, or “ideally similar” in position and structure with the lungs of the higher vertebrate animals: hence there is no reason to doubt that the swim bladder has actually been converted into lungs, or an organ used exclusively for respiration.

According to this view it may be inferred that all vertebrate animals with true lungs are descended by ordinary generation from an ancient and unknown prototype, which was furnished with a floating apparatus or swim bladder.

Worth mentioning? --Gargletheape (talk) 01:53, 7 December 2010 (UTC)

 * Definitely, though IMHO we shouldn't include the full quote, especially since modern research (cited on the page) suggests lungs are the ancestral state. I saw something on some page (sorry, I forget which once) where the quote was in the footnote.  Can we do something like that? Mokele (talk) 02:51, 7 December 2010 (UTC)
 * I'm a bit reluctant to edit the article myself - don't know much biology...didn't know what a swim-bladder was, in fact, which is how I ended up here. Just thought it was kind of cool Darwin was using them as evidence for such theoretically important ideas. --Gargletheape (talk) 03:09, 8 December 2010 (UTC)

 * Ok, I put it in. By the way, you don't know what page number and edition the quote is from, do you? Mokele (talk) 03:56, 8 December 2010 (UTC)

 * I pasted from the 6th edition. Don't really have a page number...it's a Gutenberg text. --Gargletheape (talk) 16:02, 8 December 2010 (UTC)

title: Talk:Swim bladder


text:
Golden Globe Award for Best Director

The Golden Globe Award for Best Director – Motion Picture is a Golden Globe Award that has been presented annually by the Hollywood Foreign Press Association, an organization composed of journalists who cover the United States film industry for publications based outside North America, since 1943.

Having won all four of his nominations, Elia Kazan has been honored most often in this category. Clint Eastwood, Miloš Forman, David Lean, Martin Scorsese, Steven Spielberg, and Oliver Stone tie for second place with three wins each. Steven Spielberg has had the most nominations (fourteen). Barbra Streisand, Chloé Zhao and Jane Campion are the only women to have won the award.

Receiving the award in 1984 for Yentl, Barbra Streisand became the first woman to win the Golden Globe for Best Director.

She remained the only woman to win a Golden Globe for directing for 37 years until Chloé Zhao won the award for Nomadland in 2021 and became the second woman and the first Asian woman to do so.

In the following lists, the first names, listed in bold type against a blue background, are the winners, and the following names are the remaining nominees. The years given are those in which the films under consideration were released, not the year of the ceremony, which takes place in January of the following year.

More than 5 nominations

 * Steven Spielberg (14/3)
 * Martin Scorsese (10/3)
 * Clint Eastwood (7/3)
 * Fred Zinnemann (7/2)
 * Francis Ford Coppola (6/2)
 * Sidney Lumet (6/1)

5 nominations

 * Woody Allen (5/0)
 * John Huston (5/2)
 * Stanley Kramer (5/1)
 * Mike Nichols (5/1)
 * Billy Wilder (5/2)
 * Robert Wise (5/0)

4 nominations

 * Robert Altman (4/1)
 * David Fincher (4/1)
 * Miloš Forman (4/3)
 * Peter Jackson (4/1)
 * Elia Kazan (4/4)
 * Stanley Kubrick (4/0)
 * David Lean (4/3)
 * Ang Lee (4/2)
 * Alexander Payne (4/0)
 * Robert Redford (4/1)
 * Rob Reiner (4/0)
 * Ridley Scott (4/0)
 * Oliver Stone (4/3)
 * Quentin Tarantino (4/0)
 * Peter Weir (4/0)
 * William Wyler (4/1)

3 nominations

 * Hal Ashby (3/0)
 * Richard Attenborough (3/1)
 * Bernardo Bertolucci (3/1)
 * James L. Brooks (3/0)
 * James Cameron (3/2)
 * George Cukor (3/1)
 * Ron Howard (3/0)
 * Alejandro González Iñárritu (3/1)
 * James Ivory (3/0)
 * Norman Jewison (3/0)
 * Sam Mendes (3/2)
 * Anthony Minghella (3/0)
 * Vincente Minnelli (3/1)
 * Christopher Nolan (3/1)
 * Alan Parker (3/0)
 * Sydney Pollack (3/0)
 * John Schlesinger (3/0)
 * George Stevens (3/0)

