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Euler's rotation theorem

In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point. It also means that the composition of two rotations is also a rotation. Therefore the set of rotations has a group structure, known as a rotation group.

The theorem is named after Leonhard Euler, who proved it in 1775 by means of spherical geometry. The axis of rotation is known as an Euler axis, typically represented by a unit vector $ê$. Its product by the rotation angle is known as an axis-angle vector. The extension of the theorem to kinematics yields the concept of instant axis of rotation, a line of fixed points.

In linear algebra terms, the theorem states that, in 3D space, any two Cartesian coordinate systems with a common origin are related by a rotation about some fixed axis. This also means that the product of two rotation matrices is again a rotation matrix and that for a non-identity rotation matrix one eigenvalue is 1 and the other two are both complex, or both equal to −1. The eigenvector corresponding to this eigenvalue is the axis of rotation connecting the two systems.

Euler's theorem (1776)
Euler states the theorem as follows:

Theorema. Quomodocunque sphaera circa centrum suum conuertatur, semper assignari potest diameter, cuius directio in situ translato conueniat cum situ initiali.

or (in English): "When a sphere is moved around its centre it is always possible to find a diameter whose direction in the displaced position is the same as in the initial position."

Proof
Euler's original proof was made using spherical geometry and therefore whenever he speaks about triangles they must be understood as spherical triangles.

Previous analysis
To arrive at a proof, Euler analyses what the situation would look like if the theorem were true. To that end, suppose the yellow line in Figure 1 goes through the center of the sphere and is the axis of rotation we are looking for, and point $O$ is one of the two intersection points of that axis with the sphere. Then he considers an arbitrary great circle that does not contain $O$ (the blue circle), and its image after rotation (the red circle), which is another great circle not containing $O$. He labels a point on their intersection as point $O$. (If the circles coincide, then $A$ can be taken as any point on either; otherwise $A$ is one of the two points of intersection.)

Now $A$ is on the initial circle (the blue circle), so its image will be on the transported circle (red). He labels that image as point $α$. Since $a$ is also on the transported circle (red), it is the image of another point that was on the initial circle (blue) and he labels that preimage as $A$ (see Figure 2). Then he considers the two arcs joining $AO$ and $A$ to $A$. These arcs have the same length because arc $a$ is mapped onto arc $A$. Also, since $α$ is a fixed point, triangle $α$ is mapped onto triangle $a$, so these triangles are isosceles, and arc $A$ bisects angle $αA$.

Construction of the best candidate point
Let us construct a point that could be invariant using the previous considerations. We start with the blue great circle and its image under the transformation, which is the red great circle as in the Figure 1. Let point $Aa$ be a point of intersection of those circles. If $O$’s image under the transformation is the same point then $αOA$ is a fixed point of the transformation, and since the center is also a fixed point, the diameter of the sphere containing $AOa$ is the axis of rotation and the theorem is proved.

Otherwise we label $AO$’s image as $∠αAa$ and its preimage as $O$, and connect these two points to $O′$ with arcs $O′$ and $O$. These arcs have the same length. Construct the great circle that bisects $A$ and locate point $A$ on that great circle so that arcs $A$ and $A$ have the same length, and call the region of the sphere containing $A$ and bounded by the blue and red great circles the interior of $a$. (That is, the yellow region in Figure 3.) Then since $α$ and $A$ is on the bisector of $αA$, we also have $Aa$.

Proof of its invariance under the transformation
Now let us suppose that $∠αAa$ is the image of $O$. Then we know $AO$ and orientation is preserved, so $aO$ must be interior to $O$. Now $∠αAa$ is transformed to $αA = Aa$, so $O$. Since $∠αAa$ is also the same length as $αO = aO$, then $O′$ and $O$. But $∠αAO = ∠AaO′$, so $αA$ and $A$. Therefore $OA$ is the same point as $Aa$. In other words, $a$ is a fixed point of the transformation, and since the center is also a fixed point, the diameter of the sphere containing $O′a$ is the axis of rotation.

Final notes about the construction

Euler also points out that $O′$ can be found by intersecting the perpendicular bisector of $∠αAa$ with the angle bisector of $AO$, a construction that might be easier in practice. He also proposed the intersection of two planes:

 * the symmetry plane of the angle $aO′$ (which passes through the center $AO = aO′$ of the sphere), and
 * the symmetry plane of the arc $AO$ (which also passes through $aO$).

 * Proposition. These two planes intersect in a diameter. This diameter is the one we are looking for.

 * Proof. Let us call $aO = aO′$ either of the endpoints (there are two) of this diameter over the sphere surface. Since $∠AaO = ∠aAO$ is mapped on $∠αAO = ∠aAO$ and the triangles have the same angles, it follows that the triangle $∠αAO = ∠AaO$ is transported onto the triangle $∠AaO = ∠AaO′$. Therefore the point $O′$ has to remain fixed under the movement.

 * Corollaries. This also shows that the rotation of the sphere can be seen as two consecutive reflections about the two planes described above. Points in a mirror plane are invariant under reflection, and hence the points on their intersection (a line: the axis of rotation) are invariant under both the reflections, and hence under the rotation.

Another simple way to find the rotation axis is by considering the plane on which the points $O$, $O$, $O$ lie. The rotation axis is obviously orthogonal to this plane, and passes through the center $O$ of the sphere.

