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Cartesian coordinate system

In geometry, a Cartesian coordinate system in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, called coordinate lines, coordinate axes or just axes (plural of axis) of the system. The point where they meet is called the origin and has $(0, 0)$ as coordinates.

Similarly, the position of any point in three-dimensional space can be specified by three Cartesian coordinates, which are the signed distances from the point to three mutually perpendicular planes. More generally, $n$ Cartesian coordinates specify the point in an $n$-dimensional Euclidean space for any dimension $n$. These coordinates are the signed distances from the point to $n$ mutually perpendicular fixed hyperplanes.

Cartesian coordinates are named for René Descartes, whose invention of them in the 17th century revolutionized mathematics by allowing the expression of problems of geometry in terms of algebra and calculus. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by equations involving the coordinates of points of the shape. For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates $x$ and $y$ satisfy the equation $x^{2} + y^{2} = 4$; the area, the perimeter and the tangent line at any point can be computed from this equation by using integrals and derivatives, in a way that can be applied to any curve.

Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing.

History
The adjective Cartesian refers to the French mathematician and philosopher René Descartes, who published this idea in 1637 while he was resident in the Netherlands. It was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. The French cleric Nicole Oresme used constructions similar to Cartesian coordinates well before the time of Descartes and Fermat.

Both Descartes and Fermat used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes's work.

The development of the Cartesian coordinate system would play a fundamental role in the development of the calculus by Isaac Newton and Gottfried Wilhelm Leibniz. The two-coordinate description of the plane was later generalized into the concept of vector spaces.

Many other coordinate systems have been developed since Descartes, such as the polar coordinates for the plane, and the spherical and cylindrical coordinates for three-dimensional space.

One dimension
An affine line with a chosen Cartesian coordinate system is called a number line. Every point on the line has a real-number coordinate, and every real number represents some point on the line.

There are two degrees of freedom in the choice of Cartesian coordinate system for a line, which can be specified by choosing two distinct points along the line and assigning them to two distinct real numbers (most commonly zero and one). Other points can then be uniquely assigned to numbers by linear interpolation. Equivalently, one point can be assigned to a specific real number, for instance an origin point corresponding to zero, and an oriented length along the line can be chosen as a unit, with the orientation indicating the correspondence between directions along the line and positive or negative numbers. Each point corresponds to its signed distance from the origin (a number with an absolute value equal to the distance and a $+$ or $−$ sign chosen based on direction).

A geometric transformation of the line can be represented by a function of a real variable, for example translation of the line corresponds to addition, and scaling the line corresponds to multiplication. Any two Cartesian coordinate systems on the line can be related to each-other by a linear function (function of the form $x \mapsto ax + b$) taking a specific point's coordinate in one system to its coordinate in the other system. Choosing a coordinate system for each of two different lines establishes an affine map from one line to the other taking each point on one line to the point on the other line with the same coordinate.

Two dimensions
A Cartesian coordinate system in two dimensions (also called a rectangular coordinate system or an orthogonal coordinate system ) is defined by an ordered pair of perpendicular lines (axes), a single unit of length for both axes, and an orientation for each axis. The point where the axes meet is taken as the origin for both, thus turning each axis into a number line. For any point P, a line is drawn through P perpendicular to each axis, and the position where it meets the axis is interpreted as a number. The two numbers, in that chosen order, are the Cartesian coordinates of P. The reverse construction allows one to determine the point P given its coordinates.

The first and second coordinates are called the abscissa and the ordinate of P, respectively; and the point where the axes meet is called the origin of the coordinate system. The coordinates are usually written as two numbers in parentheses, in that order, separated by a comma, as in (3, −10.5). Thus the origin has coordinates (0, 0), and the points on the positive half-axes, one unit away from the origin, have coordinates (1, 0) and (0, 1).

In mathematics, physics, and engineering, the first axis is usually defined or depicted as horizontal and oriented to the right, and the second axis is vertical and oriented upwards. (However, in some computer graphics contexts, the ordinate axis may be oriented downwards.) The origin is often labeled O, and the two coordinates are often denoted by the letters X and Y, or x and y. The axes may then be referred to as the X-axis and Y-axis. The choices of letters come from the original convention, which is to use the latter part of the alphabet to indicate unknown values. The first part of the alphabet was used to designate known values.

A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane. In a Cartesian plane, one can define canonical representatives of certain geometric figures, such as the unit circle (with radius equal to the length unit, and center at the origin), the unit square (whose diagonal has endpoints at (0, 0) and (1, 1)), the unit hyperbola, and so on.

The two axes divide the plane into four right angles, called quadrants. The quadrants may be named or numbered in various ways, but the quadrant where all coordinates are positive is usually called the first quadrant.

If the coordinates of a point are (x, y), then its distances from the X-axis and from the Y-axis are $|y|$ and $|x|$, respectively; where | denotes the absolute value of a number.

Three dimensions

A Cartesian coordinate system for a three-dimensional space consists of an ordered triplet of lines (the axes) that go through a common point (the origin), and are pair-wise perpendicular; an orientation for each axis; and a single unit of length for all three axes. As in the two-dimensional case, each axis becomes a number line. For any point P of space, one considers a hyperplane through P perpendicular to each coordinate axis, and interprets the point where that hyperplane cuts the axis as a number. The Cartesian coordinates of P are those three numbers, in the chosen order. The reverse construction determines the point P given its three coordinates.

Alternatively, each coordinate of a point P can be taken as the distance from P to the hyperplane defined by the other two axes, with the sign determined by the orientation of the corresponding axis.

Each pair of axes defines a coordinate hyperplane. These hyperplanes divide space into eight octants. The octants are:

$$ \begin{align} (+x,+y,+z) && (-x,+y,+z) && (+x,-y,+z) && (+x,+y,-z) \\ (+x,-y,-z) && (-x,+y,-z) && (-x,-y,+z) && (-x,-y,-z) \end{align} $$

The coordinates are usually written as three numbers (or algebraic formulas) surrounded by parentheses and separated by commas, as in $x = 2$ or $y = 3$. Thus, the origin has coordinates $z = 4$, and the unit points on the three axes are $(2, 3, 4)$, $(3, −2.5, 1)$, and $(t, u + v, π/2)$.

There are no standard names for the coordinates in the three axes (however, the terms abscissa, ordinate and applicate are sometimes used). The coordinates are often denoted by the letters X, Y, and Z, or x, y, and z. The axes may then be referred to as the X-axis, Y-axis, and Z-axis, respectively. Then the coordinate hyperplanes can be referred to as the XY-plane, YZ-plane, and XZ-plane.

In mathematics, physics, and engineering contexts, the first two axes are often defined or depicted as horizontal, with the third axis pointing up. In that case the third coordinate may be called height or altitude. The orientation is usually chosen so that the 90-degree angle from the first axis to the second axis looks counter-clockwise when seen from the point $(0, 0, 0)$; a convention that is commonly called the right-hand rule.

Higher dimensions
Since Cartesian coordinates are unique and non-ambiguous, the points of a Cartesian plane can be identified with pairs of real numbers; that is, with the Cartesian product $$\R^2 = \R\times\R$$, where $$\R$$ is the set of all real numbers. In the same way, the points in any Euclidean space of dimension n be identified with the tuples (lists) of n real numbers; that is, with the Cartesian product $$\R^n$$.

Generalizations
The concept of Cartesian coordinates generalizes to allow axes that are not perpendicular to each other, and/or different units along each axis. In that case, each coordinate is obtained by projecting the point onto one axis along a direction that is parallel to the other axis (or, in general, to the hyperplane defined by all the other axes). In such an oblique coordinate system the computations of distances and angles must be modified from that in standard Cartesian systems, and many standard formulas (such as the Pythagorean formula for the distance) do not hold (see affine plane).

Notations and conventions
The Cartesian coordinates of a point are usually written in parentheses and separated by commas, as in (10, 5) or (3, 5, 7). The origin is often labelled with the capital letter O. In analytic geometry, unknown or generic coordinates are often denoted by the letters (x, y) in the plane, and (x, y, z) in three-dimensional space. This custom comes from a convention of algebra, which uses letters near the end of the alphabet for unknown values (such as the coordinates of points in many geometric problems), and letters near the beginning for given quantities.