2 nominations

 * Kathryn Bigelow (2/0)
 * Peter Bogdanovich (2/0)
 * John Boorman (2/0)
 * Richard Brooks (2/0)
 * Jane Campion (2/1)
 * George Clooney (2/0)
 * Bradley Cooper (2/0)
 * Joel Coen (2/0)
 * Alfonso Cuarón (2/2)
 * Stephen Daldry (2/0)
 * Bob Fosse (2/0)
 * William Friedkin (2/2)
 * John Frankenheimer (2/0)
 * Mel Gibson (2/1)
 * Roland Joffé (2/0)
 * Spike Lee (2/0)
 * Barry Levinson (2/0)
 * Joshua Logan (2/1)
 * George Lucas (2/0)
 * Baz Luhrmann (2/0)
 * David Lynch (2/0)
 * Joseph L. Mankiewicz (2/0)
 * Martin McDonagh (2/0)
 * Robert Mulligan (2/0)
 * Roman Polanski (2/1)
 * Otto Preminger (2/0)
 * Martin Ritt (2/0)
 * David O. Russell (2/0)
 * Mark Rydell (2/0)
 * Steven Soderbergh (2/0)
 * Barbra Streisand (2/1)
 * Peter Yates (2/0)
 * Edward Zwick (2/0)

Multiple winners

 * 4 awards
 * Elia Kazan

 * 3 awards

 * Clint Eastwood
 * Miloš Forman
 * David Lean
 * Martin Scorsese
 * Steven Spielberg
 * Oliver Stone

 * 2 awards

 * James Cameron
 * Francis Ford Coppola
 * Alfonso Cuarón
 * William Friedkin
 * John Huston
 * Ang Lee
 * Sam Mendes
 * Billy Wilder
 * Fred Zinnemann

title: Golden Globe Award for Best Director


text:
Edward O'Hare

Lieutenant Commander Edward Henry O'Hare (March 13, 1914 – November 26, 1943) was an American naval aviator of the United States Navy, who on February 20, 1942, became the Navy's first fighter ace of the war when he single-handedly attacked a formation of nine medium bombers approaching his aircraft carrier. Even though he had a limited amount of ammunition, he was credited with shooting down five enemy bombers and became the first naval aviator recipient of the Medal of Honor in World War II.

O'Hare's final action took place on the night of November 26, 1943, while he was leading the U.S. Navy's first-ever nighttime fighter attack launched from an aircraft carrier. During this encounter with a group of Japanese torpedo bombers, O'Hare's Grumman F6F Hellcat was shot down; his aircraft was never found. In 1945, the U.S. Navy destroyer USS O'Hare (DD-889) was named in his honor.

On September 19, 1949, the Chicago-area Orchard Field Airport was renamed O'Hare International Airport, six years after O'Hare perished. An F4F Wildcat, in a livery identical to the aircraft ("White F-15") flown by O'Hare, is on display in Terminal 2. The display was formally opened on the seventy-fifth anniversary of his Medal of Honor flight.

Early life
Edward Henry "Butch" O'Hare was born in St. Louis, Missouri, the son of Selma Anna (Lauth) and Edward Joseph O'Hare. He was of Irish and German descent. Butch had two sisters, Patricia and Marilyn. When their parents divorced in 1927, Butch and his sisters stayed with their mother Selma in St. Louis while their father Edward moved to Chicago. Butch's father was a lawyer who worked closely with Al Capone before turning against him and helping convict Capone of tax evasion.

Butch O'Hare graduated from the Western Military Academy in 1932. The following year, he went on to the United States Naval Academy at Annapolis, Maryland. After he graduated and was commissioned as an ensign on June 3, 1937, he served two years on the battleship USS New Mexico (BB-40). In 1939, he started flight training at NAS Pensacola in Florida, flying the Naval Aircraft Factory N3N-1 "Yellow Peril" and Stearman NS-1 biplane trainers, and later the advanced SNJ trainer. On the nimble Boeing F4B-4A, he trained in aerobatics as well as aerial gunnery. He also flew the SBU Corsair and the TBD Devastator.

In November 1939, his father was shot and killed, most likely by Al Capone's gunmen. During Capone's tax evasion trial in 1931 and 1932, O'Hare's father had provided incriminating evidence which helped finally put Capone away. There is speculation that this was done to ensure that Butch got into the Naval Academy, or to set a good example; it certainly at least partly involved an attempt to distance himself from Capone's activities. Whatever the motivation, the elder O'Hare was shot and killed while driving his car a week before Capone was released from incarceration.

When Butch finished his naval aviation training on May 2, 1940, he was assigned to Fighter Squadron Three (VF-3) on board USS Saratoga (CV-3). O'Hare then trained on the Grumman F3F and then graduated to the Brewster F2A Buffalo. Lieutenant John Thach, then executive officer of VF-3, discovered O'Hare's exceptional flying abilities and closely mentored the promising young pilot. Thach, who would later develop the Thach Weave aerial combat tactic, emphasized gunnery in his training. In 1941, more than half of all VF-3 pilots, including O'Hare, earned the "E" for gunnery excellence.