Given that for a rigid body any movement that leaves an axis invariant is a rotation, this also proves that any arbitrary composition of rotations is equivalent to a single rotation around a new axis.

Matrix proof
A spatial rotation is a linear map in one-to-one correspondence with a 3 × 3 rotation matrix $Aa$ that transforms a coordinate vector $∠αAa$ into $∠αAa$, that is $C$. Therefore, another version of Euler's theorem is that for every rotation $Aa$, there is a nonzero vector $C$ for which $O$; this is exactly the claim that $αA$ is an eigenvector of $Aa$ associated with the eigenvalue 1. Hence it suffices to prove that 1 is an eigenvalue of $OαA$; the rotation axis of $OAa$ will be the line $O$, where $α$ is the eigenvector with eigenvalue 1.

A rotation matrix has the fundamental property that its inverse is its transpose, that is

\mathbf{R}^\mathsf{T}\mathbf{R} = \mathbf{R}\mathbf{R}^\mathsf{T} = \mathbf{I}, $$ where $A$ is the 3 × 3 identity matrix and superscript T indicates the transposed matrix.

Compute the determinant of this relation to find that a rotation matrix has determinant ±1. In particular,
 * $$\begin{align}

1 = \det(\mathbf{I}) &= \det\left(\mathbf{R}^\mathsf{T}\mathbf{R}\right) = \det\left(\mathbf{R}^\mathsf{T}\right)\det(\mathbf{R}) = \det(\mathbf{R})^2 \\ \Longrightarrow\qquad \det(\mathbf{R}) &= \pm 1. \end{align}$$

A rotation matrix with determinant +1 is a proper rotation, and one with a negative determinant −1 is an improper rotation, that is a reflection combined with a proper rotation.

It will now be shown that a proper rotation matrix $a$ has at least one invariant vector $C$, i.e., $R$. Because this requires that $x$, we see that the vector $X$ must be an eigenvector of the matrix $Rx = X$ with eigenvalue $R$. Thus, this is equivalent to showing that $n$.

Use the two relations
 * $$ \det(-\mathbf{A}) = (-1)^{3} \det(\mathbf{A}) = - \det(\mathbf{A}) \quad$$

for any 3 × 3 matrix A and
 * $$ \det\left(\mathbf{R}^{-1} \right) = 1 \quad$$

(since $Rn = n$) to compute
 * $$\begin{align}

&\det(\mathbf{R} - \mathbf{I}) = \det\left((\mathbf{R} - \mathbf{I})^\mathsf{T}\right) \\ {}={} &\det\left(\mathbf{R}^\mathsf{T} - \mathbf{I}\right) = \det\left(\mathbf{R}^{-1} - \mathbf{R}^{-1}\mathbf{R}\right) \\ {}={} &\det\left(\mathbf{R}^{-1}(\mathbf{I} - \mathbf{R})\right) = \det\left(\mathbf{R}^{-1}\right) \, \det(-(\mathbf{R} - \mathbf{I})) \\ {}={} &-\det(\mathbf{R} - \mathbf{I}) \\[3pt] \Longrightarrow\ 0 ={} &\det(\mathbf{R} - \mathbf{I}). \end{align}$$

This shows that $n$ is a root (solution) of the characteristic equation, that is,

\det(\mathbf{R} - \lambda \mathbf{I}) = 0\quad \hbox{for}\quad \lambda=1. $$

In other words, the matrix $R$ is singular and has a non-zero kernel, that is, there is at least one non-zero vector, say $R$, for which

(\mathbf{R} - \mathbf{I}) \mathbf{n} = \mathbf{0} \quad \Longleftrightarrow \quad \mathbf{R}\mathbf{n} = \mathbf{n}. $$

The line $R$ for real $μ$ is invariant under $μn$, i.e., $n$ is a rotation axis. This proves Euler's theorem.

Equivalence of an orthogonal matrix to a rotation matrix
Two matrices (representing linear maps) are said to be equivalent if there is a change of basis that makes one equal to the other. A proper orthogonal matrix is always equivalent (in this sense) to either the following matrix or to its vertical reflection:

\mathbf{R} \sim \begin{pmatrix} \cos\phi & -\sin\phi & 0 \\ \sin\phi & \cos\phi & 0 \\ 0 & 0 & 1\\ \end{pmatrix}, \qquad 0\le \phi \le 2\pi. $$

Then, any orthogonal matrix is either a rotation or an improper rotation. A general orthogonal matrix has only one real eigenvalue, either +1 or −1. When it is +1 the matrix is a rotation. When −1, the matrix is an improper rotation.

If $I$ has more than one invariant vector then $R$ and $n$. Any vector is an invariant vector of $Rn = n$.

Excursion into matrix theory
In order to prove the previous equation some facts from matrix theory must be recalled.

