Changing Variables

In computer graphics and geometric modeling, changing the variables is a useful technique that appears in many guises. It lets us change how we traverse curves or surfaces, modify their derivatives or place, and interpret textures or other properties associated with them. It is shown that while changing variables of a parametric curve or surface leaves its geometric shape untouched, changing the way we associate parameters to curves or surfaces does have an impact: computer graphics deals with sampled geometry, and the sampled geometry changes. We can take advantage of the change of variables to improve how the surface is textured. Moreover, change of variables makes it possible to combine curve segments or curved surface patches into a smooth ensemble when the shape is more general than the graph of a function.