These conventional names are often used in other domains, such as physics and engineering, although other letters may be used. For example, in a graph showing how a pressure varies with time, the graph coordinates may be denoted p and t. Each axis is usually named after the coordinate which is measured along it; so one says the x-axis, the y-axis, the t-axis, etc.

Another common convention for coordinate naming is to use subscripts, as (x1, x2, ..., xn) for the n coordinates in an n-dimensional space, especially when n is greater than 3 or unspecified. Some authors prefer the numbering (x0, x1, ..., xn−1). These notations are especially advantageous in computer programming: by storing the coordinates of a point as an array, instead of a record, the subscript can serve to index the coordinates.

In mathematical illustrations of two-dimensional Cartesian systems, the first coordinate (traditionally called the abscissa) is measured along a horizontal axis, oriented from left to right. The second coordinate (the ordinate) is then measured along a vertical axis, usually oriented from bottom to top. Young children learning the Cartesian system, commonly learn the order to read the values before cementing the x-, y-, and z-axis concepts, by starting with 2D mnemonics (for example, 'Walk along the hall then up the stairs' akin to straight across the x-axis then up vertically along the y-axis).

Computer graphics and image processing, however, often use a coordinate system with the y-axis oriented downwards on the computer display. This convention developed in the 1960s (or earlier) from the way that images were originally stored in display buffers.

For three-dimensional systems, a convention is to portray the xy-plane horizontally, with the z-axis added to represent height (positive up). Furthermore, there is a convention to orient the x-axis toward the viewer, biased either to the right or left. If a diagram (3D projection or 2D perspective drawing) shows the x- and y-axis horizontally and vertically, respectively, then the z-axis should be shown pointing "out of the page" towards the viewer or camera. In such a 2D diagram of a 3D coordinate system, the z-axis would appear as a line or ray pointing down and to the left or down and to the right, depending on the presumed viewer or camera perspective. In any diagram or display, the orientation of the three axes, as a whole, is arbitrary. However, the orientation of the axes relative to each other should always comply with the right-hand rule, unless specifically stated otherwise. All laws of physics and math assume this right-handedness, which ensures consistency.

For 3D diagrams, the names "abscissa" and "ordinate" are rarely used for x and y, respectively. When they are, the z-coordinate is sometimes called the applicate. The words abscissa, ordinate and applicate are sometimes used to refer to coordinate axes rather than the coordinate values.

Quadrants and octants

The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called quadrants, each bounded by two half-axes. These are often numbered from 1st to 4th and denoted by Roman numerals: I (where the coordinates both have positive signs), II (where the abscissa is negative − and the ordinate is positive +), III (where both the abscissa and the ordinate are −), and IV (abscissa +, ordinate −). When the axes are drawn according to the mathematical custom, the numbering goes counter-clockwise starting from the upper right ("north-east") quadrant.

Similarly, a three-dimensional Cartesian system defines a division of space into eight regions or octants, according to the signs of the coordinates of the points. The convention used for naming a specific octant is to list its signs; for example, (+ + +) or (− + −). The generalization of the quadrant and octant to an arbitrary number of dimensions is the orthant, and a similar naming system applies.

Distance between two points
The Euclidean distance between two points of the plane with Cartesian coordinates $$(x_1, y_1)$$ and $$(x_2, y_2)$$ is

$$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}.$$

This is the Cartesian version of Pythagoras's theorem. In three-dimensional space, the distance between points $$(x_1,y_1,z_1)$$ and $$(x_2,y_2,z_2)$$ is

$$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2+ (z_2-z_1)^2} ,$$

which can be obtained by two consecutive applications of Pythagoras' theorem.

Euclidean transformations
The Euclidean transformations or Euclidean motions are the (bijective) mappings of points of the Euclidean plane to themselves which preserve distances between points. There are four types of these mappings (also called isometries): translations, rotations, reflections and glide reflections.

Translation
Translating a set of points of the plane, preserving the distances and directions between them, is equivalent to adding a fixed pair of numbers (a, b) to the Cartesian coordinates of every point in the set. That is, if the original coordinates of a point are (x, y), after the translation they will be

$$(x', y') = (x + a, y + b) .$$

Rotation
To rotate a figure counterclockwise around the origin by some angle $$\theta$$ is equivalent to replacing every point with coordinates (x,y) by the point with coordinates (x ' ,y ' ), where

$$ \begin{align} x' &= x \cos \theta - y \sin \theta \\ y' &= x \sin \theta + y \cos \theta. \end{align} $$

Thus:

$$(x',y') = ((x \cos \theta - y \sin \theta\,), (x \sin \theta + y \cos \theta\,)) .$$

Reflection
If (x, y) are the Cartesian coordinates of a point, then (−x, y) are the coordinates of its reflection across the second coordinate axis (the y-axis), as if that line were a mirror. Likewise, (x, −y) are the coordinates of its reflection across the first coordinate axis (the x-axis). In more generality, reflection across a line through the origin making an angle $$\theta$$ with the x-axis, is equivalent to replacing every point with coordinates (x, y) by the point with coordinates (x′,y′), where

$$ \begin{align} x' &= x \cos 2\theta + y \sin 2\theta \\ y' &= x \sin 2\theta - y \cos 2\theta. \end{align} $$

Thus: $$(x',y') = ((x \cos 2\theta + y \sin 2\theta\,), (x \sin 2\theta - y \cos 2\theta\,)) .$$

Glide reflection
A glide reflection is the composition of a reflection across a line followed by a translation in the direction of that line. It can be seen that the order of these operations does not matter (the translation can come first, followed by the reflection).

General matrix form of the transformations
All affine transformations of the plane can be described in a uniform way by using matrices. For this purpose, the coordinates $$(x,y)$$ of a point are commonly represented as the column matrix $$\begin{pmatrix}x\\y\end{pmatrix}.$$ The result $$(x', y')$$ of applying an affine transformation to a point $$(x,y)$$ is given by the formula $$\begin{pmatrix}x'\\y'\end{pmatrix} = A \begin{pmatrix}x\\y\end{pmatrix} + b,$$ where $$A = \begin{pmatrix} A_{1,1} & A_{1,2} \\ A_{2,1} & A_{2,2} \end{pmatrix}$$ is a 2×2 matrix and $$b=\begin{pmatrix}b_1\\b_2\end{pmatrix}$$ is a column matrix. That is, $$ \begin{align} x' &= x A_{1,1} + y A_{1,1} + b_{1} \\ y' &= x A_{2,1} + y A_{2, 2} + b_{2}. \end{align} $$

Among the affine transformations, the Euclidean transformations are characterized by the fact that the matrix $$A$$ is orthogonal; that is, its columns are orthogonal vectors of Euclidean norm one, or, explicitly, $$A_{1,1} A_{1, 2} + A_{2,1} A_{2, 2} = 0$$ and $$A_{1, 1}^2 + A_{2,1}^2 = A_{1,2}^2 + A_{2, 2}^2 = 1.$$

This is equivalent to saying that $(1, 0, 0)$ times its transpose is the identity matrix. If these conditions do not hold, the formula describes a more general affine transformation.

The transformation is a translation if and only if $(0, 1, 0)$ is the identity matrix. The transformation is a rotation around some point if and only if $(0, 0, 1)$ is a rotation matrix, meaning that it is orthogonal and $$ A_{1, 1} A_{2, 2} - A_{2, 1} A_{1, 2} = 1 .$$

A reflection or glide reflection is obtained when, $$ A_{1, 1} A_{2, 2} - A_{2, 1} A_{1, 2} = -1 .$$

Assuming that translations are not used (that is, $$b_1=b_2=0$$) transformations can be composed by simply multiplying the associated transformation matrices. In the general case, it is useful to use the augmented matrix of the transformation; that is, to rewrite the transformation formula $$\begin{pmatrix}x'\\y'\\1\end{pmatrix} = A' \begin{pmatrix}x\\y\\1\end{pmatrix},$$ where $$A' = \begin{pmatrix} A_{1,1} & A_{1,2}&b_1 \\ A_{2,1} & A_{2,2}&b_2\\0&0&1 \end{pmatrix}.$$ With this trick, the composition of affine transformations is obtained by multiplying the augmented matrices.

Affine transformation

Affine transformations of the Euclidean plane are transformations that map lines to lines, but may change distances and angles. As said in the preceding section, they can be represented with augmented matrices: $$\begin{pmatrix} A_{1,1} & A_{2,1} & b_{1} \\ A_{1,2} & A_{2,2} & b_{2} \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ 1 \end{pmatrix} = \begin{pmatrix} x' \\ y' \\ 1 \end{pmatrix}.$$

The Euclidean transformations are the affine transformations such that the 2×2 matrix of the $$A_{i,j}$$ is orthogonal.