An $(R − I)n = 0$ matrix $n$ has $R$ orthogonal eigenvectors if and only if $λ = 1$ is normal, that is, if $det(R − I) = 0$. This result is equivalent to stating that normal matrices can be brought to diagonal form by a unitary similarity transformation:

\mathbf{A}\mathbf{U} = \mathbf{U}\; \operatorname{diag}(\alpha_1,\ldots,\alpha_m)\quad \Longleftrightarrow\quad \mathbf{U}^\dagger \mathbf{A}\mathbf{U} = \operatorname{diag}(\alpha_1,\ldots,\alpha_m), $$ and $det(R) = 1$ is unitary, that is,

\mathbf{U}^\dagger = \mathbf{U}^{-1}. $$ The eigenvalues $λ = 1$ are roots of the characteristic equation. If the matrix $R − I$ happens to be unitary (and note that unitary matrices are normal), then

\left(\mathbf{U}^\dagger\mathbf{A} \mathbf{U}\right)^\dagger = \operatorname{diag}\left(\alpha^*_1,\ldots,\alpha^*_m\right) = \mathbf{U}^\dagger\mathbf{A}^{-1} \mathbf{U} = \operatorname{diag}\left(\frac{1}{\alpha_1},\ldots,\frac{1}{\alpha_m}\right) $$ and it follows that the eigenvalues of a unitary matrix are on the unit circle in the complex plane:

\alpha^*_k = \frac{1}{\alpha_k} \quad\Longleftrightarrow\quad \alpha^*_k\alpha_k = \left|\alpha_k\right|^2 = 1,\qquad k=1,\ldots,m. $$ Also an orthogonal (real unitary) matrix has eigenvalues on the unit circle in the complex plane. Moreover, since its characteristic equation (an $m$th order polynomial in $λ$) has real coefficients, it follows that its roots appear in complex conjugate pairs, that is, if $α$ is a root then so is $n$. There are 3 roots, thus at least one of them must be purely real (+1 or −1).

After recollection of these general facts from matrix theory, we return to the rotation matrix $μn$. It follows from its realness and orthogonality that we can find a $R$ such that:

\mathbf{R} \mathbf{U} = \mathbf{U} \begin{pmatrix} e^{i\phi} & 0 & 0 \\ 0 & e^{-i\phi} & 0 \\ 0 & 0 & \pm 1 \\ \end{pmatrix} $$ If a matrix $μn$ can be found that gives the above form, and there is only one purely real component and it is −1, then we define $$\mathbf{R}$$ to be an improper rotation. Let us only consider the case, then, of matrices R that are proper rotations (the third eigenvalue is just 1). The third column of the 3 × 3 matrix $R$ will then be equal to the invariant vector $φ = 0$. Writing $R = I$ and $I$ for the first two columns of $m × m$, this equation gives

\mathbf{R}\mathbf{u}_1 = e^{i\phi}\, \mathbf{u}_1 \quad\hbox{and}\quad \mathbf{R}\mathbf{u}_2 = e^{-i\phi}\, \mathbf{u}_2. $$ If $A$ has eigenvalue 1, then $m$ and $A$ has also eigenvalue 1, which implies that in that case $A^{†}A = AA^{†}$. In general, however, as $$ (\mathbf{R}-e^{i\phi}\mathbf{I})\mathbf{u}_1 = 0 $$ implies that also $$ (\mathbf{R}-e^{-i\phi}\mathbf{I})\mathbf{u}^*_1 = 0 $$ holds, so $$ \mathbf{u}_2 = \mathbf{u}^*_1 $$ can be chosen for $$ \mathbf{u}_2 $$. Similarly, $$ (\mathbf{R}-\mathbf{I})\mathbf{u}_3 = 0 $$ can result in a $$ \mathbf{u}_3 $$  with real entries only, for a proper rotation matrix $$\mathbf{R}$$. Finally, the matrix equation is transformed by means of a unitary matrix,

\mathbf{R} \mathbf{U} \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\ \frac{1}{\sqrt{2}} & \frac{-i}{\sqrt{2}} & 0 \\ 0 & 0 & 1\\ \end{pmatrix} = \mathbf{U} \underbrace{ \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\ \frac{1}{\sqrt{2}} & \frac{-i}{\sqrt{2}} & 0 \\ 0 & 0 & 1\\ \end{pmatrix} \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ \frac{-i}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\ 0 & 0 & 1\\ \end{pmatrix} }_{=\;\mathbf{I}} \begin{pmatrix} e^{i\phi} & 0 & 0 \\ 0 & e^{-i\phi} & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\ \frac{1}{\sqrt{2}} & \frac{-i}{\sqrt{2}} & 0 \\ 0 & 0 & 1\\ \end{pmatrix} $$ which gives

\mathbf{U'}^\dagger \mathbf{R} \mathbf{U'} = \begin{pmatrix} \cos\phi & -\sin\phi & 0 \\ \sin\phi & \cos\phi & 0 \\ 0 & 0 & 1\\ \end{pmatrix} \quad\text{ with }\quad \mathbf{U'} = \mathbf{U} \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\ \frac{1}{\sqrt{2}} & \frac{-i}{\sqrt{2}} & 0 \\ 0 & 0 & 1\\ \end{pmatrix}. $$ The columns of $†$ are orthonormal as it is a unitary matrix with real-valued entries only, due to its definition above, that $$ \mathbf{u}_1 $$ is the complex conjugate of $$ \mathbf{u}_2 $$ and that $$ \mathbf{u}_3 $$ is a vector with real-valued components. The third column is still $$ \mathbf{u}_3 =$$ $U$, the other two columns of $α_{1}, ..., α_{m}$ are perpendicular to $A$. We can now see how our definition of improper rotation corresponds with the geometric interpretation: an improper rotation is a rotation around an axis (here, the axis corresponding to the third coordinate) and a reflection on a plane perpendicular to that axis. If we only restrict ourselves to matrices with determinant 1, we can thus see that they must be proper rotations. This result implies that any orthogonal matrix $α^{∗}$ corresponding to a proper rotation is equivalent to a rotation over an angle $φ$ around an axis $R$.

