3D coordiante sytems

In a 3D Cartesian coordinate system, a point P is referred to by three real numbers (coordinates), indicating the positions of the perpendicular projections from the point to three fixed, perpendicular, graduated lines, called the axes which intersect at the origin. Often the x-axis is imagined to be horizontal and pointing roughly toward the viewer (out of the page), the y-axis is also horizontal and pointing to the right, and the z-axis is vertical, pointing up. The system is called right-handed if it can be rotated so that the three axes are in the position as shown in the figure above. The x-coordinate of of the point P in the figure is a, the y-coordinate is b, and the z-coordinate is c.


To define a cylindrical coordinate system, we take an axis (usually called the z-axis) and a perpendicular plane, on which we choose a ray (the initial ray) originating at the intersection of the plane and the axis (the origin). The coordinates of a point P are the polar coordinates (r, f) of the projection of P on the plane, and the coordinate z of the projection of P on the z-axis. The coordinate r is always positive and the range of f is from 0 to 2p (360o).

To transform from Cartesian to cylindrical coordinates and vice versa, we use the transformation equations

x = r cosf, y = r sinf, z = z,

r = (x2 + y2)1/2, f = tan-1(y/x), z = z.


To define spherical coordinates, we take an axis (the polar axis) and a perpendicular plane (the equatorial plane), on which we choose a ray (the initial ray) originating at the intersection of the plane and the axis (the origin O). The coordinates of a point P are the distance r from P to the origin; the angle q (zenith) between the line OP and the positive polar axis; and the angle f (azimuth) between the initial ray and the projection of OP onto the equatorial plane. The range of f is from 0 to 2p (360o)., and the range of q is from 0 to p (180o).

To transform from Cartesian to spherical coordinates and vice versa, we use the transformation equations

x = r sinq cosf, y = r sinq sinf, z = r cosq,

r = (x2 + y2 + z2)1/2, q =tan-1(z/(x2+y2)1/2), f = tan-1(y/x).