The augmented matrix that represents the composition of two affine transformations is obtained by multiplying their augmented matrices.

Some affine transformations that are not Euclidean transformations have received specific names.

Scaling
An example of an affine transformation which is not Euclidean is given by scaling. To make a figure larger or smaller is equivalent to multiplying the Cartesian coordinates of every point by the same positive number m. If (x, y) are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates

$$(x',y') = (m x, m y).$$

If m is greater than 1, the figure becomes larger; if m is between 0 and 1, it becomes smaller.

Shearing
A shearing transformation will push the top of a square sideways to form a parallelogram. Horizontal shearing is defined by:

$$(x',y') = (x+y s, y)$$

Shearing can also be applied vertically:

$$(x',y') = (x, x s+y)$$

In two dimensions

Fixing or choosing the x-axis determines the y-axis up to direction. Namely, the y-axis is necessarily the perpendicular to the x-axis through the point marked 0 on the x-axis. But there is a choice of which of the two half lines on the perpendicular to designate as positive and which as negative. Each of these two choices determines a different orientation (also called handedness) of the Cartesian plane.

The usual way of orienting the plane, with the positive x-axis pointing right and the positive y-axis pointing up (and the x-axis being the "first" and the y-axis the "second" axis), is considered the positive or standard orientation, also called the right-handed orientation.

A commonly used mnemonic for defining the positive orientation is the right-hand rule. Placing a somewhat closed right hand on the plane with the thumb pointing up, the fingers point from the x-axis to the y-axis, in a positively oriented coordinate system.

The other way of orienting the plane is following the left-hand rule, placing the left hand on the plane with the thumb pointing up.

When pointing the thumb away from the origin along an axis towards positive, the curvature of the fingers indicates a positive rotation along that axis.

Regardless of the rule used to orient the plane, rotating the coordinate system will preserve the orientation. Switching any one axis will reverse the orientation, but switching both will leave the orientation unchanged.

In three dimensions

Once the x- and y-axes are specified, they determine the line along which the z-axis should lie, but there are two possible orientations for this line. The two possible coordinate systems, which result are called 'right-handed' and 'left-handed'. The standard orientation, where the xy-plane is horizontal and the z-axis points up (and the x- and the y-axis form a positively oriented two-dimensional coordinate system in the xy-plane if observed from above the xy-plane) is called right-handed or positive.

The name derives from the right-hand rule. If the index finger of the right hand is pointed forward, the middle finger bent inward at a right angle to it, and the thumb placed at a right angle to both, the three fingers indicate the relative orientation of the x-, y-, and z-axes in a right-handed system. The thumb indicates the x-axis, the index finger the y-axis and the middle finger the z-axis. Conversely, if the same is done with the left hand, a left-handed system results.

Figure 7 depicts a left and a right-handed coordinate system. Because a three-dimensional object is represented on the two-dimensional screen, distortion and ambiguity result. The axis pointing downward (and to the right) is also meant to point towards the observer, whereas the "middle"-axis is meant to point away from the observer. The red circle is parallel to the horizontal xy-plane and indicates rotation from the x-axis to the y-axis (in both cases). Hence the red arrow passes in front of the z-axis.

Figure 8 is another attempt at depicting a right-handed coordinate system. Again, there is an ambiguity caused by projecting the three-dimensional coordinate system into the plane. Many observers see Figure 8 as "flipping in and out" between a convex cube and a concave "corner". This corresponds to the two possible orientations of the space. Seeing the figure as convex gives a left-handed coordinate system. Thus the "correct" way to view Figure 8 is to imagine the x-axis as pointing towards the observer and thus seeing a concave corner.

Representing a vector in the standard basis
A point in space in a Cartesian coordinate system may also be represented by a position vector, which can be thought of as an arrow pointing from the origin of the coordinate system to the point. If the coordinates represent spatial positions (displacements), it is common to represent the vector from the origin to the point of interest as $$\mathbf{r}$$. In two dimensions, the vector from the origin to the point with Cartesian coordinates (x, y) can be written as:

$$ \mathbf{r} = x \mathbf{i} + y \mathbf{j},$$

where $$\mathbf{i} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$$ and $$\mathbf{j} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$$ are unit vectors in the direction of the x-axis and y-axis respectively, generally referred to as the standard basis (in some application areas these may also be referred to as versors). Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates $$(x,y,z)$$ can be written as: $$ \mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k},$$

where $$\mathbf{i} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix},$$ $$\mathbf{j} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix},$$ and $$\mathbf{k} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}.$$

There is no natural interpretation of multiplying vectors to obtain another vector that works in all dimensions, however there is a way to use complex numbers to provide such a multiplication. In a two-dimensional cartesian plane, identify the point with coordinates (x, y) with the complex number z = x + iy. Here, i is the imaginary unit and is identified with the point with coordinates (0, 1), so it is not the unit vector in the direction of the x-axis. Since the complex numbers can be multiplied giving another complex number, this identification provides a means to "multiply" vectors. In a three-dimensional cartesian space a similar identification can be made with a subset of the quaternions.

title: Cartesian coordinate system


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Commandant of the United States Marine Corps

The commandant of the Marine Corps (CMC) is normally the highest-ranking officer in the United States Marine Corps. It is a four-star general position and a member of the Joint Chiefs of Staff. The CMC reports directly to the secretary of the Navy and is responsible for ensuring the organization, policy, plans, and programs for the Marine Corps as well as advising the president, the secretary of defense, the National Security Council, the Homeland Security Council, and the secretary of the Navy on matters involving the Marine Corps. Under the authority of the secretary of the Navy, the CMC designates Marine personnel and resources to the commanders of unified combatant commands. The commandant performs all other functions prescribed in Section 8043 in Title 10 of the United States Code or delegates those duties and responsibilities to other officers in his administration in his name. As with the other joint chiefs, the commandant is an administrative position and has no operational command authority over United States Marine Corps forces.

The commandant is nominated for appointment by the president, for a four-year term of office, and must be confirmed by the Senate. The commandant can be reappointed to serve one additional term, but only during times of war or national emergency declared by Congress. By statute, the commandant is appointed as a four-star general while serving in office. "The commandant is directly responsible to the Secretary of the Navy for the total performance of the Marine Corps. This includes the administration, discipline, internal organization, training, requirements, efficiency, and readiness of the service. The Commandant is also responsible for the operation of the Marine Corps material support system." Since 1806, the official residence of the commandant has been located in the Marine Barracks in Washington, D.C., and his main offices are in Arlington County, Virginia.

The 39th and current commandant is General Eric M. Smith.

Responsibilities
The responsibilities of the commandant are outlined in Title 10, Section 5043, the United States Code and the position is "subject to the authority, direction, and control of the Secretary of the Navy". As stated in the U.S. Code, the commandant "shall preside over the Headquarters, Marine Corps, transmit the plans and recommendations of the Headquarters, Marine Corps, to the Secretary and advise the Secretary with regard to such plans and recommendations, after approval of the plans or recommendations of the Headquarters, Marine Corps, by the Secretary, act as the agent of the Secretary in carrying them into effect, exercise supervision, consistent with the authority assigned to commanders of unified or specified combatant commands under chapter 6 of this title, over such of the members and organizations of the Marine Corps and the Navy as the Secretary determines, perform the duties prescribed for him by section 171 of this title and other provisions of law and perform such other military duties, not otherwise assigned by law, as are assigned to him by the President, the Secretary of Defense, or the Secretary of the Navy".

List of commandants
39 men have served as the commandant of the Marine Corps. The first commandant was Samuel Nicholas, who took office as a captain, though there was no office titled "Commandant" at the time, and the Second Continental Congress had authorized that the senior-most Marine could take a rank up to Colonel. The longest-serving was Archibald Henderson, sometimes referred to as the "Grand old man of the Marine Corps" due to his thirty-nine-year tenure. In the history of the United States Marine Corps, only one Commandant has ever been fired from the job: Anthony Gale, as a result of a court-martial in 1820.