Equivalence classes
The trace (sum of diagonal elements) of the real rotation matrix given above is $U$. Since a trace is invariant under an orthogonal matrix similarity transformation,

\mathrm{Tr}\left[\mathbf{A} \mathbf{R} \mathbf{A}^\mathsf{T}\right] = \mathrm{Tr}\left[ \mathbf{R} \mathbf{A}^\mathsf{T}\mathbf{A}\right] = \mathrm{Tr}[\mathbf{R}]\quad\text{ with }\quad \mathbf{A}^\mathsf{T} = \mathbf{A}^{-1}, $$ it follows that all matrices that are equivalent to $U$ by such orthogonal matrix transformations have the same trace: the trace is a class function. This matrix transformation is clearly an equivalence relation, that is, all such equivalent matrices form an equivalence class.

In fact, all proper rotation 3 × 3 rotation matrices form a group, usually denoted by SO(3) (the special orthogonal group in 3 dimensions) and all matrices with the same trace form an equivalence class in this group. All elements of such an equivalence class share their rotation angle, but all rotations are around different axes. If $U$ is an eigenvector of $n$ with eigenvalue 1, then $u_{1}$ is also an eigenvector of $u_{2}$T, also with eigenvalue 1. Unless $U$, $u_{1}$ and $φ = 0$ are different.

Generators of rotations
Suppose we specify an axis of rotation by a unit vector $u_{2}$, and suppose we have an infinitely small rotation of angle $R = I$ about that vector. Expanding the rotation matrix as an infinite addition, and taking the first order approach, the rotation matrix $U′$ is represented as:

\Delta R = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} + \begin{bmatrix} 0 & z & -y \\ -z & 0 &  x \\ y & -x & 0 \end{bmatrix}\,\Delta \theta = \mathbf{I} + \mathbf{A}\,\Delta \theta. $$

A finite rotation through angle $θ$ about this axis may be seen as a succession of small rotations about the same axis. Approximating $n$ as $U′$ where $n$ is a large number, a rotation of $θ$ about the axis may be represented as:

 * $$R = \left(\mathbf{1}+\frac{\mathbf{A}\theta}{N}\right)^N \approx e^{\mathbf{A}\theta}.$$

It can be seen that Euler's theorem essentially states that all rotations may be represented in this form. The product $R$ is the "generator" of the particular rotation, being the vector $n$ associated with the matrix $1 + 2 cos φ$. This shows that the rotation matrix and the axis–angle format are related by the exponential function.

One can derive a simple expression for the generator $R$. One starts with an arbitrary plane (in Euclidean space) defined by a pair of perpendicular unit vectors $n$ and $R$. In this plane one can choose an arbitrary vector $An$ with perpendicular $ARA$. One then solves for $A = I$ in terms of $n$ and substituting into an expression for a rotation in a plane yields the rotation matrix $An$ which includes the generator $[x, y, z]$$T$$Δθ$$T$.

 * $$\begin{align}

\mathbf{x} &= \mathbf{a}\cos\alpha + \mathbf{b}\sin\alpha \\ \mathbf{y} &= -\mathbf{a}\sin\alpha + \mathbf{b}\cos\alpha \\[8pt] \cos\alpha &= \mathbf{a}^\mathsf{T}\mathbf{x} \\ \sin\alpha &= \mathbf{b}^\mathsf{T}\mathbf{x} \\[8px] \mathbf{y} &= -\mathbf{ab}^\mathsf{T}\mathbf{x} + \mathbf{ba}^\mathsf{T}\mathbf{x} = \left( \mathbf{ba}^\mathsf{T} - \mathbf{ab}^\mathsf{T} \right)\mathbf{x} \\[8px] \mathbf{x}' &= \mathbf{x}\cos\beta + \mathbf{y}\sin\beta \\ &= \left( \mathbf{I}\cos\beta + \left( \mathbf{ba}^\mathsf{T} - \mathbf{ab}^\mathsf{T} \right) \sin\beta \right)\mathbf{x} \\[8px] \mathbf{R} &= \mathbf{I}\cos\beta + \left( \mathbf{ba}^\mathsf{T} - \mathbf{ab}^\mathsf{T} \right)\sin\beta \\ &= \mathbf{I}\cos\beta + \mathbf{G}\sin\beta \\[8px] \mathbf{G} &= \mathbf{ba}^\mathsf{T} - \mathbf{ab}^\mathsf{T} \end{align}$$

To include vectors outside the plane in the rotation one needs to modify the above expression for $ΔR$ by including two projection operators that partition the space. This modified rotation matrix can be rewritten as an exponential function.

