Representation of implicit curves for procedural surfaces

摘要

Compact and accurate piecewise parametric representations of implicit functions may be achieved by iteratively selecting ranges of parameterizing regions and testing each for satisfying an intervalized super convergence test. In one aspect, the implicit function is represented as a compact form of one or more representations of such convergence regions. In yet another aspect, iteration is begun with applying the intervalized convergence test to an entire pameterization region. In yet another aspect, the range being tested for super convergence is iteratively sub-divided to generate other ranges for testing. In one aspect, such sub-dividing comprises dividing the selected ranges by half. In one further aspect, Newton iterate steps are applied to selected ranges to change such ranges for further testing of super convergence of such ranges. Parametric representations that use such representations of convergence regions to express implicit functions consume far less memory for storage than conventional representations. Algorithms are described herein for quickly calculating such representations.