Eric M. Smith (born c. 1964) During his hospitalization after suffering a heart attack, Lieutenant General Karsten Heckl (30 October – 3 November 2023) and General Christopher J. Mahoney (3 November 2023 – 5 March 2024) served as acting commandants.
 * style=background:#e6e6aa; align="center" | -
 * rowspan="2" | Gen Smith Official Photo V1.jpg
 * data-sort-value="Smith, Eric" style="text-align:center" rowspan="2" | General
 * data-sort-value="Smith, Eric" style="text-align:center" rowspan="2" | General
 * style=background:#e6e6aa; align="center" | 10 July 2023
 * style=background:#e6e6aa; align="center" | 22 September 2023
 * style=background:#e6e6aa; align="center" |
 * rowspan="2" style="text-align:center" | Served as acting commandant after Berger's retirement amid Senator Tommy Tuberville's hold on military nominations, and sworn in following confirmation.
 * style="text-align:center | 39
 * style="text-align:center | 22 September 2023
 * style="text-align:center | Incumbent
 * style="text-align:center |
 * }
 * }

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


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Talk:Cartesian coordinate system

Cartesian coordinate system and analytic geometry
To let you know, coordinate systems in general, and Cartesian coordinate systems in particular, are topics in Analytic Geometry. Just like any other mathematical topic, they are used in other branches of mathematics and have applications in many other fields of science. You have stated that “Cartesian coordinates are used in almost all parts of geometry and its applications”. However, this does not mean that “Cartesian coordinates” do not belong to a more specific branch of mathematics. Belonging to a specific branch of mathematics does not restrict the application of a mathematical topic in other fields. Coordinate systems are, in fact, a topic in Analytic Geometry, as it combines geometry with algebra. By the way, coordinate systems are NOT used in plane geometry. That’s another reason why coordinate systems are not generally a topic in “geometry” but are actually a topic in “analytic geometry”. Persianwise (talk) 10:05, 28 August 2023 (UTC)

 * Thank for this course on a topic that I know well since many years. I know that analytic geometry is a part of geometry, although, nowadays, it is more the name of courses in colleges, than that of a specific area of geometry; this is not the problem here. The problem is that the first phrase "In geometry" is not aimed to categorize the subject (this is the object of the categories appearing at the end of the article). It is aimed to tell to readers whether they could be interested to the article. Here, everybody knows geometry, but  people who know analytic geometry, know generally what are Cartesian coordinates, and thus your first phrase is not useful for them. On the other hand, many people encounter Cartesian coordinates in their studies or their work, without knowing that their use is called analytic geometry. This is the case, for example, of students in physics. This is for such people that the first phrase is intended. D.Lazard (talk) 10:33, 28 August 2023 (UTC)
 * You have stated: "... nowadays, it [analytic geometry] is more the name of courses in colleges, than that of a specific area of geometry."
 * However, not just analytic geometry is the name of a course. Algebra, calculus, topology, complex analysis, and almost every other mathematical subject are also the names of courses, and this has always been the case.
 * Your other statements are subjective, chaotic, and do not aim to make the article accurate.
 * For your information, although analytic geometry combines geometry and algebra, it is neither a branch of geometry nor algebra. If you were correct in saying that analytic geometry is a branch of geometry, then you could also say it is a branch of algebra too, which is not the case.
 * Furthermore, Wikipedia articles are written for all audiences, not just for physics students, and should be as accurate as possible for all readers.
 * That's why "geometry" should be changed to "analytic geometry" in order to start the article with correct phrasing. Persianwise (talk) 13:58, 28 August 2023 (UTC)

End of the moved discussion. D.Lazard (talk) 11:25, 31 August 2023 (UTC)
 * By "[analytic geometry] is more the name of courses in colleges, than that of a specific area of geometry", I meant that, nowadays, the phrase "analytic geometry" is rarely used outside college studies. In fact, it has been proved (See Cantor–Dedekind axiom) that the synthetic geometry (the geometry defined by geometric axioms) is strictly equivalent to analytic geometry, in the sense that every theorem of synthetic geometry can be proved by means of analytic geometry, and vice-versa.
 * It is not accurate to say that analytic geometry "combines" geometry and algebra. The correct statement is that analytic geometry is the use of algebra in geometry (or, more precisely, the use of real numbers and their properties). So analytic geometry is definitively a part of geometry.
 * About the first phrase: "In analytic geometry" is not wrong, but it may be confusing to many readers, who are not supposed to know what is analytic geometry: for knowing analytic geometry, one must know Cartesion coordinates, and this article if for people who do not know what are Cartesian coordinates (otherwise, they do not need this article). Also, people who have never heard of analytic geometry may have encountered Cartesian coordinates in other contexts, and they may understand the phrase "In analytic geometry" as an indication that the article is not for them. D.Lazard (talk) 17:24, 1 September 2023 (UTC)

Axes and number lines
I reverted this edit of, with the edit summary. In turn, he reverted my revert with the edit summary, and posted on my talk.

After that, I tried to get a compromise, but I was reverted again. Thinking about that, I got convinced that the use of (coordinate) axis for a number line is very uncommon. So, I'll revert the section to the old version where the word  "axis" did not appear.

By the way, I'll revert the other edits of the same user for reasons given in the preceding thread and in my edit summaies. D.Lazard (talk) 14:29, 2 September 2023 (UTC)

 * In the same paragraph, it is stated: ‘A line with a chosen Cartesian system is called a number line.’ It should be clarified that any coordinate system, including Cartesian coordinate systems, have axes. A number line is essentially a one-dimensional coordinate system. If the number line isn’t considered an axis, then where would the axis be located? If you answer this question correctly, you would understand that number lines and axes are two different names referring to the same concept.
 * DO not make any unwarranted revertions.
 * Persianwise (talk) Persianwise (talk) 23:20, 17 September 2023 (UTC)
 * You have not provided any reliable source that uses "axis" to refer to a numbeer line. So, your edit is a personal opinion. This is called WP:original research in Wikipedia, and not allowed in articles per policy WP:NOR. D.Lazard (talk) 11:20, 18 September 2023 (UTC)
 * Every axis is a number line but not every number line is an axis; the two are not synonyms, and your edit incorrectly proposed that they are.
 * Almost every part of D.Lazard's most recent edit was an improvement. In my opinion, there are two exceptions: (1) the cn tag needs double curly braces instead of single curly-braces, and (2) I think the spelling out of the meaning of bijection (that each point corresponds to a real number and each real number corresponds to a point) was a less technical way of conveying the point, so is likely worth preserving. 100.36.106.199 (talk) 13:22, 18 September 2023 (UTC)

It is "Analytic Geometry" that Introduced "Coordinate Systems".
Analytic Geometry is the branch of mathematics that introduced the concept of Coordinate Systems. Euclidean or non-Euclidean Geometries, by themselves, do NOT use Coordinate Systems. Therefore, it would be more accurate to change ‘geometry’ to ‘analytic geometry’, as this is what analytic geometry entails.

's comment:

"analytic geometry is much too restrictive, as it may prevent people interested in computational geometry, physics, mechanics, astronomy, etc. to read this article (all know geometry, but many may not know analytic geometry."

is subjective and lacks meaningful content. It is, in fact, a fallacy.

He has been warned before on his talk page, and he is hereby warned again to cease any attempts to make unwarranted reversions and to stop introducing inaccuracies due to insufficient mathematical knowledge. Persianwise (talk) 22:47, 17 September 2023 (UTC)

 * It is not analytic geometry that introduced coordinates systems, but it is René Descartes, a long time before the term analytic geometry was coined. Moreover, it is not geometries but geometers that uses coordinates systems. This is documented in the article Geometry, where coordinates systems and analytic geometry are clearly mentioned as parts of geometry.
 * Please, stop using accusations ("fallacy" and "insufficient mathematical knowledge") for trying to support your personal views. Use WP:reliable sources, if you can. D.Lazard (talk) 11:11, 18 September 2023 (UTC)
 * You are again using a fallacy by playing with words.
 * When we say that a branch or field of science introduced (or uses) some concept, it implicitly states that someone (a mathematician, scientist, etc.) invented that thing by introducing (or using) that concept.
 * For example, the invention of analytic geometry is generally credited to René Descartes, who combined algebra and geometry by using the concept of coordinate systems. (Geometry, by itself, was known for many thousands of years before the invention of analytic geometry.)
 * Analytic geometry cannot exist without a coordinate system, which paved the way for using algebra alongside geometry.
 * Therefore, it is more precise to say "In analytic geometry, a Cartesian coordinate system ..." than "In geometry, a Cartesian coordinate system ...." This is because coordinate systems are used analytic geometry, but they are not used in all types of geometry.
 * Unlike what you said previously, precision is not restriction. In fact, precision is often necessary for clear and accurate description. Persianwise (talk) 00:52, 26 September 2023 (UTC)