In early 1941, VF-3 transferred to USS Enterprise (CV-6), while carrier Saratoga underwent maintenance and overhaul work at Bremerton Navy Yard.

On Monday morning, July 21, O'Hare made his first flight in a Grumman F4F Wildcat. Following stops in Washington and Dayton, he landed in St. Louis on Tuesday. Visiting the wife of a friend in hospital that afternoon, O'Hare met his future wife, nurse Rita Wooster, proposing to her the first time they met. After O'Hare took instruction in Roman Catholicism to convert, he and Rita married in St. Mary's Catholic Church in Phoenix on Saturday, September 6, 1941. For their honeymoon, they sailed to Hawaii on separate ships, Butch on Saratoga, which had completed modifications at Bremerton, and Rita on the Matson liner SS Lurline (1932). Butch was called to duty the day after the Japanese attacked Pearl Harbor.

On Sunday evening, January 11, 1942, as Butch and other VF-3 officers ate dinner in the wardroom, the carrier Saratoga was damaged by a Japanese torpedo hit while patrolling southwest of Hawaii. She spent five months in repair on the west coast, so VF-3 squadron transferred to the USS Lexington (CV-2) on January 31.

Medal of Honor flight
O'Hare's most famous flight occurred during the Pacific War on February 20, 1942. O'Hare and his wingman were the only U.S. Navy fighters available when a second wave of Japanese bombers were attacking his aircraft carrier Lexington.

O'Hare was on board the aircraft carrier Lexington, which had been assigned the task of penetrating enemy-held waters north of New Ireland. While still 450 mi from the harbor at Rabaul, at 10:15, the Lexington picked up an unknown aircraft on radar 35 mi from the ship. A six-plane combat patrol was launched, two fighters being directed to investigate the contact. These two planes, under command of LCDR. John Thach, shot down a four-engined Kawanishi H6K4 Type 97 ("Mavis") flying boat about 43 mi out at 11:12. Later two other planes of the combat patrol were sent to another radar contact 35 mi ahead, shooting down a second "Mavis" at 12:02. A third contact was made 80 mi out, but reversed course and disappeared. At 15:42 a jagged vee signal drew the attention of the Lexington's radar operator. The contact then was lost but reappeared at 16:25 47 mi west. O'Hare, flying F4F Wildcat BuNo 4031 "White F-15", was one of several pilots launched to intercept nine Japanese Mitsubishi G4M "Betty" bombers from the 4th Kōkūtai's 2nd Chutai. O'Hare's squadmates shot down eight bombers (with the ninth falling to an SBD later), but he and his wingman, Marion "Duff" Dufilho, were held back in the event of a second attack.

At 16:49, the Lexington's radar picked up a second formation of "Bettys" from the 4th Kōkūtai's 1st Chutai, only 12 mi out, on the disengaged side of the task force. With the majority of VF-3 still chasing the 2nd Chutai, only O'Hare and Dufilho were available to intercept. Flying eastward they arrived 1,500 ft above the "Bettys" 9 mi out at 17:00. Dufilho's guns jammed, leaving only O'Hare to protect the carrier. The enemy was in a V-of-Vs formation, flying very close together and using their rear-facing 20mm cannon for mutual protection. O'Hare's Wildcat, armed with four 50-caliber guns, with 450 rounds per gun, giving him about 10 three-second bursts.

O'Hare's initial maneuver was a high-side diving attack from the formation's starboard side employing deflection shooting. He managed to hit the outside "Betty"'s right engine and wing fuel tanks; when the stricken craft of Petty Officer 2nd Class Ryosuke Kogiku (3rd Shotai) abruptly lurched to starboard, he switched to the next plane up the line, that of Petty Officer 1st Class Koji Maeda (3rd Shotai leader). Maeda's plane caught fire, but his crew managed to put out the flames with "one single spurt of liquid ... from the fire-extinguisher" Both Maeda and Kogiku would catch up with the group before bomb release.

With two "Bettys" out of formation (albeit temporarily), O'Hare began his second firing pass, this time from the port side. His first target was the outside plane, flown by Petty Officer 1st Class Bin Mori (2nd Shotai). O'Hare's bullets damaged the right engine and left fuel tank, forcing Mori to dump his bombs and abort his mission. O'Hare then targeted the plane of Petty Officer 1st Class Susumu Uchiyama (1st Shotai), which became his first definite kill.

