Non‐linear transformation of freeform curves and surfaces for CAD

Non‐linear transformation of freeform curves and surfaces is useful in computer‐aided design and computer graphics. It is highly desirable that the original and transformed curves or surfaces are defined using the same representation. But freeform curves and surfaces defined by control points are invariant only under affine transformations, and not so under non‐linear transformations. This paper develops a method that can perform non‐linear transformations of freeform curves to specific accuracies, while retaining the same representation. It involves first applying the transformation to the control points and then modifying them so that the resulting curve and the exact transformed curve are equal at a specific number of points, which is the number of control points. Refinement to the approximation is made by increasing the number of control points. A method for measuring the maximum positional error has been implemented and this is used to facilitate an algorithm for automatic refinement. Extension of the method for surfaces is also given.