Protected
I have fully protected the article for a short period so the edit warring can be resolved on this talk page. Please review WP:DR and seek assistance at a relevant wikiproject. If necessary, and per WP:DR, use an WP:RFC if necessary. I will watch this page for a while. Feel free to ping me or any administrator if the article is ready to be unprotected due to consensus here (remembering that WP:CONSENSUS is not unanimity). Johnuniq (talk) 09:43, 18 September 2023 (UTC)

 * It should be mentioned that the page Cartesian coordinate system is about a subject of mathematics. To edit a mathematical subject, like any other scientific field, one must have the proper knowledge. Without the proper knowledge, any attempt to edit or make changes leads to inaccuracies that would confuse and misinform the readers. Notice that the user D.Lazard, due to lack of sufficient mathematical knowledge, is haphazardly changing sentences and publishing:
 * Cartesian coordinates are named for René Descartes, whose invention of them in the 17th century revolutionized mathematics by providing the first systematic link between geometry and algebra.
 * The correct and precise statement should be:
 * Cartesian coordinates are named for René Descartes, whose invention of them in the 17th century revolutionized mathematics by combining algebra and geometry.
 * The explanation for this was presented on his talk page: User_talk:D.Lazard.
 * But then he replaced it with:
 * Cartesian coordinates are named for René Descartes, whose invention of them in the 17th century revolutionized mathematics by allowing the expression of every problem of geometry in terms of algebra and calculus.
 *  This is wrong because calculus was not even invented at that time .
 * (Analytic gemetry invented by René Descartes in 1637], but Calculaus was invented in 1665 by Isaac Newton.
 * The user is evidently not just initiating the edit warring but also is actively engaged in sabotaging the article and preventing further improvement to the article. He should be blocked from editing mathematical related pages.
 * More information regarding his other unwarranted reversions will be given. Persianwise (talk) 17:03, 18 September 2023 (UTC)
 * The talk page of an article is not the correct place to discuss other editors. The correct page is WP:ANI but please examine WP:BOOMERANG before thinking about that. It is very common for disagreements to occur at Wikipedia and participants must follow standard procedures to avoid sanctions. Handling disputes is covered at WP:DR. Do not mention other editors or their alleged shortcomings on this page. If you want independent opinions on the accuracy of what I have written, ask at WP:Teahouse. The point is that editors must focus on content not on contributors. Have you asked for opinions at the wikiproject (see headings at top of this page)? I have not done more than glance at the issue and will take no part in discussing what should be done. However, you might bear in mind that articles generally focus on current practice, that is, what do people mean when they talk about the Cartesian coordinate system now? Please do not reply to this: I just raised an issue for you to consider. I suggest starting a new section with a brief outline of what problem you see in the current article and what should be done to fix it, with a brief explanation. Then try to get other opinions per WP:DR. Johnuniq (talk) 04:41, 19 September 2023 (UTC)
 * The talk page of an article is not the correct place to discuss other editors. The correct page is WP:ANI but please examine WP:BOOMERANG before thinking about that. It is very common for disagreements to occur at Wikipedia and participants must follow standard procedures to avoid sanctions. Handling disputes is covered at WP:DR. Do not mention other editors or their alleged shortcomings on this page. If you want independent opinions on the accuracy of what I have written, ask at WP:Teahouse. The point is that editors must focus on content not on contributors. Have you asked for opinions at the wikiproject (see headings at top of this page)? I have not done more than glance at the issue and will take no part in discussing what should be done. However, you might bear in mind that articles generally focus on current practice, that is, what do people mean when they talk about the Cartesian coordinate system now? Please do not reply to this: I just raised an issue for you to consider. I suggest starting a new section with a brief outline of what problem you see in the current article and what should be done to fix it, with a brief explanation. Then try to get other opinions per WP:DR. Johnuniq (talk) 04:41, 19 September 2023 (UTC)

 * "Allowing the expression of every problem of geometry in terms of algebra and calculus" may perhaps not be ideal wording, but the essential point is not invalidated by the fact that calculus had not yet been invented, any more than "the invention of the wheel has had revolutionary effects on transport, such as enabling the manufacture of motor cars" is invalidated by the fact that the motor car was not invented until long after the invention of the wheel. JBW (talk) 20:33, 20 September 2023 (UTC)
 * I find both first and second versions acceptable, but the third... It's too strong to assume that "every" problem of geometry is reduced to algebra and calculus. 慈居 (talk) 08:49, 21 September 2023 (UTC)
 * Well, in the context of Cartesian coordinate "geometry" clearly means Euclidean geometry, and Euclidean space is isomorphic to linear algebra in a vector space over the real numbers, and therefore every proposition about geometry in Euclidean space can indeed be expressed as an equivalent proposition in real algebra and calculus That is what was said in the quoted passage, which refers to "the expression of every problem of geometry in terms of algebra and calculus" (my emphasis) which is certainly possible. That is true whatever you may have in mind when you refer to "reducing" every problem of geometry to algebra and calculus, and whether or not such "reduction" in your intended sense is possible. JBW (talk) 17:07, 21 September 2023 (UTC)
 * I think you made a fair point. Instead of a boring argument, I'd like to mention the Jordan curve theorem, which says that every closed curve partitions the whole plane into two parts. This looks trivial (despite its lengthy complicated proof), but it yields nontrival conclusions when it's applied to some crazy curves like the Hilbert curve. This is how the word "every" is powerful. Even for classical Euclidean geometry, there was a famous (meta-)problem asking whether there is a hyperbolic geometry, and certainly this is not something expressable only with algebra and calculus (at least with the method of coordinate geometry). So in my opinion, both three sentences make sense (as you have pointed out for the third one), but I'd pick the first two of them as the better choices. 慈居 (talk) 20:11, 21 September 2023 (UTC)
 * OK, prompted by your comments,, I have thought some more about this. What do we mean by algebra? My post above in effect treats "algebra" as including anything which can be expressed in terms of an algebraic structure (in this case a finite dimensional real linear algebra) but that is a dubious interpretation; for example the metric and topological properties of such a space are not usually regarded as "algebraic". One could argue about exactly what features are added to the situation by the words "and calculus", but again topological properties are not, I think, generally regarded as being part of calculus. Above, I took issue with your statement on the basis that the quote attributed above to (but I haven't checked that attribution) refers not to reducing every problem in geometry to algebra and calculus, but to expressing it as such, but on reflection I think I was wrong: that was based on an interpretation of "algebra" as "any property of an algebraic space", but that is an unhelpfully broad interpretation of the word. Indeed, your example is highly pertinent: it would be difficult to express the Jordan curve theorem in terms which could reasonably be regarded as belonging to algebra and calculus. In the context in which this appears in the article, it is probably reasonably true that the kinds of geometric properties considered at the time when Cartesian coordinates were introduced can be expressed in terms of algebra and calculus, but since the statement in the form in which it is given is not justifiable, on reflection I think probably the best thing to do is to remove the word "every". I therefore intend to remove that word once I have posted this message, but of course I will be willing and interested to read any arguments in favour of its retention. (Well, any arguments by anyone who comes anywhere near to knowing what they are talking about.) JBW (talk) 21:15, 21 September 2023 (UTC)
 * About "calculus": I used "calculus" instead of analysis, because it is known by a wider audience. However "analysis" would be a proper term since, AFAIK, it denoted initially Descartes's method of coordinates and led to the term Analytic geometry, before becoming an alternative name of calculus.
 * About Jordan curve theorem: This is not a good example, as it involves the concept of "continuous curve" which cannot be defined in terms of pure Euclidean geometry. I do not imagine a way to define continuous curves without using coordinates, at least implicitly.
 * Nevertheless, I agree with the removal of "every", which would require either a source or an accurate definition of "algebra and calculus". D.Lazard (talk) 09:54, 22 September 2023 (UTC)
 * I also agree with the removal of "every". By the way @D.Lazard, the Jordan curve theorem was not at all intended to be a good example, or even an example. It was to illustrate how an assumption can be unnecessarily strong with the word "every". 慈居 (talk) 10:36, 22 September 2023 (UTC)

Cartesian coordinate system
You are starting an WP:edit war for trying to impose the idea that "number line" is a synonymous of "axis". It is true that the axes of a coordinate system are number lines, but this does not mean that the words are synonyms. Otherwise, this would be said in number line. In any case, Wikipedia rule are that any new assertion like yours must be supported by a reliable source (in this case of elementary mathematics, a widely used textbook). Otherwise, it is forbidden to add the assertion, per the policy WP:OR.

