ABSTRACT
To characterize a specific point in space, for example, on the surface of a specimen, we necessitate coordinates; the simplest ones are Cartesian coordinates “length, width, height” being denoted by x, y, z or x1, x2, x3 (xi, i = 1, 2, 3), respectively. Figure 2.1 shows a (right-handed1) Cartesian coordinate system with the particular coordinates x0, y0, z0 of a spatial point P0(x0, y0, z0). The location of that point is known if the three figures x0, y0, and z0 are known under the assumption of an arbitrary but fixed coordinate origin and the arbitrary but fixed orientation of the coordinate axes. Figure 2.1 also displays that P0 can be equally characterized by the knowledge of cylindrical r0,ϕ0, z0 or spherical coordinates R0,ϑ0,ϕ0. The following coordinate transforms are immediately obvious:
x0 = r0 cosϕ0, y0 = r0 sinϕ0, (2.1) z0 = z0;
x0 = R0 sinϑ0 cosϕ0, y0 = R0 sinϑ0 sinϕ0, (2.2) z0 = R0 cosϑ0.
