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In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in “the x-coordinate”. The coordinates are taken to be real numbers in elementary mathematics but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry.
Most of us are familiar with the Cartesian coordinate system and have used it in plotting points and graphing equations. But there are many other coordinate systems that are widely used in Mathematics and other fields.
Different Types of Coordinate Systems
In the past, different types of coordinate systems are developed. According to the application, a suitable coordinate system can be selected. Here we will look into 10 different types of coordinate systems.
1. Number Line
The simplest example of a coordinate system is the identification of points on a line with real numbers using the number line. In this system, an arbitrary point O (the origin) is chosen on a given line. The coordinate of a point P is defined as the signed distance from O to P, where the signed distance is the distance taken as positive or negative depending on which side of the line P lies. Each point is given a unique coordinate and each real number is the coordinate of a unique point.
Writing down points on a number line makes it easy for us to compare them. There are three parts of a number line which are – the negative side, zero, and positive side. Points to the left of 0 are negative and points to the right of 0 are all positive.
2. Cartesian Coordinate System
The Cartesian coordinate system allows one to specify the location of a point in the plane (two-dimensional), or in three-dimensional space. The Cartesian coordinates (also called rectangular coordinates) of a point are a pair of numbers (in two dimensions) or a triplet of numbers (in three dimensions) that specified signed distances from the coordinate axis.
Cartesian Coordinates in a Plane
The Cartesian coordinates in the Cartesian coordinate system are specified in terms of the x-coordinates axis and the y-coordinate axis, as illustrated in the below figure. The origin is the intersection of the x and y axes. The Cartesian coordinates of a point in the plane are written as (x, y). The first number x is called the x-coordinate (or x-component), as it is the signed distance from the origin in the direction along the x-axis.
In the Cartesian coordinate system the x-coordinate specifies the distance to the right (if x is positive), or to the left (if x is negative) of the y-axis. Similarly, the second number y is called the y-coordinate (or y-component), as it is the signed distance from the origin in the direction along the y-axis. The y-coordinate specifies the distance above (if y is positive) or below (if y is negative) the x-axis. In the following figure, the point has coordinates (-3, 2), as the point is three units to the left and two units up from the origin.
Cartesian Coordinates in Three-Dimensional Space
In three-dimensional space, the Cartesian coordinate system is based on three mutually perpendicular coordinate axes: the x-axis, the y-axis, and the z-axis. The three axes intersect at the point called the origin. You can imagine the origin being the point where the walls in the corner of a room meet the floor.
The x-axis is the horizontal line along which the wall to your left and the floor intersect. The y-axis is the horizontal line along which the wall to your right and the floor intersect. The z-axis is the vertical line along which the walls intersect. The parts of the lines that you see while standing in the room are the positive portion of each of the axes, illustrated by the halves of each axis labeled by x, y, and z. The negative part of these axes would be the continuations of the lines outside of the room, illustrated by the unlabeled halves of each axis, below.
In addition to the three coordinate axes, we often refer to three coordinate planes in a Cartesian coordinate system. The xy-plane is the horizontal plane spanned by the x and y-axes. It is identical to the two-dimensional coordinate plane and contains the floor in the room analogy. Similarly, the xz-plane is the vertical plane spanned by the x and z-axes and contains the left wall in the room analogy. Lastly, the yz-plane is the vertical plane spanned by the y and z axes and contains the right wall in the room analogy.
The Cartesian coordinates of a point in a three-dimensional Cartesian coordinate system are a triplet of numbers (x, y, z). The three numbers, or coordinates, specify the signed distance from the origin along the x, y, and z-axes, respectively. They can be visualized by forming the box with edges parallel to the coordinate axis and opposite corners at the origin and the given point. The Cartesian coordinates (x, y, z) of a point in a three-dimensional Cartesian coordinate system specify the signed distance from the origin along with the x, y, and z-axes respectively.
3. Polar Coordinate System
The polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. When we think about plotting points in the plane, we usually think of rectangular coordinates (x, y) in the Cartesian coordinate system. However, there are other ways of writing a coordinate pair and other types of grid systems. Polar coordinates are points labeled (r, 𝛳) and plotted on a polar grid. The polar grid is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane.