Please, remember also that edit warring may lead to an edit block (see WP:Edit warring). D.Lazard (talk) 18:00, 30 August 2023 (UTC)

 * In mathematics, a number line and an axis are equivalent concepts. They are often used interchangeably and can be regarded as mathematically synonymous, but they are not considered synonyms in the dictionary.
 * You stated that
 * “It is true that the axes of a coordinate system are number lines.”
 * However, on your own talk page, you stated that
 * "A number line and an axis are two different things."
 * This is contradictory to mathematical facts. Your actions evidently constitute Edit warring, as you have repeatedly reverted my correct assertion that a number line and an axis are the same thing. If you continue to engage in this behavior, I will report you as a Disruptive user. Persianwise (talk) 21:42, 30 August 2023 (UTC)
 * I don't see the logic of what you say. A cat is an animal, but "cat" and "animal" are not the same thing. There is no contradiction there; nor is there any contradiction in what you quote as saying. JBW (talk) 15:52, 20 September 2023 (UTC)
 * Your use of analogy is not valid here. The fact that "not all animals are cats" is because the set of all cats is a proper subset of the set of all animals.
 * However, my assertion is that the set of all number-lines and the set of all one-dimensional coordinate systems are identical.
 * This mathematical fact is even visually evident if you draw a number-line and a coordinate system in one dimension, both on the same geometric straight line, on a piece of paper. You will notice that a number-line is exactly a one-dimensional coordinate system because the axis in the one-dimensional coordinate system coincides with the number-line.
 * So, there is a bijection between the set of all points on the number-line and the set of all points on the axis; hence, they are the same.
 * Otherwise, where do you think the axis and the number-line are located relative to the aforementioned geometric straight line ?  Persianwise (talk) 18:19, 20 September 2023 (UTC)
 * That is completely missing the point. You said that there was a contradiction between the statement that axes of a coordinate system are number lines and the statement that a number line and an axis are two different things. Whether it is or is not true that a number line and an axis are both the same thing does not have any bearing on whether or not the statement (whether true or false) that they aren't is inconsistent with the statement (whether true or false) that the axes of a coordinate system are number lines. And that is quite apart from the fact that there being a bijection between the elements of two sets is not at all the same thing as their being the same thing. If you really can't see those two points then you should probably not be trying to improve the logic of the treatment of mathematical topics in articles. JBW (talk) 19:15, 20 September 2023 (UTC)
 * For clarification, it should be noted that the set of real numbers is a Field. It is obvious that the structure of a number-line defined by the same operations of addition and multiplication makes the two isomorphic - See Isomorphism. To mathematicians, two isomorphic fields (or any other relevant structures) are the same thing.
 * Analogously, the same thing can be said about the axis (more precisely, the set of points on the axis, with the familiar operations).
 * So all three are isomorphic, there are no differences between them, except the naming. They are all the same thing! Persianwise (talk) 20:48, 20 September 2023 (UTC)
 * NOTE: By “This is contradictory”, I meant “This [the second statement] is contradictory [to the established mathematical facts] (and not to the first statement). Persianwise (talk) 21:07, 20 September 2023 (UTC)
 * Oh, Christ. if you are going to say things and then say you meant something completely different than what any reasonable person would have thought you meant, then I may as well give up even trying. JBW (talk) 21:18, 20 September 2023 (UTC)
 * “If you noticed, I had stated ‘a number line and an axis are the same thing’, but the other user had said ‘A number line and an axis are two different things.’ which was contradictory to my assertion, and to mathematical facts. When ambiguity arises, why not explain things further to make things precise to get rid of ambiguity? Isn’t that the right thing to do?” Persianwise (talk) 22:01, 20 September 2023 (UTC)
 * You are saying that the x-axis and the y-axis of a Cartesian coordinate system are the "same thing" since they are both isomorphic to the field of the real numbers, and thus they are isomorphic. However, these two axes are different, and thus they cannot be the same thing. D.Lazard (talk) 22:08, 20 September 2023 (UTC)
 * The main meaning of isomorphism is the assertion that two seemingly different things are actually the same thing, except possibly for the names of their elements and the way the operations defined on them work. This is an advanced topic, hard to grasp for those who lack in-depth mathematical knowledge. I suggest you read the topic on this page that I posted at 18:19, 20 September 2023 (UTC). Anyone can see how to visualize why a number line and an axis (in a coordinate system) coincide, both representing the real number system. If you cannot see it, it is then crucial to answer the question that was asked there:
 *  “Otherwise, where do you think the axis and the number line are located relative to the aforementioned geometric straight line?”  Persianwise (talk) 22:46, 20 September 2023 (UTC)
 * How kind of you to help Daniel Lazard by explaining to him ideas which you understand but which are beyond him because of his being one of "those who lack in-depth mathematical knowledge". What a pity that nobody thought to teach him some elementary mathematics before he started his career. Hmm. Or... JBW (talk) 13:36, 21 September 2023 (UTC)
 * Your response is not pertinent to a mathematical discussion. He was asked this question twice:
 * "Otherwise, where do you think the axis and the number line are located relative to the aforementioned geometric straight line?”
 * and you know he failed to answer it.
 * Can you supply an answer to the aforementioned question yourself?
 * One does not even have to be a mathematician to answer it.
 * What is your answer? Persianwise (talk) 20:46, 24 September 2023 (UTC)
 * Just in passing, there would be no edit war if either of you just stop. No point in blaming each other. It might help if one restore an old version in which there was no edit dispute and seek consensus through discussion. (That worked in a dispute between me and still, D.Lazard.)
 * The difference of "number line" and "axis" is similar to that of a set and a subset. Any subset is a set and a set is always a subset (e.g. of itself). But are they the same? But anyway, you two should clarify why this matters. 慈居 (talk) 20:50, 25 September 2023 (UTC)

End of the moved discussion

It does not need to be restated that a number line can be considered its own axis. This is (a) unnecessary and redundant, and (b) a confusing distraction. (It is also not a claim supported by the provided sources, both of which just said that a number line is a line with numbers on it, and that each coordinate axis of a Cartesian plane is a number line, which is not at all the same thing as saying that "axis" and "number line" are synonyms.) –jacobolus (t) 16:49, 8 January 2024 (UTC)

 * (comment moved here from user talk:jacobolus)
 * You stated that "neither of these sources supports this claim". You are wrong! Please read both of the cited references.
 * You also stated that bold text are distractions. They are not. Bold text is for emphasis.
 * Additionally, you stated that "axis is defined to mean the principal lines through the origin". It does not mathematically make sense as "principal lines" are not defined in mathematics.
 * Persianwise (talk) 17:25, 8 January 2024 (UTC)