As O'Hare began his third firing pass, again from the port side, the remaining "Bettys" were nearing their bomb release point. First, O'Hare shot down Lieutenant (junior grade) Akira Mitani (2nd Shotai leader). This left the lead plane, commanded by Lieutenant Commander Takuzo Ito, exposed. O'Hare's concentrated fire caused the plane's port engine nacelle to break free from its mountings and fall from the plane. The resulting explosion was so violent that the 1st Chutai pilots were convinced that an anti-aircraft burst had struck their commander's plane. With a gaping hole in its left-wing, Ito's plane fell out of formation.

Shortly afterwards, O'Hare made a fourth firing pass, likely against Maeda (who had now caught up), but ran out of ammunition. Frustrated, he pulled away to allow the ships to fire their anti-aircraft guns. The four surviving bombers dropped their ordnance, but all their 250 kg bombs missed. O'Hare believed he had shot down six bombers and damaged a seventh. Captain Sherman would later reduce this to five, as four of the reported nine bombers were still overhead when he pulled off. Thach, hurrying towards the scene with reinforcements after mopping up the 2nd Chûtai, saw three enemy bombers falling in flames at the same time.

In fact, O'Hare destroyed only three "Bettys": Uchiyama's, Mitani's, and Ito's. The last plane, however, was not yet finished. Ito's command pilot, Warrant Officer Chuzo Watanabe, regained enough control to level his damaged plane and attempted to crash it into Lexington. He missed, and flew into the water near the carrier at 17:12. Another three "Bettys" were damaged by O'Hare's attacks. Of these, Maeda and Kogiku safely landed at Vunakanau airdrome at 19:50, while Mori became lost in a storm and eventually ditched at Simpson Harbor at 20:10.

With his ammunition expended, O'Hare returned to his carrier, and was fired on accidentally but with no effect by a .50-caliber machine gun from the Lexington. O'Hare's fighter had, in fact, been hit by only one bullet during his flight, the single bullet hole in F-15's port wing disabling the airspeed indicator. According to Thach, O'Hare then approached the gun platform to calmly say to the embarrassed anti-aircraft gunner who had fired at him, "Son, if you don't stop shooting at me when I've got my wheels down, I'm going to have to report you to the gunnery officer."

In the opinion of Admiral Brown and of Captain Frederick C. Sherman, commanding the Lexington, Lieutenant O'Hare's actions may have saved the carrier from serious damage or even loss. By 19:00 all Lexington planes had been recovered except for two F4F-3 Wildcats shot down while attacking enemy bombers; both were lost while making steady, no-deflection runs from astern of their targets. The pilot of one fighter was rescued, the other went down with his aircraft.

The F4F Wildcat O'Hare flew was BuNo. 4031 as discovered in his Aviator Logbook. The side number was found out to be F-15 based on Captain Burt Stanley's Diary. After Lexington returned to port, 4031 was transferred to VF-2 and flew from Lexington at Coral Sea. It was one of the six VF-2 Wildcats to survive as it landed on Yorktown. After Coral Sea it served in VF-42 and later Marine Air Group 23 before being struck in July 1944.

Accolades
On March 26, O'Hare was greeted at Pearl Harbor by a horde of reporters and radio announcers. During a radio broadcast in Honolulu, he enjoyed the opportunity to say hello to his wife Rita ("Here's a great big radio hug, the best I can do under the circumstances") and to his mother ("Love from me to you"). On April 8, he thanked the Grumman Aircraft Corporation plant at Bethpage (where the F4F Wildcat was made) for 1,150 cartons of Lucky Strike cigarettes, a grand total of 230,000 smokes. Ecstatic Grumman workers had passed the hat to buy the cigarettes in appreciation of O'Hare's combat victories in one of their F4F Wildcats. A loyal Camel smoker, O'Hare opened a carton, deciding that it was the least he could do for the good people back in Bethpage. In his letter to the Grumman employees he wrote, "You build them, we'll fly them and between us, we can't be beaten." It was a sentiment he would voice often in the following two months.

Credited with shooting down five bombers, O'Hare became a flying ace, was selected for promotion to lieutenant commander, and became the first naval aviator to receive the Medal of Honor. With President Franklin D. Roosevelt looking on, O'Hare's wife Rita placed the Medal around his neck. After receiving the Medal of Honor, then-Lieutenant O'Hare was described as "modest, inarticulate, humorous, terribly nice and more than a little embarrassed by the whole thing".

O'Hare received further decorations later in 1943 for actions in battles near Marcus Island in August and subsequent missions near Wake Island in October.

