Change of variables

Changing variables is a useful tool that appears in many guises in computer graphics and geometric modeling. Changing the variables allows us to change the way we traverse a curve or surface, change their derivatives or place or interpret textures and other properties associated with them. In calculus, changing variables miraculously converts hard integration problems into simple standard ones and in algebra, seemingly hard root finding problems turn into an easy exercises. Even change of coordinates can be interpreted as a change of variables, albeit linear and therefore not our focus below.