Cartesian coordinate system
The Cartesian coordinate system can pick out a single point on a flat page using only two numbers, such as 3 and minus 10.5. Each number is a signed distance, measured from one of two fixed lines that cross at a right angle. Together the pair is enough to tell that point apart from every other point on the plane.
A system this plain rarely gets a second glance. Yet it has a name, a disputed origin story, and a surprising number of rules about which way is up. Who first dreamed it up, and who got the credit? Why do mathematicians and computer programmers sometimes draw the vertical axis in opposite directions? What can a left hand tell you about a coordinate system? And how can one grid describe shapes sliding, spinning and stretching across a page?
In 1637, René Descartes, a French mathematician and philosopher living in the Netherlands, published the idea that gave us the adjective Cartesian. Pierre de Fermat arrived at the same idea independently, and he also worked in three dimensions, but he never published it. The French cleric Nicole Oresme had used constructions much like these well before either man.
Both Descartes and Fermat worked with a single axis, measuring a varying length against it. The pair of axes came afterward, once Frans van Schooten and his students translated Descartes's La Géométrie into Latin in 1649. Those commentators were trying to clarify a difficult text, and they introduced several new concepts along the way.
A circle of radius 2, centered at the origin, shows why the invention mattered so much. Its points are exactly the ones whose coordinates x and y satisfy one equation. From that equation, the area, the perimeter and the tangent line at any point can be found with integrals and derivatives. The same method works for any curve, so problems of geometry became problems of algebra and calculus.
Isaac Newton and Gottfried Wilhelm Leibniz developed calculus with this system playing a fundamental role. The graph of a function is its most familiar offspring. Linear algebra, complex analysis, differential geometry, multivariate calculus and group theory all gain geometric interpretations from it. Astronomy, physics and engineering depend on it, and it is the most common system in computer graphics and computer-aided geometric design. The two-coordinate description of the plane later grew into the concept of vector spaces.
Descartes's grid was only the first of many. Polar coordinates for the plane, and spherical and cylindrical coordinates for three-dimensional space, followed.
Zero and one are the usual anchors of a Cartesian system on a single line. Such a line is called a number line, and every real number matches exactly one of its points. Choosing the system involves two degrees of freedom. One way is to pick two distinct points and assign them two distinct real numbers, and every other point then gets its number by linear interpolation. The other way is to put the origin at zero and choose an oriented unit length, so each point's number is its signed distance from the origin. Any two such systems on a line are related by a linear function of the form ax plus b. Translating the line corresponds to addition, and scaling it corresponds to multiplication.
Two axes meeting at a right angle and sharing one unit of length define the plane version. Their meeting point is the origin, with coordinates (0, 0), and each axis becomes a number line. To read a point P, draw a line through it perpendicular to each axis and note where each line meets its axis. The two numbers, in order, are the coordinates of P. The first is called the abscissa and the second the ordinate. The points one unit out along the positive half-axes sit at (1, 0) and (0, 1).
Four quadrants appear when the two axes cut the plane, and they are numbered with the Roman numerals I through IV. In quadrant I both coordinates are positive. In quadrant II the abscissa is negative and the ordinate positive, in III both are negative, and in IV the abscissa is positive and the ordinate negative. In the usual drawing, the numbering runs counter-clockwise from the upper right. On this plane, mathematicians define standard figures. The unit circle has a radius equal to the length unit and its center at the origin. The unit square has a diagonal running from (0, 0) to (1, 1).
Eight octants appear once a third axis joins the other two, because three mutually perpendicular planes divide space into eight regions. A point is read the same way as before, by passing a plane through it perpendicular to each axis. The third coordinate is called the applicate, and when the third axis points up it may be called the height or altitude. An example point is (3, minus 2.5, 1), and the origin is (0, 0, 0). An octant is named by listing the signs of its coordinates, such as (+ + +) or ( minus + minus ).
Beyond three dimensions, n coordinates specify a point in n-dimensional Euclidean space. They are the signed distances from n mutually perpendicular fixed hyperplanes. Points of the plane match pairs of real numbers, and points in n-dimensional space match lists of n real numbers. The quadrant and the octant generalize to the orthant for any number of dimensions.