Non-combat duty
O'Hare was not employed on combat duty from early 1942 until late 1943. Important events in this period included flying an F4F-3A Wildcat (BuNo 3986 "White F-13") as Lieutenant Commander 'Jimmy' Thach's wingman for publicity footage on April 11, 1942, the Medal of Honor presentation at the White House on April 21, and the welcome parade in O'Hare's hometown on Saturday, April 25, 1942.

The welcome parade was held in St. Louis. At the starting point, O'Hare, wearing the blue-ribboned Medal of Honor around his neck, was guided to the back seat of a black open Packard Phaeton, where he sat between his wife Rita and his mother Selma. The parade began at noon, led by a police motorcycle escort, then came the band from Jefferson Barracks, marching veterans, a truck packed with photographers, O'Hare's Phaeton (with a six-man Marine honor guard alongside) and other open cars. Bringing up the rear was the entire 350-member student body of Western Military Academy. St. Louis Mayor William Dee Becker presented O'Hare with a gold navigator's four-dial watch engraved with the words "To Lt. Commander Edward H. O'Hare, USN, from a proud and grateful City of St. Louis, April 25, 1942". As O'Hare's mother and his sisters clipped newspaper stories and photos the following days, his place in history began to dawn on them. A newspaper headline read, "60,000 give O'Hare a hero's welcome here." The United States in 1942 badly needed a live hero, and Butch O'Hare was a young, handsome naval aviator, so he participated in several war bond tours the following months.

On June 19, 1942, O'Hare assumed command of VF-3, relieving Lieutenant Commander Thach. He was relocated to Maui, Hawaii, to instruct other pilots in combat tactics. U.S. Navy policy was to use its best combat pilots to train newer pilots, in contrast to the Japanese practice of keeping their best pilots flying combat missions. Ensign Edward L. "Whitey" Feightner, who served with O'Hare in July 1942, later said that one of the best pieces of information O'Hare passed on to him, was: "If you ever jump one of these Zeros and you surprise him, remember, the first thing he's going to do is a loop. Don't follow him into it! By the time you go into it a second time, he'll be behind you. The first thing you should do when he starts up the loop is to make a hard right turn and keep turning. You'll come right around, and when he bottoms out of the loop, you'll be right on his tail!'"

O'Hare also related

"First of all, remember, in today's world, whenever you take off and engage the enemy, you're going to be outnumbered. If you want to survive this war, you have to look behind you every chance you get. Even when you pull the trigger, be sure to look behind because there's gonna be someone back there."

An anecdote about O'Hare, serving as an instructor on Hawaii mid-1942:

"[O'Hare] was a great swimmer and spearfisherman, and he insisted that the squadron swim with him. Swimming with Butch O'Hare meant that at eight o'clock in the morning, you swam out into the ocean off Maui; he would still be out there at three in the afternoon! If he got hungry, Butch would roll over and dive, and the next thing you knew, he would come up with a fish of some sort. Then he'd just roll over and lie on his back like an otter and eat the thing raw! He really impressed us with that! One day, he came back to the surface with an octopus draped over his arm. He said, 'Now, you have to learn how to kill these things, boys: you bite 'em right behind the eye.' And with that, he chomped down! The octopus has some sort of spinal cord there, and biting it there does kill it! Then we had to go back to the beach where Butch would put these things in a frying pan with a little oil and some salt and stir them around. He enjoyed them, but they tasted like old rubber tires to me!" On March 2, 1943, O'Hare met Rita and hugged his one-month-old daughter, Kathleen, for the first time. His family resided in Coronado at 549 Orange Avenue, near North Island NAS. At the end of March 1943, O'Hare made Ensign Alexander Vraciu, a young Naval Reservist just out of flight school, his wingman. On July 15, 1943, VF-3 swapped designations with VF-6 squadron.

