Topological and differential-equation methods for surface intersections

Abstract

An implicit representation of the intersection of two parametric surface patches is formulated in terms of the zero level set of the oriented distance function of one surface from the other. The topological properties of the vector field defined as the gradient of the oriented distance function (and specifically the concepts of the rotation number of a vector field and the Poincaré index theory) are next used to formulate a condition for the detection of critical points of the field, which may be used to identify internal loops and singularities of the solution. An adaptive search guided by this condition and direct numerical techniques are used to detect and compute critical points. Tensorial differential equations that rely on the implicit representation of the intersection are then developed, and are used to trace intersection segments. The tracing scheme relies on a characterization procedure for singular points that allows the computation of the tangent directions to the intersection curve at such points. The method also allows the tracing of degenerate intersections between surfaces that have a common tangent plane along the intersection curve. Examples drawn from rational spline surface intersections illustrate the above techniques.