Perpendicular axes and equal units are not strictly required. In an oblique system, each coordinate comes from projecting the point onto one axis along a direction parallel to the others, and the units may differ from axis to axis. The price is that calculations of distances and angles must change. Many standard formulas, including the Pythagorean formula for distance, no longer hold.
The letters x and y come from an old algebra custom of using the end of the alphabet for unknown values and the beginning for known ones. Generic coordinates are written (x, y) in the plane and (x, y, z) in space, and the origin is often labeled with a capital O. Each axis is usually named after the quantity measured along it. A graph of pressure against time can therefore have a p-axis and a t-axis.
Subscripts, as in x1, x2 and on to xn, serve when the number of dimensions exceeds 3 or is left unspecified. Some authors prefer to count from x0 up to x of n minus 1. This habit pays off in computer programming. Storing a point's coordinates as an array, rather than a record, lets the subscript act as an index.
"Walk along the hall then up the stairs" is a mnemonic that young children use to learn the order of the values. It teaches them to read straight across and then vertically, before they have settled the ideas of the x-, y- and z-axes.
The 1960s, or possibly earlier, gave computer graphics a habit of its own. Image processing often points the y-axis downward on the display. The convention grew out of the way images were originally stored in display buffers.
Three-dimensional pictures follow their own habits. The xy-plane is drawn flat, with the z-axis marking height and positive values upward. The x-axis is usually oriented toward the viewer, leaning either right or left. If a diagram shows the x- and y-axes horizontally and vertically, the z-axis points out of the page toward the viewer or camera. On paper it then appears as a line or ray running down and to the left or right. In such diagrams the words abscissa and ordinate are rarely used for x and y. When they are used, they are sometimes applied to the axes rather than to the coordinate values.
The orientation of the three axes as a whole is arbitrary in any picture. Their orientation relative to one another is not.
A single decision remains once the x-axis is chosen. The y-axis must be the perpendicular through the point marked 0, so the only open question is which of its two half-lines counts as positive. Each choice gives a different orientation, also called handedness, to the Cartesian plane.
The usual plane has the positive x-axis pointing right and the positive y-axis pointing up. It is called the positive, standard or right-handed orientation. A mnemonic is the right-hand rule. Lay a somewhat closed right hand on the plane with the thumb pointing up, and the fingers sweep from the x-axis to the y-axis. Doing the same with the left hand gives the other orientation. When the thumb points along an axis away from the origin, the curl of the fingers shows the positive direction of rotation about that axis. Rotating the whole coordinate system never changes its orientation. Switching one axis reverses it, while switching both leaves it unchanged.
Three dimensions offer the same fork for the z-axis, and the two results are called right-handed and left-handed. In the standard right-handed system the xy-plane is horizontal and the z-axis points up. The right hand gives a second recipe. Point the index finger forward, bend the middle finger inward at a right angle to it, and place the thumb at a right angle to both. The thumb marks the x-axis, the index finger the y-axis and the middle finger the z-axis. Repeating this with the left hand produces a left-handed system.
A drawing of a right-handed system can flip in the viewer's mind between a convex cube and a concave corner. A three-dimensional object shown on a two-dimensional screen is ambiguous, and the two views match the two possible orientations of space. Seeing the convex cube gives a left-handed system. The correct reading is a concave corner, with the x-axis pointing toward the observer.
That is why diagrams obey the right-hand rule unless they say otherwise, since all laws of physics and math assume right-handedness.
Pythagoras's theorem takes a Cartesian form in the distance between two points. In the plane, that distance is the square root of the sum of two squared differences, one for the x-values and one for the y-values. In three-dimensional space a third squared difference joins the sum. The result follows from applying Pythagoras's theorem twice in a row.
Four kinds of Euclidean transformation move the plane while preserving distances. They are translations, rotations, reflections and glide reflections. Also called Euclidean motions or isometries, they are the bijective mappings of the plane to itself that keep the distances between points unchanged.
A fixed pair of numbers (a, b) added to the coordinates of every point produces a translation. A point at (x, y) lands at (x plus a, y plus b). Distances and directions between the points do not change.
Rotating a figure counter-clockwise about the origin by an angle theta replaces each point (x, y) with a new point. Its new x-coordinate is x cosine theta minus y sine theta, and its new y-coordinate is x sine theta plus y cosine theta.