Return to combat
Equipped with the highly successful follow-on to the Wildcat, the new Grumman F6F-3 Hellcat, two-thirds of VF-6 (twenty-four F6F-3s) under O'Hare's command embarked on August 22, 1943, on the light carrier USS Independence (CVL-22). The arrival of the F6Fs with their powerful Pratt & Whitney R-2800 radial engines in late 1943 combined with the deployment of the new Essex-class aircraft carrier carriers and the Independence-class aircraft carrier carriers immediately gave the U.S. Pacific Fleet air supremacy wherever the Fast Carrier Force operated. The Hellcat's first combat mission occurred on August 31, 1943, in a strike against Marcus Island. The F6F did well against Japanese fighters and proved that with the right tactics and teamwork the Japanese Zero need not be considered a superior enemy. VF-6's combat debut on the Independence also went reasonably well. For his actions in battles near Marcus Island on August 31, 1943, O'Hare was awarded the Distinguished Flying Cross. For his actions in subsequent missions near Wake Island on October 5, 1943, O'Hare was awarded a Gold Star in lieu of a second Distinguished Flying Cross. On October 10, 1943, O'Hare flew with VF-6 again in the airstrikes against Wake Island. On this mission, the future ace Lt.(jg) Alex Vraciu was his wingman – both Butch and Vraciu shot down one enemy plane that day. When they came across an enemy formation Butch took the outside aeroplane and Vraciu took the inside plane. Butch went below the clouds to get a Japanese Mitsubishi Zero and Vraciu lost him, so Vraciu kept an eye on a second Zero that went to Wake Island and landed. Vraciu strafed the Zero on the ground, then saw a "Betty" bomber and shot it down. Upon returning to the carrier, O'Hare asked Vraciu where he went and Vraciu knew then that he should have definitely stayed with his leader. Alex Vraciu later said after the war, "O'Hare taught many of the squadron members little things that would later save their lives. One example was to swivel your neck before starting a strafing run to make sure enemy fighters were not on your tail." Vraciu also learned from O'Hare the "high side pass" used for attacking the G4M "Betty" bombers. The high side technique was used to avoid the lethal 20 mm fire of the "Betty"'s tail gunner. The Wake Island raid would be the last occasion Butch would lead VF-6 in battle. According to orders dated September 17, 1943, October found O'Hare as Commander Air Group (CAG) commanding Air Group Six, embarked on USS Enterprise. Functioning as CAG, O'Hare was given command of the entire Enterprise air group: Grumman F6F Hellcat fighters, Douglas SBD Dauntless dive bombers, Grumman TBF Avenger torpedo bombers and 100 pilots.

Now overseeing three squadrons, O'Hare still insisted that everyone call him "Butch". O'Hare's VF-6 squadron would "still stay broken up" among three light aircraft carriers, the squadron had made itself just too useful filling out the light carrier air groups, and AirPac had no well-trained replacements on hand. As a result, Fighting Squadron Two (VF-2) boarded the USS Enterprise from November 1943 and became Butch's new Fighting Squadron. While he readied his new air group, he suffered what he intended as only a temporary separation from his beloved VF-6 "Felix the Cat" Squadron. The news, that the commanding officer had to leave them, hit the men of VF-6 hard. O'Hare first flew a TBM-1 Avenger as CAG-6 command aircraft with bombardier Del Delchamps, AOM1/c and radioman Hal Coleman as crew members. With its good radio facilities, docile handling, and long-range, the Grumman Avenger made an ideal command aircraft for Air Group Commanders (CAGs), but Butch considered the Grumman torpedo bomber as a 'lame turkey' compared to the Grumman F6F Hellcat fighter.

Later Rear Admiral Arthur W. Radford honored a request from O'Hare to take a fighter as command aircraft instead of the Avenger, so O'Hare in a fateful decision happily drew Grumman F6F-3 Hellcat Bureau Number 66168 from the fleet pool to become his principal CAG plane, numbered "00". From 20 – November 23, 1943, the U.S. forces landed in the Gilberts (Tarawa and Makin), and the Enterprise joined in providing close air support to the Marines landing on Makin Island. Equipped with the Grumman F6F Hellcat, the U.S. Navy fighter pilots could protect the fleet from attacking Japanese aircraft.

Final mission and death
Faced with U.S. daylight air superiority, the Japanese quickly developed tactics to send torpedo-armed "Betty" bombers on night missions from their bases in the Marianas against the U.S. aircraft carriers. In late November they launched these low-altitude strikes almost nightly to attack Enterprise and other American ships, so Rear Admiral Arthur W. Radford, O'Hare and Commander Tom Hamilton, CV-6 air officer, were deeply involved in developing ad hoc counter-tactics, the first carrier-based night fighter operations of the U.S. Navy. O'Hare's plan required the carrier's Fighter Director Officer (FDO) to spot incoming enemy formations at a distance and send a "Bat Team" section consisting of a Grumman TBF Avenger torpedo bomber and two Grumman F6F Hellcat fighters toward the Japanese intruders. Although improvements in new types of aviation radar were soon forthcoming from the engineers at MIT and the electronic industry, the available primitive radars in 1943 were very bulky, attributed to the fact that they contained vacuum tube technology. Radars were carried only on the roomy TBF Avengers, but not on the smaller and faster Hellcats, so the radar-equipped TBF Avenger would lead the Hellcats into position behind the incoming bombers, close enough for the F6F pilots to spot visually the blue exhaust flames of the Japanese bombers. Finally, the Hellcats would close in and shoot down the torpedo-carrying bombers.