 * To elaborate a bit, these sources (an "Anonymous LibreText" and "Lesson 28" by Curriculum Associates, LLC, neither of which has obvious authorship or really seems like a "reliable source" by Wikipedia standards) claim that "The rectangular coordinate system consists of two real number lines that intersect at a right angle. The horizontal number line is called the x-axis, and the vertical number line is called the y-axis." and "When a horizontal number line and a vertical number line are lined up so that the 0s meet, a coordinate plane is formed. [...] The horizontal number line is called the x-axis. The vertical number line is called the y-axis. The point where the x- and y-axes meet is the origin.", respectively. These sources are saying that, in the context of the Cartesian plane, each axis is a number line.
 * Neither of these sources makes any claim that a one-dimensional line is, per se, an "axis", or that the term axis is a synonym of number line. It would be logically supportable based on an n-dimensional definition, albeit trivial, to call a number line its own axis. But this is unnecessary to make explicit, and essentially nobody ever describes it that way in practice.
 * Aside: The word axis originates from the line which remains fixed for a spinning object like a wheel, lathe, or the Earth (related to "axle"). Later the sense was broadened to include discrete kinds of spinning such as reflection across a line in the plane ("axis of symmetry"). When there is only one line involved as the entire space, as in the case of the number line, a linear axis is entirely trivial; you can't spin or reflect anything about it, because there is nothing outside of it. In the context of a line, the only relevant kind of transformation is reflection in a point, but a point center of reflection is rarely if ever called the "axis" either. I'm not quite sure when the term axis first started getting applied to the lines in a rectangular coordinate system with one coordinate equal to zero. –jacobolus (t) 17:19, 8 January 2024 (UTC)
 * NOTE:The edit wars mentioned below, which initiated by user:D.Lazard and user:JBW, were already terminated by correctly asserting that the terms "number line" and "number axis" (or simply axis) are the same. These are the terminology used in mathematics. You (user:Jacobolus) are restarting the same edit-war. Even school pupils know that these terms are interchangeable.
 * Persianwise 21:33, 8 January 2024 (UTC)
 * @Persianwise You don't need to make your text extra large or colorful. The folks participating here are all capable of reading ordinary-sized black text. You still have not provided any sources which validate your claim. The two sources you did provide say something entirely different, as I explained. I don't know what you mean by "used in mathematics", but in a quick literature search (using Google scholar) I can't find any example of the term "axis" being defined to mean the same as "number line". –jacobolus (t) 21:42, 8 January 2024 (UTC)
 * From your statement that "the horizontal number line is called the x-axis," it logically follows that a number line is indeed an axis . Persianwise (talk) 22:04, 8 January 2024 (UTC)
 * "the horizontal number line is called the x-axis" is not Jacobulus's statement; it is a statement of one of your sources. The logical consequence of this statement is that some number lines are axes, not that all number lines are axes as you pretend. In any case, bolding your assertions do not make them true. D.Lazard (talk) 22:16, 8 January 2024 (UTC)
 * While I don't think any published sources are likely to say this explicitly, I think it would be defensible to say something along the lines of "a number line is its own axis", in the sense that as a one-dimensional affine space it can be said to have one coordinate axis, coinciding with itself. I just don't think this article benefits in any way from making this (rather pedantic) point. It's also not at all the same as claiming that the two terms number line and axis mean the same (which is not a defensible claim in my opinion). –jacobolus (t) 22:50, 8 January 2024 (UTC)
 * You said:
 * "It does not need to be restated that a number line can be considered its own axis."
 * You statement does not mathematically make any sense.
 * You said:
 * "each coordinate axis of a Cartesian plane is a number line".
 * Don't you realize that "coordinate axis" is another name for "number axis" or just "axis". All these terms that you think are different objects, are actually the same thing. They are mathematically synonyms. Persianwise (talk) 21:55, 8 January 2024 (UTC)
 * Please stop to repeat again and again the same assertions without any source or argument for supporting them. This is WP:Disruptive editing, and this is not accepted in Wikipedia. D.Lazard (talk) 22:22, 8 January 2024 (UTC)
 * Anyone with basic knowledge of mathematics, even school pupils, learns that "number line," "coordinate line," "number axis," and simply "axis" are the same mathematical objects. References supporting this fact were provided on the following webpage and were also mentioned on your talk page.
 * Special:diff/1178434751
 * Persianwise (talk) 23:03, 8 January 2024 (UTC)
 * These are the same two sources we have been discussing, which I directly quoted above; neither one of them says that that "axis" is a synonym for "number line". (Aside, neither one of these seems like a "reliable source" by Wikipedia standards. Neither one has a listed author, and neither seems to have been formally published.) Everyone here has well beyond a "basic knowledge of mathematics"; adopting a condescending tone is tedious and does not make your argument any more convincing. –jacobolus (t) 23:12, 8 January 2024 (UTC)
 * If someone lacks basic mathematical understanding, it's advisable for them to consult relevant expertise before editing articles in that field. It's Common_sense. Persianwise (talk) 23:25, 8 January 2024 (UTC)
 * "Each cow in the barn is a farm animal" does not mean that "cow" and "farm animal" mean precisely the same thing. –jacobolus (t) 22:43, 8 January 2024 (UTC)
 * For your information, in mathematics, definitions are bi-directional. The terms "coordinate line," "number axis," and simply "axis" refer to the same thing. In your statement, you are not defining the cow. You are merely saying where the cows are. Persianwise (talk) 23:14, 8 January 2024 (UTC)
 * "in mathematics, definitions are bi-directional" – This is not correct, but this is not really the place to argue about it. Perhaps take it up on a forum about logic or philosophy. "In your statement, you are not defining the cow." – that is true; neither of the quoted passages from the sources you cited is making a definition either. Let me give a more explicitly analogous example: Imagine I am standing in a barn with you, and I point to a cow, saying "the farm animal by the water trough is called a cow". Can you see how that is not the same as a statement that every farm animal is a cow? (And also not a definition of "cow" or "farm animal".) –jacobolus (t) 23:18, 8 January 2024 (UTC)
 * In mathematics, ALL definitions are bi-directional.
 * Number lines are by definition are number axes. And number axes are number lines. Persianwise (talk) 23:29, 8 January 2024 (UTC)
 * "Number axis" is not a common term in English. A search turns up mostly results from Russian or Chinese sources, for example "Approximate solution of hypersingular integral equations on the number axis" (in Russian, where "number axis" is translated from "числовой оси"). –jacobolus (t) 23:54, 8 January 2024 (UTC)
 * Hmm, how about a "set" and a "subset"? Every subset of a set is a set, and every set is a subset of some other (e.g., of itself). Does that make sets and subsets the same thing for you? 慈居 (talk) 00:01, 9 January 2024 (UTC)
 * If isomorphic objects (some of which might be, e.g., disjoint) are the same, what about categorically equivalent objects? Is it acceptable to you to say that a group and a non-group (e.g., a groupoid) might be the same? 慈居 (talk) 00:05, 9 January 2024 (UTC)
 * I think these distinctions are a bit beyond the level of conversation we're engaged in here. Overall, this thread seems like a waste of time.
 * @Persianwise why don't you come back when you have (a) some sources which actually support your claim, and also (b) a more collaborative attitude, with some willingness to listen and learn from other people? In the mean time, please stop with the edit warring. –jacobolus (t) 00:10, 9 January 2024 (UTC)
 * I agree; can't believe this goes again. 慈居 (talk) 00:18, 9 January 2024 (UTC)
 * Number axis is a term used in elementary mathematics with the same meaning as number line. Although mathematical definitions are universally valid and transcend language, their presentation can vary across educational levels. My references from secondary school websites naturally reflect terminology introduced at that stage, even though the concepts behind number line and number axis are synonymous and readily grasped by most. Therefore, including a clarification within the article about the interchangeable use of these terms would benefit those encountering them across different educational resources, even those less mathematically inclined. This concise explanation should dispel any potential confusion and enhance the reader's understanding. Persianwise (talk) 01:55, 9 January 2024 (UTC)
 * If you disagree with my assertion that "number line" and "number axis" are synonyms, I encourage you to review the reputable secondary school resources I cited, or relevant mathematical literature that clarifies their equivalence. Persianwise (talk) 02:03, 9 January 2024 (UTC)
 * You were previously blocked over this. Coming back months later to restart the same issue is not a good idea. Time to WP:DROPTHESTICK Meters (talk) 02:04, 9 January 2024 (UTC)
 * @User:Meters Your statement lacks mathematical content. Do you concur with user:D.Lazard, user:JBW, and user:慈居 in asserting that "number line" and "number axis" are distinct objects? Persianwise (talk) 02:16, 9 January 2024 (UTC)
 * I have bothered to give you a mathematical content and yet expecting an answer from you. Why are sets and subsets different concept with your logic? 慈居 (talk) 02:55, 9 January 2024 (UTC)
 * @user:慈居 Your question, even if assumed to be correctly stated, is irrelevant to the issue caused by the edit war initially started by user:D.Lazard, and user:JBW and then restarted by user:jacobolus. Please stay on topic.
 * Do you disagree that "number line" and "number axis" are the same thing? Persianwise (talk) 08:49, 9 January 2024 (UTC)
 * That depends on what you mean by "the same"; you are free to define it whatever you like, but they don't apply to others. 慈居 (talk) 10:01, 9 January 2024 (UTC)
 * 1+1 is the same as two. Obviously, number lines are number axes; they are the same. Any other question? Persianwise (talk) 15:55, 9 January 2024 (UTC)
 * Just chiming in to concur that Persianwise has not convinced me. All squares are rectangles, but not all rectangles are squares. All axes may be number lines, but it does not follow that all number lines are axes. EducatedRedneck (talk) 17:19, 9 January 2024 (UTC)
 * I'm just chiming in to note that Persianwise has been blocked indefinitely for continuing to edit war after having been previously blocked for it. Wikipedia is not a game to be won - we settle disputes by constructive discussion, not by wearing down opponents with legalesque interrogation and thinly-veiled insults. Ivanvector (Talk/Edits) 17:42, 9 January 2024 (UTC)
 * Just chiming in to concur that Persianwise has not convinced me. All squares are rectangles, but not all rectangles are squares. All axes may be number lines, but it does not follow that all number lines are axes. EducatedRedneck (talk) 17:19, 9 January 2024 (UTC)
 * I'm just chiming in to note that Persianwise has been blocked indefinitely for continuing to edit war after having been previously blocked for it. Wikipedia is not a game to be won - we settle disputes by constructive discussion, not by wearing down opponents with legalesque interrogation and thinly-veiled insults. Ivanvector (Talk/Edits) 17:42, 9 January 2024 (UTC)

