Reverse the other coordinate map, and the inverse coordinate transformation results.

The composite map corresponds to the coordinate transformation equations:



Example

Suppose is a Cartesian coordinate map on a three dimensional space for which Euclidean geometry works. Choose a point and use Euclidean geometry to build a polar coordinate map about that point.

From the right triangles in the diagram and a little trigonometry
These equations provide the Cartesian coordinates (x,y,z) of the point .

Another to way to say that is that they provide .

Still another way to say it is that they provide

The equations provide the coordinate transformation from polar coordinates to Cartesian coordinates.

Now invert the equations to obtain the coordinate transformation from Cartesian coordinates to polar coordinates.

Notice that the inverse transformation fails for the Cartesian Coordinates x = x1, y = y1, z = z1, . The point labeled by that set of Cartesian coordinates does not have polar coordinates. Similarly, the inverse trigonometric functions are multiple valued unless the ranges of the angles are restricted:

Those restrictions mean that a half-plane of points with Cartesian coordinates x > x1 and y = y1 do not have polar coordinates.
Note that this example does not use the polar coordinate angle definitions that are normally used in physics applications.