One of the four "Bat Team" fighter pilots to conduct these experimental night fighter operations to intercept and destroy enemy bombers attacking Allied landing forces was then-LT Roy Marlin Voris, who after the war founded and commanded the Navy's flight demonstration squadron, the Blue Angels.

On the night of November 26, 1943, the Enterprise introduced the experiment in the co-operative control of Avengers and Hellcats for night fighting, when the three-plane team from the ship broke up a large group of land-based bombers attacking Task Group TG 50.2. O'Hare volunteered to lead this mission to conduct the first-ever Navy nighttime fighter attack from an aircraft carrier to intercept a large force of enemy torpedo bombers. When the call came to man the fighters, Butch O'Hare was eating. He grabbed up part of his supper in his fist and started running for the ready room. He was dressed in loose marine coveralls. The night fighter unit consisting of 1 VT and 2 VF was catapulted between 17:58 and 18:01. The pilots for this flight were Butch O'Hare and Ensign Warren Andrew "Andy" Skon of VF-2 in F6Fs and the Squadron Commander of VT-6, LCDR John C. Phillips in a TBF-1C. The crew of the TBF torpedo plane consisted of LTJG Hazen B. Rand, a radar specialist and Alvin Kernan, A. B., AOM1/c. The "Black Panthers", as the night fighters were dubbed, took off before dusk and flew out into the incoming mass of Japanese planes.

Confusion and complications endangered the success of the mission. The Hellcats first had trouble finding the Avenger, the FDO had difficulty guiding any of them on the targets. O'Hare and Ensign W. Skon in their F6F Hellcats finally got into position behind the Avenger. Butch O'Hare had been well aware of the deadly danger of friendly fire in this situation – he radioed to the Avenger pilot of his section, "Hey, Phil, turn those running lights on. I want to be sure it's a yellow devil I'm drilling."

O'Hare was last seen at the 5 o'clock position of the TBF. About that time, the turret gunner of the TBF, Alvin Kernan (AOM1/c) noticed a Japanese G4M "Betty" bomber above and almost directly behind O'Hare's 6 o'clock position. Kernan opened fire with the TBF's .50 cal. machine gun in the dorsal turret and a Japanese gunner fired back. O'Hare's F6F Hellcat apparently was caught in a crossfire. Seconds later O'Hare's F6F slid out of formation to port, pushing slightly ahead at about 160 kn and then vanished in the dark. The Avenger pilot, Lieutenant Commander Phillips, called repeatedly to O'Hare, but received no reply. Ensign Skon responded: "Mr Phillips, this is Skon. I saw Mr O'Hare's lights go out and, at the same instant, he seemed to veer off and slant down into darkness." Phillips later asserted, as the Hellcat dropped out of view, it seemed to release something that fell almost vertically at a speed too slow for anything but a parachute. Then something "whitish-gray" appeared below, perhaps the splash of the aircraft plunging into the sea.

Lieutenant Commander Phillips reported the position (1.43333°N, -171.93333°W) to the ship. After dawn, a three-plane search was made, but no trace of O'Hare or his aircraft was found. On November 29 a PBY Catalina flying boat also conducted a search with no positive result, and O'Hare was reported missing in action.

For 54 years there was no definitive answer as to whether he had been brought down by friendly fire or the Japanese bomber's nose gunner. In 1997, Fateful Rendezvous: The Life of Butch O'Hare, by Steve Ewing and John B. Lundstrom (see References below) shed new light. Ewing and Lundstrom state that Japanese guns, and not Kernan's, killed O'Hare.

In Chapter 16, "What Happened to Butch", the authors write, "Butch fell to his old familiar adversary, a Betty. Most likely he died from or was immediately disabled by, a lucky shot from the forward observer crouched in the rikko's [Betty's] forward glassed-in nose ... the nose gunner's 7.7 mm slugs very likely penetrated Butch's cockpit from above on the port side and ahead of the F6F's armor plate." In the index, Ewing and Lundstrom flatly state that Kernan is "wrongly accused of shooting down Butch."

Ewing and Lundstrom point out that the "most influential and oft-cited" account of O'Hare's last mission came in a 1962 history of the Enterprise by CDR Edward P. Stafford, which relied on action reports and recollections of former Enterprise crew, but did not contain interviews with any of the living participants. By contrast, Ewing and Lundstrom came to their conclusions on what happened to Butch after interviewing the still-living survivors of O'Hare's last mission: F6F pilot Skon, TBF radar officer Rand, and TBF gunner Kernan. Ewing and Lundstrom write, "Through Stafford and other accounts based largely on the action reports, Butch has wrongly become known as one of America's most famous 'friendly fire' casualties."

