ELIMINATING EXTRANEOUS SOLUTIONS IN CURVE AND SURFACE OPERATIONS

We study exact representations for offset curves and surfaces, for equal-distance curves and surfaces, mid for fixed- and variable-radius blending surfaces. The representations are systems of nonlinear equations that define the curves and surfaces as natural projections from a higher-dimensional space into 3-space. We show that the systems derived by naively translating the geometric constraints defining the curves and surfaces can entail degeneracies that result in additional solutions that have no geometric significance. We characterize these extraneous solution points geometrically, and then augment the systems with auxiliary equations of a uniform structure that exclude all extraneous solutions. Thereby, we arrive at representations that capture the geometric intent of the curve and surface definitions precisely.