The point (x, y) mirrored across the y-axis lands at (minus x, y), as if that axis were a mirror. Across the x-axis it lands at (x, minus y). A reflection across any line through the origin that makes an angle theta with the x-axis uses twice that angle. The new x-coordinate is x cosine 2 theta plus y sine 2 theta, and the new y-coordinate is x sine 2 theta minus y cosine 2 theta.
A glide reflection is a reflection across a line followed by a translation in the direction of that line. The order of the two steps does not matter, because the translation can come first.
All four motions, and more, share one tidy description in the language of matrices.
A 2 by 2 matrix called A and a column of two numbers called b describe every affine transformation of the plane. The coordinates (x, y) of a point are written as a column matrix. The result of the transformation is A times that column, plus b.
The word orthogonal separates the Euclidean motions from the rest. The columns of A must be orthogonal vectors of Euclidean norm one, which is the same as saying that A times its transpose is the identity matrix. If that fails, the formula describes a more general affine transformation. The transformation is a translation exactly when A is the identity matrix. It is a rotation about some point when A is orthogonal and its determinant equals 1. A determinant of minus 1 signals a reflection or a glide reflection.
When b is zero, meaning no translations are used, transformations compose by simply multiplying their matrices. Translations need a trick. The augmented matrix has three rows and three columns, with a bottom row of 0, 0, 1 and the column b on the right. It acts on the column (x, y, 1). Composing two affine transformations then means multiplying their augmented matrices.
Affine transformations send lines to lines, but they may change distances and angles. Scaling is the simplest example that is not Euclidean. Multiply both coordinates of every point by the same positive number m. If m is greater than 1, the figure becomes larger, and if m lies between 0 and 1, it becomes smaller.
Shearing pushes the top of a square sideways to form a parallelogram. Horizontal shearing sends (x, y) to (x plus y times s, y). Vertical shearing sends it to (x, x times s plus y).
Every one of these maps treats a point as a column of numbers, and that same column can also be read as an arrow drawn from the origin.
An arrow from the origin to a point is called the position vector of that point, and it is often written as r. In two dimensions, r equals x times i plus y times j. Here i is the pair (1, 0) and j is the pair (0, 1). These unit vectors point along the x-axis and y-axis and form the standard basis. In some application areas they are called versors. In three dimensions a third unit vector, k, equal to (0, 0, 1), joins them, and r equals x times i plus y times j plus z times k.
No natural way exists to multiply two vectors and get another vector in every dimension. In the plane, complex numbers offer a workaround. Identify the point (x, y) with a complex number. The imaginary unit, written i, sits at the point (0, 1), so despite the shared letter it is not the unit vector along the x-axis. Complex numbers can be multiplied to give another complex number, and so this identification gives a way to multiply vectors. In three-dimensional space, a similar identification can be made with a subset of the quaternions.
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Common questions
What is a Cartesian coordinate system?
It is a way of specifying each point in a plane by a pair of real numbers. These are the signed distances from the point to two fixed perpendicular lines called axes. The axes meet at the origin, which has coordinates (0, 0).
Who invented Cartesian coordinates?
René Descartes published the idea in 1637 while living in the Netherlands, and the system carries his name. Pierre de Fermat discovered it independently but did not publish it. The French cleric Nicole Oresme used similar constructions well before either of them.
What are the abscissa and the ordinate?
They are the names for the first and second coordinates of a point in the plane. The abscissa is measured along the horizontal axis and the ordinate along the vertical axis. In three dimensions the third coordinate is called the applicate.
What are quadrants and octants?
The two axes of a plane divide it into four quadrants, numbered I to IV counter-clockwise from the upper right. Three axes divide space into eight octants, named by the signs of the coordinates. The general version for any number of dimensions is the orthant.
What does right-handed mean for a coordinate system?
It describes the standard orientation. In the plane, the positive x-axis points right and the positive y-axis points up. In space, the right hand's thumb, index finger and middle finger indicate the x-, y- and z-axes. Switching the direction of one axis gives the opposite, left-handed orientation.
Why does the y-axis point down in some computer graphics?
Computer graphics and image processing often orient the y-axis downward on the display. The convention developed in the 1960s, or earlier, from the way images were originally stored in display buffers.
All sources
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