The reference point (analogous to the origin of a Cartesian coordinate system) is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is called the angular coordinate, polar angle, or azimuth. The radial coordinate is often denoted by r or ⍴ and the angular coordinate by 𝛳, ɸ or t.
Angles in polar notation are generally expressed in either degrees or radians (2π = 360o). Degrees are traditionally used in navigation, surveying, and many applied disciplines, while radians are more common in mathematics and mathematical physics. In many contexts, a positive angular coordinate means that the angle ϕ is measured counterclockwise from the axis. In mathematical literature, the polar axis is often drawn horizontally and points to the right.
Plotting Points Using Polar Coordinates
The polar grid is scaled as the unit circle with the positive x-axis now viewed as the polar axis and the origin as the pole. The first coordinate r is the radius or length of the directed line segment from the pole. The angle 𝛳 measured in radians indicates the direction of r. We move counterclockwise from the polar axis by an angle of 𝛳. Even though we measure 𝛳 first and then r, the polar point is written with the r-coordinate first. For example, to plot the point (2, π/4), we would move π/4 units in the counterclockwise direction and then a length of 2 from the pole.
4. Homogeneous Coordinate System
Homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, are a system of coordinates used in projective geometry, as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian coordinate system counterparts. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix.
If homogeneous coordinates of a point are multiplied by a non-zero scalar then the resulting coordinates represent the same point. Since homogeneous coordinates are also given to points at infinity, the number of coordinates required to allow this extension is one more than the dimension of the projective space being considered. For example, two homogeneous coordinates are required to specify a point on the projective line and three homogeneous coordinates are required to specify a point in the projective plane.
The real projective plane can be thought of as the Euclidean plane with additional points added, which are called points at infinity and are considered to lie on a new line, the line at infinity. There is a point at infinity corresponding to each direction (numerically given by the slope of a line), informally defined as the limit of a point that moves in that direction away from the origin. Parallel lines in the Euclidean plane are said to intersect at a point at infinity corresponding to their common direction.
Given a point (x, y) on the Euclidean plane, for any non-zero real number Z, the triple (xZ, yZ, Z) is called a set of homogeneous coordinates for the point. By this definition, multiplying the three homogeneous coordinates by a common, non-zero factor gives a new set of homogeneous coordinates for the same point.
In particular, (x, y, 1) is such a system of homogeneous coordinates for the point (x, y). For example, the Cartesian point (1, 2) can be represented in homogeneous coordinates as (1, 2, 1) or (2, 4, 2). The original coordinates in a Cartesian coordinate system are recovered by dividing the first two positions by the third. Thus unlike Cartesian coordinates, a single point can be represented by infinitely many homogeneous coordinates.
The equation of a line through the origin (0, 0) may be written nx + my = 0 where n and m are not both 0. In parametric form this can be written x = mt, y = −nt. Let Z = 1/t, so the coordinates of a point on the line may be written (m/Z, −n/Z). In homogeneous coordinates, this becomes (m, −n, Z).
In the limit, as t approaches infinity, in other words, as the point moves away from the origin, Z approaches 0 and the homogeneous coordinates of the point become (m, −n, 0). Thus we define (m, −n, 0) as the homogeneous coordinates of the point at infinity corresponding to the direction of the line nx + my = 0. Like any line of the Euclidean plane is parallel to a line passing through the origin, and since parallel lines have the same point at infinity, the infinite point on every line of the Euclidean plane has been given homogeneous coordinates.
Important points about a homogeneous coordinate system are:
- Any point in the projective plane is represented by a triple (X, Y, Z), called homogeneous coordinates or projective coordinates of the point, where X, Y, and Z are not all 0.
- The point represented by a given set of homogeneous coordinates is unchanged if the coordinates are multiplied by a common factor.
- Conversely, two sets of homogeneous coordinates represent the same point if and only if one is obtained from the other by multiplying all the coordinates by the same non-zero constant.
- When Z is not 0 the point represented is the point (X/Z, Y/Z) in the Euclidean plane.
- When Z is 0 the point represented is a point at infinity.
5. Curvilinear Coordinate System
Curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible (a one-to-one map) at each point. This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back. The name curvilinear coordinates, coined by the French mathematician Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved.