On December 9, the official word arrived that O'Hare was missing in action. His mother Selma left for San Diego to be with his wife Rita and his daughter Kathleen. LCDR Bob Jackson wrote to Rita O'Hare from the Enterprise to describe the extensive but unsuccessful search for her husband. In the letter, LCDR Jackson quoted RADM Arthur W. Radford saying of Butch O'Hare that he "never saw one individual so universally liked." The hardest thing O'Hare's former wingman Alex Vraciu had to do was to talk to O'Hare's wife Rita after returning stateside. On December 20, 1943, a Solemn Pontifical Mass of Requiem was offered for Butch O'Hare at the St. Louis Cathedral.

As O'Hare went missing on November 26, 1943, and was declared dead a year later, his widow Rita received her husband's posthumous decorations, a Purple Heart and the Navy Cross on November 26, 1944.

Medal of Honor citation

"For conspicuous gallantry and intrepidity in aerial combat, at grave risk of his life above and beyond the call of duty, as section leader and pilot of Fighting Squadron 3 on February 20, 1942. Having lost the assistance of his teammates, Lieutenant O'Hare interposed his fighter between his ship and an advancing enemy formation of 9 attacking twin-engine heavy bombers. Without hesitation, alone and unaided, he repeatedly attacked this enemy formation, at close range in the face of intense combined machine gun and cannon fire. Despite this concentrated opposition, Lieutenant O'Hare, by his gallant and courageous action, his extremely skilful marksmanship in making the most of every shot of his limited amount of ammunition, shot down 5 enemy bombers and severely damaged a sixth before they reached the bomb release point. As a result of his gallant action—one of the most daring, if not the most daring, single action in the history of combat aviation—he undoubtedly saved his carrier from serious damage."

Navy Cross citation
"The President of the United States of America takes pride in presenting the Navy Cross (Posthumously) to Lieutenant Commander Edward Henry 'Butch' O'Hare (NSN: 0-78672), United States Navy, for extraordinary heroism in operations against the enemy while serving as Pilot of a carrier-based Navy Fighter Plane in Fighting Squadron TWO (VF-2), attached to the U.S.S. ENTERPRISE (CV-6), and deployed over Tarawa in the Gilbert Islands, in action against enemy Japanese forces on 26 November 1943. When warnings were received of the approach of a large force of Japanese torpedo bombers, Lieutenant Commander O'Hare volunteered to lead a fighter section of aircraft from his carrier, the first time such a mission had been attempted at night, in order to intercept the attackers. He fearlessly led his three-plane group into combat against a large formation of hostile aircraft and assisted in shooting down two Japanese aeroplanes and dispersed the remainder. Lieutenant Commander O'Hare's outstanding courage, daring airmanship and devotion to duty were in keeping with the highest traditions of the United States Naval Service. He gallantly gave his life for his country."

USS O'Hare
On January 27, 1945, the United States Navy named a Gearing-class destroyer USS O'Hare (DD-889) in his honor. The ship was launched June 22, 1945, with his mother, Selma O'Hare, as the sponsor. O'Hare was decommissioned on October 31, 1973, then transferred on loan and later sold to the Spanish Navy. In 1992, the Spanish Navy decommissioned and scrapped the ship.

O'Hare International Airport
Colonel Robert R. McCormick, publisher of the Chicago Tribune, suggested that the name of Chicago's Orchard Depot Airport be changed as a tribute to O'Hare. On September 19, 1949, the airport was renamed O'Hare International Airport to honor O'Hare's bravery. The airport displays a Grumman F4F-3 like the one flown during the Medal of Honor action.

The Grumman F4F-3 Wildcat on display was recovered virtually intact from the bottom of Lake Michigan, where it sank after a training accident in 1943 when it went off the training aircraft carrier USS Wolverine (IX-64). In 2001, the Air Classics Museum remodeled the aircraft to replicate the F4F-3 Wildcat that O'Hare flew on his Medal of Honor flight. The restored Wildcat is exhibited in the west end of Terminal 2 behind the security checkpoint to honor O'Hare International Airport's namesake.

Other honors
The same month, O'Hare's name was engraved on the National Memorial Cemetery of the Pacific "Wall of the Missing" in Honolulu. In March 1963, President John F. Kennedy performed a wreath-laying ceremony at O'Hare Airport to honor Butch O'Hare. The Patriots Point Naval and Maritime Museum is honoring O'Hare with an F4F-3A on display and a plaque dedicated by the USS Yorktown CV-10 association, "May Butch O'Hare rest in peace ...".

title: Edward O'Hare