 * $$\begin{align}

\mathbf{P_{ab}} &= -\mathbf{G}^2 \\ \mathbf{R} &= \mathbf{I} - \mathbf{P_{ab}} + \left( \mathbf{I} \cos \beta + \mathbf{G} \sin \beta \right)\mathbf{P_{ab}} = e^{\mathbf{G}\beta } \end{align}$$

Analysis is often easier in terms of these generators, rather than the full rotation matrix. Analysis in terms of the generators is known as the Lie algebra of the rotation group.

Quaternions
It follows from Euler's theorem that the relative orientation of any pair of coordinate systems may be specified by a set of three independent numbers. Sometimes a redundant fourth number is added to simplify operations with quaternion algebra. Three of these numbers are the direction cosines that orient the eigenvector. The fourth is the angle about the eigenvector that separates the two sets of coordinates. Such a set of four numbers is called a quaternion.

While the quaternion as described above, does not involve complex numbers, if quaternions are used to describe two successive rotations, they must be combined using the non-commutative quaternion algebra derived by William Rowan Hamilton through the use of imaginary numbers.

Rotation calculation via quaternions has come to replace the use of direction cosines in aerospace applications through their reduction of the required calculations, and their ability to minimize round-off errors. Also, in computer graphics the ability to perform spherical interpolation between quaternions with relative ease is of value.

Generalizations
In higher dimensions, any rigid motion that preserves a point in dimension $Δθ$ or $θ⁄N$ is a composition of at most $n$ rotations in orthogonal planes of rotation, though these planes need not be uniquely determined, and a rigid motion may fix multiple axes. Also, any rigid motion that preserves $N$ linearly independent points, which span an $Aθ$-dimensional body in dimension $(x,y,z)$ or $A$, is a single  plane of rotation. To put it another way, if two rigid bodies, with identical geometry, share at least $G$ points of 'identical' locations within themselves, the convex hull of which is $a$-dimensional, then a single planar rotation can bring one to cover the other accurately in dimension $b$ or $x$.

A rigid motion in three dimensions that does not necessarily fix a point is a "screw motion". This is because a composition of a rotation with a translation perpendicular to the axis is a rotation about a parallel axis, while composition with a translation parallel to the axis yields a screw motion; see screw axis. This gives rise to screw theory.

title: Euler's rotation theorem


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Onogami, Gunma

Onogami (小野上村) was a village located in Kitagunma District, Gunma Prefecture, Japan.

As of 2003, the village had an estimated population of 2,067 and a density of 72.88 persons per km2. The total area was 28.36 km2.

On February 20, 2006, Onogami, along with the town of Ikaho, the village of Komochi (all from Kitagunma District), and the villages of Akagi and Kitatachibana (both from Seta District), was merged into the expanded city of Shibukawa.

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

title: Onogami, Gunma


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Hydrozincite

Hydrozincite, also known as zinc bloom or marionite, is a white carbonate mineral consisting of Zn5(CO3)2(OH)6. It is usually found in massive rather than crystalline form.

It occurs as an oxidation product of zinc ores and as post mine incrustations. It occurs associated with smithsonite, hemimorphite, willemite, cerussite, aurichalcite, calcite and limonite.

It was first described in 1853 for an occurrence in Bad Bleiberg, Carinthia, Austria and named for its chemical content.

title: Hydrozincite


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Book of Days

Book of Days may refer to:

Literature

 * Fasti (poem), known in English as the "Book of Days", a Latin poem by Ovid
 * Chambers Book of Days, by Robert Chambers
 * The Wicca Book of Days by Gerina Dunwich
 * Gorillas Among Us: A Primate Ethnographer's Book of Days by Dawn Prince-Hughes
 * Men Are from Mars, Women Are from Venus Book of Days by John Gray
 * Book of Days by China Bayles
 * The diary of Stanislaus Joyce that he called his "Book of Days”
 * Book of Days, a 2000 play by Lanford Wilson

Music

 * The Celtic Book of Days, a 1998 album David Arkenstone
 * "Book of Days" (song), a 1992 single by Enya
 * Book of Days (The Psychedelic Furs album), 1989
 * Book of Days (Nektar album), 2008
 * Book of Days (Meredith Monk album), 1990
 * Book of Days, a 2003 album by Susan Crowe
 * Book of Days, a music video DVD by The Brian Jonestown Massacre 2008

Other

 * Book of Days (film), a 2003 film, starring Wil Wheaton and Isaac Hayes

title: Book of Days


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Talk:Book of Days

I usually make the mistake of thinking that a Book of Hours is actually called a book of days. However, I suspect the term could also be used to refer modern books of aniversaries or on this day type lists, in which case feel free to replace this redirect with an article. -- Solipsist 11:24, 29 Jul 2004 (UTC)

dab page criteria
I have not read the dab page rules in a while, but I think the general convention is it is for navigation purposes, not to list every possible usage .. so if there is no article for it, it may not be appropriate for a dab page. -- Stbalbach 03:16, 30 May 2006 (UTC)

title: Talk:Book of Days


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Wikipedia:Articles for deletion/Morbidity rate

On 29 July 2004, Morbidity rate was proposed for deletion. This page is an archive of the discussion about the proposed deletion. This page is no longer live. Further comments should be made on the article's talk page rather than here so that this page is preserved as an historic record. The result of the debate was KEEP. Rossami 00:23, 5 Aug 2004 (UTC)