Well-known examples of curvilinear coordinate systems in three-dimensional Euclidean space (R3) are cylindrical and spherical coordinates. A Cartesian coordinate surface in this space is a coordinate plane; for example, z = 0 defines the x–y plane. In the same space, the coordinate surface r = 1 in spherical coordinates is the surface of a unit sphere, which is curved. The formalism of curvilinear coordinates provides a unified and general description of the standard coordinate systems.
Curvilinear coordinates are often used to define the location or distribution of physical quantities which may be, for example, scalars, vectors, or tensors. Mathematical expressions involving these quantities in vector calculus and tensor analysis (such as the gradient, divergence, curl, and Laplacian) can be transformed from one coordinate system to another, according to transformation rules for scalars, vectors, and tensors. Such expressions then become valid for any curvilinear coordinate system.
A curvilinear coordinate system may be simpler to use than the Cartesian coordinate system for some applications. The motion of particles under the influence of central forces is usually easier to solve in spherical coordinates than in Cartesian coordinates; this is true of many physical problems with spherical symmetry defined in R3. Equations with boundary conditions that follow coordinate surfaces for a particular curvilinear coordinate system may be easier to solve in that system.
While one might describe the motion of a particle in a rectangular box using Cartesian coordinates, the motion in a sphere is easier with spherical coordinates. Spherical coordinates are the most common curvilinear coordinate systems and are used in Earth sciences, cartography, quantum mechanics, relativity, and engineering.
6. Log-Polar Coordinate System
Log-polar coordinates (or logarithmic polar coordinates) is a coordinate system in two dimensions, where a point is identified by two numbers, one for the logarithm of the distance to a certain point, and one for an angle. Log-polar coordinates are closely connected to polar coordinates, which are usually used to describe domains in the plane with some sort of rotational symmetry. In areas like harmonic and complex analysis, the log-polar coordinates are more canonical than polar coordinates.
Log-polar coordinates in the plane consist of a pair of real numbers (ρ,θ), where ρ is the logarithm of the distance between a given point and the origin and θ is the angle between a line of reference (the x-axis) and the line through the origin and the point.
The angular coordinate is the same as for polar coordinates, while the radial coordinate is transformed according to the rule r = eρ, where r is the distance to the origin. The formulas for transformation from Cartesian coordinates to log-polar coordinates are given by ρ = log(sqrt(x2+y2)) and =tan-1(y/x), if x > 0
and the formulas for transformation from log-polar to Cartesian coordinates are x=e⍴cos𝜃 and y=e⍴sin𝜃
By using complex numbers (x, y) = x + iy, the transformation can be written as x+iy=e⍴+i𝜃
7. Barycentric Coordinate System
In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle for points in a plane, a tetrahedron for points in three-dimensional space, etc.). The barycentric coordinates of a point can be interpreted as masses placed at the vertices of the simplex, such that the point is the center of mass (or barycenter) of these masses. These masses can be zero or negative; they are all positive if and only if the point is inside the simplex.
Every point has barycentric coordinates, and its sum is not zero. Two tuples of barycentric coordinates specify the same point if and only if they are proportional; that is to say, if one tuple can be obtained by multiplying the elements of the other tuple by the same non-zero number. Therefore, barycentric coordinates are either considered to be defined up to multiplication by a nonzero constant or normalized for summing to unity.
Barycentric coordinates are particularly useful in triangle geometry for studying properties that do not depend on the angles of the triangle, such as Ceva’s theorem. In computer-aided design, they are useful for defining some kinds of Bézier surfaces.
8. Trilinear Coordinate System
The trilinear coordinates x:y:z of a point relative to a given triangle describes the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio x:y is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices A and B respectively; the ratio y:z is the ratio of the perpendicular distances from the point to the sidelines opposite vertices B and C respectively; and likewise for z:x and vertices C and A.
In the diagram above, the trilinear coordinates of the indicated interior point are the actual distances (a’, b’, c’ ), or equivalently in ratio form, ka’:kb’:kc’ for any positive constant k. If a point is on a sideline of the reference triangle, its corresponding trilinear coordinate is 0. If an exterior point is on the opposite side of a sideline from the interior of the triangle, its trilinear coordinate associated with that sideline is negative. It is impossible for all three trilinear coordinates to be non-positive.