Orthogonal Coordinate System
A coordinate system is a way of naming the points of n-dimensional space with n-tuples of real numbers so that no two points have the same name. For example, a line can be assigned coordinates be naming one of the points 0 (also called the origin) and deciding which side of 0 is positive and which is negative. Any point that is on the positive side is assigned the number equal to its distance from 0. A number on the left side is assigned the number equal to minus its distance from 0. These coordinates are called the Cartesian coordinates of the line. Cartesian coordinates can also be extended to the plane and n-dimensional space for any positive integer n. In polar coordinates, each point (x, y) in the plane is given the coordinates (r,a) in which r is the distance (0, 0) to (x, y) and a is the angle between the ray containing (0, 0) and (1, 0) and the ray containing (0, 0) and (x, y). In other words, r = (x2 + y2) and a = arctan (y/x). In cylindrical coordinates, each point (x, y, z) of three-dimensional space is assigned the coordinates (r, a, z) in which, as in polar coordinates, r = (x^2 + y^2) and a = arctan (y/x). In spherical coordinates, each point (x, y, z) is assigned the point (r, a, b) in which r is the distance from (0, 0, 0) to (x, y, z), a is, again, the angle between the ray containing (0, 0) and (1, 0) and the ray containing (0, 0) and (x, y), and b is the angle between the ray containing (0, 0, 0) and (0, 0, 1) and the ray containing (0, 0, 0) and (x, y, z). In each of these examples, there are circles, cylinders, or spheres defined the equation r = constant. In some sense, these coordinates are based upon these objects. Coordinate systems can be based on ellipsoids, paraboloids, torii, hyperboloids, or other surfaces.
Orthogonality is a generalization of perpendicularity that uses tangent spaces. Two intersecting curves in the plane are orthogonal if, near the point of intersection, the two curves together almost form a plus sign. To be precise requires some differential calculus. Suppose the two curves are given by differentiable functions f and g from the real numbers to the plane such that f(0) = g(0) is the point of intersection. Then the lines consisting of all points of the form f'(0)t + f(0) and g'(0)t + g(0) for all real numbers t, are called the tangent lines to f and g (at the point of intersection) respectively. Here f' and g' are the derivatives of f and g. If these tangent lines intersect at right angles, then the curves are said to be orthogonal.
Two planes in three-dimensional space are orthogonal if their union is congruent to the union of the xy plane with the xz plane. In other words, if we look at one plane from its side so that it appears as a line, the other plane appears as a line, too and these two lines are perpendicular.
Two (differentiable) surfaces are orthogonal if in a small neighborhood of any point of intersection, the intersection is "looks" like the intersection of two orthogonal planes. If P is a point on a surface and f is a differentiable curve from the real numbers to the surface, then the tangent line to f at the point P is called a tangent line to S at P. The tangent plane of the surface at P is the union of all such tangent lines. Two surfaces are said to be orthogonal if at every point of intersection, their tangent planes are orthogonal. Orthogonality can be defined more generally for (n-1)-dimensional manifolds in n-dimensional space in a similar manner.
An orthogonal coordinate system for three-dimensional space is a triplet of functions (u1,u2,u3), say, that satisfies the following properties.
- (1) Each ui maps three-dimensional space to the real line.
- (2) The function that sends (x, y, z) to (u1(x, y, z), u2(x, y, z), u3(x, y, z)) is a one-to-one, differentiable function.
- (3) For every point (x,y,z), the three surfaces defined by the equations u1 = x, u2 = y, and u3 = z are orthogonal. Equivalently, grad u1, grad u2, and grad u3 are orthogonal vector fields.
Orthogonal coordinate systems for n-dimensional spaces in which n is not three are defined in a similar manner.
Cylindrical coordinates, for example, are given by u1(x, y, z) = (x2 + y 2 + z2), u2(x, y, z) = arctan(y/x), u3(x, y, z) = z. The surfaces defined by the equations mentioned in (3) are just the cylinder of radius r that is perpendicular the xy-plane, the plane that contains the z-axis and the point (x, y, z), and the plane parallel to the xy-plane at height z.
Different coordinate systems are used depending on the problem one is trying to solve. For example, objects with rotational symmetry are often easier to analyze if they are given in cylindrical or spherical coordinates whereas objects built from cubes are easiest to analyze with Cartesian coordinates.
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