Dicdef. - UtherSRG 11:28, 29 Jul 2004 (UTC) This page is now preserved as an archive of the debate and, like other '/delete' pages is no longer 'live'. Subsequent comments on the issue, the deletion or on the decision-making process should be placed on the relevant 'live' pages. Please do not edit this page.
 * Delete: dictdef. Not sure if Wiktionary has it or not.  One could broaden this slightly by talking about the medical uses, but it would be insufficient to salvage the entry. Geogre 12:29, 29 Jul 2004 (UTC)
 * Could expand from stub if someone found reliable statistics on relative rates throughout the world. Dunc_Harris|☺ 13:17, 29 Jul 2004 (UTC)
 * Keep stub; suggest cleanup. Certainly a valid topic.  Smerdis of Tlön 13:42, 29 Jul 2004 (UTC)
 * Keep, valid stub. - SimonP 19:14, Jul 29, 2004 (UTC)
 * Keep. Important topic. Should link to epidemiology. Wile E. Heresiarch 19:53, 29 Jul 2004 (UTC)
 * Keep. Since being listed it has been expanded into a good stub.  SWAdair | Talk  04:58, 30 Jul 2004 (UTC)


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Ena Sharples

Ena Sharples (also Schofield) is a fictional character from the British ITV soap opera Coronation Street, played by Violet Carson. She appeared in the first episode, broadcast on 9 December 1960, and stayed with the show until 2 April 1980.

Development
Ena was the widowed caretaker of the Glad Tidings Mission Hall and spent much of her time criticising the activities and loose morals of the street's other residents. One of the main characters during the 1960s, she was featured less regularly in the 1970s due to the declining health of actress Violet Carson and was written out in 1980. Almost always wearing a double-breasted overcoat and hairnet, she spent much of her free time in the serial's early years with her two cronies, Martha Longhurst (Lynne Carol) and Minnie Caldwell (Margot Bryant), in the snug bar of local pub The Rovers Return Inn, drinking milk stout. Ena particularly criticised Elsie Tanner (Pat Phoenix), who she believed had loose morals. In turn, Elsie strongly disliked Ena for her seemingly incessant gossip and desire to know about the private lives of others, and the two confronted each other frequently.

The final episode featuring Ena was broadcast on 2 April 1980. It attracted little media or public attention, as the producers intended for her to continue in the serial and the character was merely going to stay with a friend, Henry Foster, in St Annes while her flat in Coronation Street was being renovated. However, all subsequent storylines involving Ena had to be scrapped owing to Carson developing symptoms of pernicious anemia and becoming too ill to appear.

Carson died on Boxing Day 1983 at the age of 85. No further mention of Ena was made, and it was assumed she had moved to St Annes permanently. The first reference to her after this was in the 1985 spin-off video The Jubilee Years when Ken Barlow revealed that Ena died "a couple of years ago", placing her death around the same time as that of Carson's. The first mention of Ena's death in the show itself was on 27 November 1989, during a scene in the Rovers Return when Deirdre says to builder Maurice Jones: "Ena Sharples would be turning in her grave."

In 2010, a short online video titled Ken: A Life on the Street reiterated that she was no longer alive when Ken Barlow (William Roache) told his grandson Simon (Alex Bain) that she was "long dead now". This is available as an extra on the Tram Crash DVD, released in 2011.

Backstory
Ena Schofield was born in Weatherfield on 24 November 1899. She had two elder siblings, Alice Raynold (née Schofield) (1885–1965) and Tom Schofield (d. 1973). During her school days Ena met Martha Hartley (Lynne Carol) and Minnie Carlton (Margot Bryant), establishing lifelong friendships. An attentive student with a strong Christian upbringing, Ena became devoted to her religion and developed a firm belief in the importance of rules, regulations and morals. She also quickly learned to be self-sufficient, taking on factory work when she was only eleven.

In 1915, Ena became engaged to Phil Moss, nephew of Gladys Arkwright, her mentor at the Glad Tidings Mission Hall. However, he enlisted for military service in the Great War and was killed in battle, devastating Ena. In 1914, Ena had become friends with Albert Tatlock after Dinky Low, a local boy Ena had a crush on, asked her to write to Albert. Ena and Albert became close enough that he would stay with her family while on leave but, when he asked her if she wanted to be more than friends, she declined, stating that she was still mourning Phil. Around 1917, Ena met Alfred Sharples, who had been sent home because of a leg injury. There was an instant chemistry, but Ena was upset to learn he was already married, and broke contact with him. When the war ended Alfred and Ena met up again and he told her that his marriage had been in name only – he'd had to marry the daughter of someone his father had owed money to. Alfred also informed Ena that his wife had recently died, so Ena decided to give him another chance. Ena married Alfred in 1920 and they had three children: Vera, Madge and Ian. Ian died after only two days and Alfred died in 1937 during the Depression, leaving Ena a widow with two children. As Vera and Madge grew up Ena alienated them through her judgemental and frequently interfering nature – a point of much contention for Ena. Madge and her mother became almost completely estranged after Madge emigrated in 1950.

