Spherical orthogonal coordinate system (3 dimensions) Morio Kikuchi

Spherical orthogonal coordinate system (3 dimensions) Morio Kikuchi Abstract: Product of metric coefficient and radius of round line is constant in spherical orthogonal coordinate system. Coordinates and so forth are constant in the coordinate transformation from orthogonal coordinates into spherical orthogonal coordinates if the value is special. 1. Length on round line Figure 1 shows round line on xz plane. Because a round line is a figure by inversion of a line, the top N corresponds to the foot of the perpendicular dropped from origin to the line. The length s from the top N to the point P on the round line is s = K tan θ treating K as metric coefficient of round line. Differential form is ds = K sec 2 θdθ 2. Case of two dimensions We find the condition for two round lines which are obtained by inversion of parallel lines on plane being parallel lines also on a sphere. In Figure 2, two sets of parallel lines cross at right angles each other, and their directions are x and y, and one side of parallel lines is on the coordinate axis in either set. Figure 3 shows round lines by inversion of them, and the figure which the four round lines form and x y plane are in the relation of Figure 4. In addittion, the round lines which pass through the top N of the sphere are the coordinate axes on the sphere, and the coordinates on the sphere are expressed like x = K tan θ x, y = K tan θ y 1

using * to distinguish them from orthogonal coordinates. l 1 = K tan θ 1 = K(y /K), l 2 = K φ tan θ 2 = K φ {y /(K/ cos φ)} From l 1 = l 2 K φ = K/ cos φ That is to say, metric coefficient of round line which is inclined by φ is metric coefficient of round line which is a great circle divided by cos φ. 3. Case of three dimensions We consider the equidistancy about orthogonal circles in the direction of x and z on xz plane in Figure 5, however the principle is just the same as the case of two dimensions. It is supposed that metric coefficient of an orthogonal circle of which radius is a is K a. Coordinates of an intersection of two orthogonal circles in the direction of x and z of which radius are a and b respectively are x = 2ab 2 /(a 2 + b 2 ), z = 2a 2 b/(a 2 + b 2 ) Threfore l a = K a tan θ ab K a tan θ ab = K a (a/b a/b ) = K a a(1/b 1/b ) l a = K a tan θ a b K a tan θ a b = K a (a /b a /b ) = K a a (1/b 1/b ) From l a = l a K a a = K a a The relation is realized also in K φ in the case of two dimensions. Assuming radius of round line of which metric coefficient is 1 to be r u K a a = r u 2

If b becomes infty, it becomes Figure 6. z = l a = K a tan θ ab = (r u /a)(a/b) = r u /b = r u /(z /2) = 2r u /z z is a coordinate of which origin O is at infinity, namely a spherical orthogonal coordinate, and the superior index * is used in the sense of infinity. Next, we find the condition for coordinates being constant on inversion. To simplify the matter, we suppose that inversion is in xz plane. In Figure 7, x and x z and z are in the relation of inversion. Assuming radius of inversion to be R xx = R 2, zz = R 2 are realized. K tan θ = K(z/x) = z From the equation K = x Furthermore a = x /2 = R 2 /(2x) From the two equations Ka = R 2 /2 If metric coefficient and radius of round line are in such relation 3

z = K tan θ = z = R 2 /z is realized. In this case z = 2r u /z = 2r u /(R 2 /z) = z From the equation r u = R 2 /2 is realized. Length can be shown to be constant by setting up a coordinate axis which lies at right angles to a line which includes a line segment. To simplify the matter, we suppose that the line segment is in xz plane and the coordinate axis passes through the extreme point of the line segment. In Figure 8, the length of the line segment is v, and the length of the part of the round line corresponding to it is s. s = K tan θ = K(v/u) Because Kc = R 2 /2 is required K = R 2 /(2c) = R 2 /{2(u /2)} = R 2 /(R 2 /u) = u Therefore s = v In the case where a line which includes a line segment passes through origin of orthogonal coordinate system, setting up a coordinate axis on the line, length can be shown to be constant by previously stated coordinate constancy. 4

4. Scale A scale which measures a distance of two points is unique in orthogonal coordinate system. Wherever two points are, it is useful for measure. On the other hand, spherical orthogonal coordinate system is a coordinate system which is built into orthogonal coordinate system with origin and point at infinity swapped, and its scale is not unique. The scale expands and contracts. What contributes to the flexibility is metric coefficient. Because product of metric coefficient and radius of round line is constant, in two dimensions, it changes from metric coefficient of round line which is a great circle to, and in three dimensions, it approaches 0 limitlessly if radius of round line becomes, and it becomes if radius of round line approaches 0 limitlessly. 5. Concrete example By inversion of torus in orthogonal coordinate system, tubular surface of which the size of section changes continuously, namely Dupin s surface is obtained. It is based on a viewpoint of orthogonal coordinate system that the size of section changes continuously. Because coordinates, length and so forth are constant if a viewpoint of coordinate system of which origin is at infinity, namely spherical orthogonal coordinate system is used, we may be able to call the surface torus too. References: [1] Morio Kikuchi, Spherical orthogonal coordinate system (vixra:1103.0077, 2011) [2] Morio Kikuchi, On coordinate systems by use of spheres (the 1st,..., the 4th) (2009)(in Japanese) 5

**************************************************************************************** (3 ) : 1. 1 xz N N P s K s = K tan θ ds = K sec 2 θdθ 2. 2 2 2 x y 3 4 x y 4 3 N * 6

x = K tan θ x, y = K tan θ y l 1 = K tan θ 1 = K(y /K), l 2 = K φ tan θ 2 = K φ {y /(K/ cos φ)} l 1 = l 2 K φ = K/ cos φ φ K φ K cos φ 3. 3 5 xz x z 2 a K a x a z b x = 2ab 2 /(a 2 + b 2 ), z = 2a 2 b/(a 2 + b 2 ) l a = K a tan θ ab K a tan θ ab = K a (a/b a/b ) = K a a(1/b 1/b ) l a = K a tan θ a b K a tan θ a b = K a (a /b a /b ) = K a a (1/b 1/b ) l a = l a K a a = K a a 2 K φ 1 r u K a a = r u 7

b 6 z = l a = K a tan θ ab = (r u /a)(a/b) = r u /b = r u /(z /2) = 2r u /z z O * xz 7 x x z z R xx = R 2, zz = R 2 K tan θ = K(z/x) = z K = x a = x /2 = R 2 /(2x) Ka = R 2 /2 z = K tan θ = z = R 2 /z 8

z = 2r u /z = 2r u /(R 2 /z) = z r u = R 2 /2 xz 8 v s s = K tan θ = K(v/u) Kc = R 2 /2 K = R 2 /(2c) = R 2 /{2(u /2)} = R 2 /(R 2 /u) = u s = v 4. 2 2 9

2 3 0 0 5. Dupin : [1] (vixra:1103.0077, 2011) [2] ( 1... 4 ) (2009) 10