title: Talk:Cartesian coordinate system


text:
California Department of Transportation

The California Department of Transportation (Caltrans) is an executive department of the U.S. state of California. The department is part of the cabinet-level California State Transportation Agency (CalSTA). Caltrans is headquartered in Sacramento.

Caltrans manages the state's highway system, which includes the California Freeway and Expressway System, supports public transportation systems throughout the state and provides funding and oversight for three state-supported Amtrak intercity rail routes (Capitol Corridor, Pacific Surfliner and San Joaquins) which are collectively branded as Amtrak California.

In 2015, Caltrans released a new mission statement: "Provide a safe, sustainable, integrated and efficient transportation system to enhance California's economy and livability."

History
The earliest predecessor of Caltrans was the Bureau of Highways, which was created by the California Legislature and signed into law by Governor James Budd in 1895. This agency consisted of three commissioners who were charged with analyzing the roads of the state and making recommendations for their improvement. At the time, there was no state highway system, since roads were purely a local responsibility. California's roads consisted of crude dirt roads maintained by county governments, as well as some paved streets in certain cities, and this ad hoc system was no longer adequate for the needs of the state's rapidly growing population. After the commissioners submitted their report to the governor on November 25, 1896, the legislature replaced the Bureau with the Department of Highways.

Due to the state's weak fiscal condition and corrupt politics, little progress was made until 1907, when the legislature replaced the Department of Highways with the Department of Engineering, within which there was a Division of Highways. California voters approved an $18 million bond issue for the construction of a state highway system in 1910, and the first California Highway Commission was convened in 1911. On August 7, 1912, the department broke ground on its first construction project, the section of El Camino Real between South San Francisco and Burlingame, which later became part of California State Route 82. The year 1912 also saw the founding of the Transportation Laboratory and the creation of seven administrative divisions, which are the predecessors of the 12 district offices in use. The original seven division headquarters were located in:
 * Willits Mercantile Building for Del Norte, Humboldt, Lake, and Mendocino counties
 * Redding C.R.Briggs Building for Lassen, Modoc, Shasta, Siskiyou, Tehama, and Trinity counties
 * Sacramento Forum Building for Alpine, Amador, Butte, Calaveras, Colusa, El Dorado, Glenn, Nevada, Placer, Plumas, Sacramento, San Joaquin, Sierra, Solano, Stanislaus, Sutter, Tuolumne, Yolo, and Yuba counties
 * San Francisco Rialto Building for Alameda, Contra Costa, Marin, Napa, San Francisco, Santa Clara, Santa Cruz, San Mateo, and Sonoma counties
 * San Luis Obispo Union National Bank Building for Monterey, San Benito, Santa Barbara, and San Luis Obispo counties
 * Fresno Forsythe Building for Fresno, Inyo, Kern, Kings, Madera, Mariposa, Merced, Mono, and Tulare counties
 * Los Angeles Union Oil Building for Imperial, Los Angeles, Orange, Riverside, San Bernardino, San Diego, and Ventura counties

In 1913, the California State Legislature began requiring vehicle registration and allocated the resulting funds to support regular highway maintenance, which began the next year.

In 1921, the state legislature turned the Department of Engineering into the Department of Public Works, which continued to have a Division of Highways. That same year, three additional divisions (now districts) were created, in Stockton, Bishop, and San Bernardino.

In 1933, the state legislature enacted an amendment to the State Highway Classification Act of 1927, which added over 6,700 miles of county roads to the state highway system. To help manage all the additional work created by this massive expansion, an eleventh district office was founded that year in San Diego.

The enactment of the Collier–Burns Highway Act of 1947 after "a lengthy and bitter legislative battle" was a watershed moment in Caltrans history. The act "placed California highway's program on a sound financial basis" by doubling vehicle registration fees and raising gasoline and diesel fuel taxes from 3 cents to 4.5 cents per gallon. All these taxes were again raised further in 1953 and 1963. The state also obtained extensive federal funding from the Federal-Aid Highway Act of 1956 for the construction of its portion of the Interstate Highway System. Over the next two decades after Collier-Burns, the state "embarked on a massive highway construction program" in which nearly all of the now-extant state highway system was either constructed or upgraded. In hindsight, the period from 1940 to 1969 can be characterized as the "Golden Age" of California's state highway construction program.

The history of Caltrans and its predecessor agencies during the 20th century was marked by many firsts. It was one of the first agencies in the United States to paint centerlines on highways statewide; the first to build a freeway west of the Mississippi River; the first to build a four-level stack interchange; the first to develop and deploy non-reflective raised pavement markers, better known as Botts' dots; and one of the first to implement dedicated freeway-to-freeway connector ramps for high-occupancy vehicle lanes.

In 1967, Governor Ronald Reagan formed a Task Force Committee on Transportation to study the state transportation system and recommend major reforms. One of the proposals of the task force was the creation of a State Transportation Board as a permanent advisory board on state transportation policy; the board would later merge into the California Transportation Commission in 1978. In September 1971, the State Transportation Board proposed the creation of a state department of transportation charged with responsibility "for performing and integrating transportation planning for all modes." Governor Reagan mentioned this proposal in his 1972 State of the State address, and Assemblyman Wadie P. Deddeh introduced Assembly Bill 69 to that effect, which was duly passed by the state legislature and signed into law by Reagan later that same year. AB 69 merged three existing departments to create the Department of Transportation, of which the most important was the Department of Public Works and its Division of Highways. The California Department of Transportation began official operations on July 1, 1973. The new agency was organized into six divisions: Highways, Mass Transportation, Aeronautics, Transportation Planning, Legal, and Administrative Services.

Caltrans went through a difficult period of transformation during the 1970s, as its institutional focus shifted from highway construction to highway maintenance. The agency was forced to contend with declining revenues, increasing construction and maintenance costs (especially the skyrocketing cost of maintaining the vast highway system built over the past three prior decades), widespread freeway revolts, and new environmental laws. In 1970, the enactment of the National Environmental Policy Act and the California Environmental Quality Act forced Caltrans to devote significant time, money, people, and other resources to confronting issues such as "air and water quality, hazardous waste, archaeology, historic preservation, and noise abatement." The devastating 1971 San Fernando earthquake compelled the agency to recognize that its existing design standards had not adequately accounted for earthquake stress and that numerous existing structures needed expensive seismic retrofitting. Maintenance and construction costs grew at twice the inflation rate in this era of high inflation; the reluctance of one governor after another to raise fuel taxes in accordance with inflation meant that California ranked dead last in the United States in per capita transportation spending by 1983. During the 1980s and 1990s, Caltrans concentrated on "the upgrading, rehabilitation, and maintenance of the existing system," plus occasional gap closure and realignment projects.

In 2023, Caltrans demoted one of its top officials, Jeanie Ward-Waller, because she objected to highway expansions and alleged that permits for those expansions improperly understated their adverse environmental impact.

Administration
For administrative purposes, Caltrans divides the State of California into 12 districts, supervised by district offices. Most districts cover multiple counties; District 12 (Orange County) is the only district with one county. The largest districts by population are District 4 (San Francisco Bay Area) and District 7 (Los Angeles and Ventura counties). Like many state agencies, Caltrans maintains its headquarters in Sacramento, which is covered by District 3.

title: California Department of Transportation