1960–1980
In 1960, Ena was in conflict with the Mission's new lay preacher Leonard Swindley (Arthur Lowe), who objected to her frequenting the Rovers. Ena collapsed due to the stress but walked out of hospital to return to her post so that Martha Longhurst (Lynne Carol), who had taken on her responsibilities while she recovered, could not steal her job. In 1961, Ena was sacked for spreading a rumour that Coronation Street was being demolished, which turned out to be untrue. Swindley was forced to rehire her when a suitable replacement could not be found (Ena had bribed the other candidates to turn the job down). Still unhappy with the working conditions, however, Ena walked out of the job later in the year and moved in with Minnie Caldwell (Margot Bryant), with Albert Tatlock (Jack Howarth) briefly taking on the caretaker position. She was eventually offered her job back.

Ena had a health scare in 1962 when she suffered a minor stroke, brought on by hypostatic pneumonia. She quickly regained her speech and mobility, but the following year was diagnosed with arteriosclerosis. Despite her willingness to gossip Ena was very guarded about her private life, and berated Martha for discussing her health problems with Ena's daughter, Vera Lomax (Ruth Holden).

Later that year Ena's great nephew, Tom Schofield (David Holliday), visited her and invited her to her brother's home in the US for an extended stay. Ena jumped at the chance to go abroad for the first time and meet the family she never knew she had. Upon her return, Ena was horrified to see the Mission converted into a Community Centre, with social worker Ruth Winter (Colette O'Neil) employed there full-time. She quit upon hearing the news and moved in with Minnie at No.5. Despite being out of work, Ena was soon occupied with other problems as Vera came to stay, having separated from her husband Bob Lomax. Vera had debts to pay but not the money to pay them, so Ena gave her the money – even though it was all her savings. A disoriented Ena was later caught accidentally stealing from a supermarket. In court Ena pleaded not guilty, but refused to give her age when questioned, saying only that she was over 21. She was subsequently fined 40 shillings. To offset her money problems Ena took on the job of live-in housekeeper at No.9 for Len Fairclough (Peter Adamson) although, when the Community Centre at the Mission closed down, Ena moved back into the vestry.

Vera came to stay again later in 1966, claiming to be ill. Ena did not believe her until she spoke to Vera's doctor, who said that Vera had a brain tumour and had just a month to live; however, Vera had not yet been told that her condition was terminal. Ena watched her daughter decline over several weeks until she died in Ena's bed in January 1967.

The Mission was closed permanently a year later, when it was demolished along with the factory to make way for a block of two-storey maisonettes. Ena was offered a place at an old people's nursing home, which she unsurprisingly declined, choosing to lodge with her old friend Henry Foster at St. Annes after briefly living with Minnie; nevertheless, when the maisonettes were built Ena moved into No.6, a purpose-built OAP ground floor flat. Ena was pleased, as it occupied the exact spot where the vestry had been.

In 1969, Ena became bored with the maisonette and moved into a flat above Ernest Bishop's (Stephen Hancock) camera shop. With Glad Tidings gone, Ena's closest place of worship was the Victoria Street Mission, and she kept close tabs on the comings and goings there. She was delighted in 1970 to meet young Tony Parsons (David Hill), who shared her passion for the harmonium. Recognising his talent, Ena made him her protégé and gave him lessons, while seeing about getting him a scholarship.

When the maisonettes were demolished in 1971, one of the buildings which replaced them was a community centre. Despite her age, Ena was determined to secure the position of caretaker, and scared off her competitor Hetty Thorpe (Margery Withers) by warning her about the violence in the area. With no-one else to take the job, Ena was selected for the job, and she moved into a flat adjacent to the centre. Ena's age and ability to do the work was a constant concern to the council. A co-caretaker was foisted on her (none other than Albert Tatlock), though she insisted on being called "senior caretaker". In 1973, Ena suffered two heart attacks, but refused to move away as she wanted to die in the street. When she disappeared with the centre's keys at Christmas, preventing the children from getting any presents, Ena assumed she would be sacked and left to stay with Henry Foster in St. Annes. However, she was allowed to keep her job.

Later in the 1970s, Ena alternated between Weatherfield and St. Anne's. Her friends gradually left her life – Martha had died in 1964 and Minnie left in 1976 to live at Whaley Bridge with their old friend Handel Gartside (Harry Markham). In 1977, Councillor Tattersall (George Waring) tried to sack Ena so that his niece could take her job, but Alf Roberts (Bryan Mosley) fought on Ena's behalf. The Lomaxes offered to house Ena in Hartlepool, but she declined. In 1980, Ena dumped herself on Elsie Tanner (Pat Phoenix) and later Albert, while the flat was being redecorated. Angered by the lack of progress that prevented her from moving back into her home, she left to stay in St. Annes, unsure about whether or not to return to Coronation Street.

Ena was never seen in Weatherfield again, and all subsequent references to her were in the past tense. In 1989, Deirdre Barlow (Anne Kirkbride) said that, due to recent events in Coronation Street, "Ena Sharples would be turning in her grave", confirming that she had died by this time.

