Cartesian coordinate system

Choosing a Cartesian coordinate system for a one-dimensional space—that is, for a straight line—involves choosing a point O of the line (the origin), a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive and which is negative; we then say that the line "is oriented" (or "points") from the negative half towards the positive half. Then each point P of the line can be specified by its distance from O, taken with a + or − sign depending on which half-line contains P.

A line with a chosen Cartesian system is called a number line. Every real number has a unique location on the line. Conversely, every point on the line can be interpreted as a number in an ordered continuum such as the real numbers.

A Cartesian coordinate system in two dimensions (also called a rectangular coordinate system or an orthogonal coordinate system[6]) 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.

A three dimensional Cartesian coordinate system, with origin O and axis lines X, Y and Z, oriented as shown by the arrows. The tick marks on the axes are one length unit apart. The black dot shows the point with coordinates x = 2, y = 3, and z = 4, or (2, 3, 4).

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:

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

The coordinates are usually written as three numbers (or algebraic formulas) surrounded by parentheses and separated by commas, as in (3, −2.5, 1) or (t, u + v, π/2). Thus, the origin has coordinates (0, 0, 0), and the unit points on the three axes are (1, 0, 0), (0, 1, 0), and (0, 0, 1).

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, 1); a convention that is commonly called the right hand rule.

The coordinate surfaces of the Cartesian coordinates (x, y, z). The z-axis is vertical and the x-axis is highlighted in green. Thus, the red hyperplane shows the points with x = 1, the blue hyperplane shows the points with z = 1, and the yellow hyperplane shows the points with y = −1. The three surfaces intersect at the point P (shown as a black sphere) with the Cartesian coordinates (1, −1, 1).

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 ${\displaystyle \mathbb {R} ^{2}=\mathbb {R} \times \mathbb {R} }$, where ${\displaystyle \mathbb {R} }$ 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 ${\displaystyle \mathbb {R} ^{n}}$.

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).

The four quadrants of a Cartesian coordinate system

The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called quadrants,[6] 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,[6] 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.

The Euclidean distance between two points of the plane with Cartesian coordinates ${\displaystyle (x_{1},y_{1})}$ and ${\displaystyle (x_{2},y_{2})}$ is

$${\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.}$$

This is the Cartesian version of Pythagoras's theorem. In three-dimensional space, the distance between points ${\displaystyle (x_{1},y_{1},z_{1})}$ and ${\displaystyle (x_{2},y_{2},z_{2})}$ is

$${\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}},}$$

which can be obtained by two consecutive applications of Pythagoras' theorem.[8]

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.[9]

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

$${\displaystyle (x',y')=(x+a,y+b).}$$

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

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

Thus:

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

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 ${\displaystyle \theta }$ with the x-axis, is equivalent to replacing every point with coordinates (x, y) by the point with coordinates (x′,y′), where

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

Thus:

$${\displaystyle (x',y')=((x\cos 2\theta +y\sin 2\theta \,),(x\sin 2\theta -y\cos 2\theta \,)).}$$

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).

All affine transformations of the plane can be described in a uniform way by using matrices. For this purpose the coordinates ${\displaystyle (x,y)}$ of a point are commonly represented as the column matrix ${\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}.}$ The result ${\displaystyle (x',y')}$ of applying an affine transformation to a point ${\displaystyle (x,y)}$ is given by the formula

$${\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}=A{\begin{pmatrix}x\\y\end{pmatrix}}+b,}$$

where

$${\displaystyle A={\begin{pmatrix}A_{1,1}&A_{1,2}\\A_{2,1}&A_{2,2}\end{pmatrix}}}$$

is a 2×2 matrix and ${\displaystyle b={\begin{pmatrix}b_{1}\\b_{2}\end{pmatrix}}}$ is a column matrix.[10] That is,

$${\displaystyle {\begin{aligned}x'&=xA_{1,1}+yA_{1,1}+b_{1}\\y'&=xA_{2,1}+yA_{2,2}+b_{2}.\end{aligned}}}$$

Among the affine transformations, the Euclidean transformations are characterized by the fact that the matrix ${\displaystyle A}$ is orthogonal; that is, its columns are orthogonal vectors of Euclidean norm one, or, explicitly,

$${\displaystyle A_{1,1}A_{1,2}+A_{2,1}A_{2,2}=0}$$

and

$${\displaystyle A_{1,1}^{2}+A_{2,1}^{2}=A_{1,2}^{2}+A_{2,2}^{2}=1.}$$

This is equivalent to saying that A 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 A is the identity matrix. The transformation is a rotation around some point if and only if A is a rotation matrix, meaning that it is orthogonal and

$${\displaystyle A_{1,1}A_{2,2}-A_{2,1}A_{1,2}=1.}$$

A reflection or glide reflection is obtained when,

$${\displaystyle A_{1,1}A_{2,2}-A_{2,1}A_{1,2}=-1.}$$

Assuming that translations are not used (that is, ${\displaystyle b_{1}=b_{2}=0}$) 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

$${\displaystyle {\begin{pmatrix}x'\\y'\\1\end{pmatrix}}=A'{\begin{pmatrix}x\\y\\1\end{pmatrix}},}$$

where

$${\displaystyle A'={\begin{pmatrix}A_{1,1}&A_{1,2}&b_{1}\\A_{2,1}&A_{2,2}&b_{2}\\0&0&1\end{pmatrix}}.}$$

With this trick, the composition of affine transformations is obtained by multiplying the augmented matrices.

Effect of applying various 2D affine transformation matrices on a unit square (reflections are special cases of scaling)

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:

$${\displaystyle {\begin{pmatrix}A_{1,1}&A_{2,1}&b_{1}\\A_{1,2}&A_{2,2}&b_{2}\\0&0&1\end{pmatrix}}{\begin{pmatrix}x\\y\\1\end{pmatrix}}={\begin{pmatrix}x'\\y'\\1\end{pmatrix}}.}$$

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

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

$${\displaystyle (x',y')=(mx,my).}$$

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

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

$${\displaystyle (x',y')=(x+ys,y)}$$

Shearing can also be applied vertically:

$${\displaystyle (x',y')=(x,xs+y)}$$

The right-hand rule

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.

Fig. 7 – The left-handed orientation is shown on the left, and the right-handed on the right.

Fig. 8 – The right-handed Cartesian coordinate system indicating the coordinate planes.

Once the x- and y-axes are specified, they determine the line along which the z-axis should lie, but there are two possible orientation 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.

3D Cartesian coordinate handedness

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.