Reception
In a book written about Women and Soap Opera, Ena was described as "brusque and uncompromising, refusing to adjust to the changing times". Ena has also been described as "a Coronation Street prototype, a strong, bossy woman". In an interview, Violet Carson described Ena as "the eternal busybody, the rock of ages; when things are going well, she is the old bag, when things are going wrong, she is the first one they turn to", in another interview she said that "children were not frightened of Mrs. Sharples", Violet Carson said "children will come and hold Mrs. Sharples' hand". Actor Michael Melia has claimed in an interview that he thought the character of Ena was "the most forbidding face on the street. A harridan in a hairnet with a surly expression that could stop a grown man in his tracks."

title: Ena Sharples


text:
Joseph Daul

Joseph Alexander Daul (born 13 April 1947) is a French politician who served as President of the European People's Party (EPP) from 2013 to 2019. He previously served as Leader of the EPP Group in the European Parliament from 2007 to 2014 and a Member of the European Parliament (MEP) from France from 1999 until 2014. He is a member of The Republicans, part of the EPP.

Daul had previously come to prominence as the chairman of the European Parliament's powerful Committee on Agriculture, a position he was elected to on 23 July 2004, and previously, from 2002, as chairman of the Conference of Committee Chairmen. He has been mayor of Pfettisheim (Bas-Rhin) since 1989, and is an officer of the Agricultural Order of Merit, knight of the National Order of Merit and knight of the Legion of Honour. In addition to French, he speaks standard German and Alsatian, the Alemannic variety of German spoken in his native Alsace.

Early life
Daul comes from a large family (six brothers and sisters) and at the age of 20 he took over a small family farm of 7 ha which he has since managed with his wife and, for the last few years, his son. While the operation has grown to 75 ha, it is still family run and specialises in beef production and the cultivation of sugar beet.

Career in national politics
Daul became active in the agricultural trade union movement, through the Centre National des Jeunes Agriculteurs (CNJA – National Centre for Young Farmers). He completed his professional training and general education and in 1976 he was appointed national Vice-Chairman of the CNJA with responsibility for European affairs. Daul also became a member of the Committee of Professional Agricultural Organisations (COPA) and of the European Economic and Social Committee in Brussels.

At the age of 35 he continued his trade union activities beyond the youth sector and combined his professional life as a farmer with positions of responsibility at a regional, (professional organisations, cooperatives, Chamber of Agriculture), national and European levels.

In 1997, when the 'mad cow' crisis arose, he was chair of the National Federation of Beef Producers in France and of the 'beef' group in Europe. These activities, together with his experience as mayor of his town (1000 inhabitants) led to a more active role in politics when, in 1999, he was asked to represent the farming and rural communities on the RPR list under Nicolas Sarkozy. Daul also became deputy mayor in his home town, vice-president of the Community of Councils and chairman of the Strasbourg Abattoir Management Cooperative (160 employees).

Member of the European Parliament
After having been elected in 1999, Daul joined the then EPP-ED Group, now EPP Group. Daul became a member of the Committee on Agriculture and Rural Development, appointed chairman in January 2002.

In 2002, Daul was appointed chairman of the Conference of Committee Chairmen, which is responsible for co-ordinating 'horizontal' issues affecting the 20 parliamentary committees and for preparing the draft agenda for the plenary part-sessions. As part of the reform of the European Parliament, the Conference put forward proposals for making its work more efficient. Daul also co-chaired, together with the committee chairmen concerned, the hearings of the Commissioners and transmitted the conclusions to the President of the European Parliament.

Finally, in 2005 and 2006, he and Socialist MEP Richard Corbett were Parliament's representatives for a new round of negotiations with the Council and Commission on comitology, giving Parliament, for the first time, the right to block the adoption of Commission implementing measures.

The 'Lisbon Strategy' Group headed by Daul explored questions of employment and in this capacity Daul co-chaired two meetings in 2005 and 2006 between the European Parliament and the national parliaments. As a member of the European Parliament Delegation for the WTO negotiations, Daul formed part of the European Parliament delegation to the ministerial conferences in Seattle (1999), Doha (2001) and Cancun (2003). He was involved in the preparatory work for the ministerial conferences by regularly meeting ambassadors in Geneva to represent the position of the European Parliament.

In January 2007, he was elected chairman of the EPP Group. He remains member of the Committee on Agriculture and Rural Development and substitute member of the Committee on International Trade and of the Delegation to the ACP-EU Joint Parliamentary Assembly. He is also chairman of the European Parliamentary Association (EPA).

On 12 November 2013, Daul was elected president of the European People's Party, succeeding the late Wilfried Martens, while also remaining chairman of the EPP Group in the European Parliament. However, he decided not to run again in the European elections in 2014 and to leave his position as Chairman of the EPP Group in the European Parliament to dedicate all his time to his new function. He was re-elected as President of the EPP at the Madrid Congress in October 2015. In 2019, he was succeeded by Donald Tusk.

Recognition
Among other honours, Daul is an "Officier de la Légion d'Honneur", "Chevalier de l'Ordre National du Mérite" and "Officier du Mérite Agricole".

Controversy
Daul was under investigation for misuse of public funds in France. He was part of an inquiry into the diversion of €16 million (£10.6 million) of agricultural money in the 1990s that has also ensnared three former agriculture ministers. He was placed under investigation in 2004 for allegedly diverting money to finance the farmers' union, the FNSEA, and to pay his own staff. He was not accused of benefiting personally but of "complicity and concealment of the abuse of public funds". On 8 August 2007, the court had, however, asked that all charges against him be dropped, stating that the cases made against him did not require legal punishment.

title: Joseph Daul