Thomas W. Sederberg

Free-form deformation of solid geometric models

A technique is presented for deforming solid geometric models in a free-form manner. The technique can be used with any solid modeling system, such as CSG or B-rep. It can deform surface primitives of any type or degree: planes, quadrics, parametric surface patches, or implicitly defined surfaces, for example. The deformation can be applied either globally or locally. Local deformations can be imposed with any desired degree of derivative continuity. It is also possible to deform a solid model in such a way that its volume is preserved.The scheme is based on trivariate Bernstein polynomials, and provides the designer with an intuitive appreciation for its effects.

T-spline simplification and local refinement

A typical NURBS surface model has a large percentage of superfluous control points that significantly interfere with the design process. This paper presents an algorithm for eliminating such superfluous control points, producing a T-spline. The algorithm can remove substantially more control points than competing methods such as B-spline wavelet decomposition. The paper also presents a new T-spline local refinement algorithm and answers two fundamental open questions on T-spline theory.

Conversion of complex contour line definitions into polygonal element mosaics

A simple algorithm is presented for processing complex contour arrangements to produce polygonal element mosaics which are suitable for line drawing and continuous tone display. The program proceeds by mapping adjacent contours onto the same unit square and, subject to ordering limitations, connecting nodes of one contour to their nearest neighbors in the other contour. While the mapping procedure provides a basis for branching decisions, highly ambiguous situations are resolved by user interaction. The program was designed to interface a contour definition of the components of a human brain. These brain data are a most complex definition and, as such, serve to illustrate both the capabilities and limitations of the procedures.

A physically based approach to 2–D shape blending

This paper presents a new afgorithm for smoothly blending between two 2-D polygonal shapes. The algorithm is based on a physical model wherein one of the shapes is considered to be constructed of wire, and a solution is found whereby the first shape can be bent and/or stretched into the second shape with a minimum amount of work. The resulting solution tends to associate regions on the two shapes which look alike. If the two polYgons have m and n vertices respectively, the afgorithm is O(mn). The algorithm avoids local shape inversions in whkh intermediate polygons self-intersect, if such a solution exists.

Implicit representation of parametric curves and surfaces

Abstract The following two problems are shown to have closed-form solutions requiring only the arithmetic operations of addition, subtraction, multiplication and division: (1) Given a curve or surface defined parametrically in terms of rational polynomials, find an implicit polynomial equation which defines the same curve or surface. (2) Given the Cartesian coordinates of a point on such a curve or surface, find the parameter(s) corresponding to that point. It is shown that a two-dimensional curve defined parametrically in terms of rational degree n polynomials in t can be expressed implicitly as a degree n polynomial in z and y. It is also demonstrated that a “bi-m-ic” parametric surface (where e.g., m = 3 for bicubic) can be expressed implicitly as a polynomial in x, y, z of degree 2m2. The degree of a rational bi-m-ic surface is also shown to be 2m2. The application of these results to finding curve and surface intersections is discussed.

2-D shape blending: an intrinsic solution to the vertex path problem

This paper presents an algorithmfor determiningthe paths along which corresponding vertices travel in a 2–D shape blending. Rather than considering the vertex paths explicitly, the algorithm defines the intermediate shapes by interpolating the intrinsic definitions of the initial and final shapes. The algorithm produces shape blends which generally are more satisfactory than those produced using linear or cubic curve paths. Particularly, the algorithm can avoid the shrinkage that normally occurs when rotating rigid bodies are linearly blended, and avoids kinks in the blend when there were none in the key polygons.

Ray tracing trimmed rational surface patches

This paper presents a new algorithm for computing the points at which a ray intersects a rational Bézier surface patch, and also an algorithm for determining if an intersection point lies within a region trimmed by piecewise Bézier curves. Both algorithms are based on a recent innovation known as Bézier clipping, described herein. The intersection algorithm is faster than previous methods for which published performance data allow reliable comparison. It robustly finds all intersections without requiring special preprocessing.

Non-uniform recursive subdivision surfaces

Doo-Sabin and Catmull-Clark subdivision surfaces are based on the notion of repeated knot insertion of uniform tensor product B-spline surfaces. This paper develops rules for non-uniform Doo-Sabin and Catmull-Clark surfaces that generalize non-uniform tensor product Bspline surfaces to arbitrary topologies. This added flexibility allows, among other things, the natural introduction of features such as cusps, creases, and darts, while elsewhere maintaining the same order of continuity as their uniform counterparts.

Implicitization using moving curves and surfaces

This paper presents a radically new approach to the century old problem of computing the implicit equation of a parametric surface. For surfaces without base points, the new method expresses the implicit equation in a determinant which is one fourth the size of the conventional expression based on Dixon’s resultant. If base points do exist, previous implicitization methods either fail or become much more complicated, while the new method actually simplifies.

Loop detection in surface patch intersections

Abstract A condition is presented for guaranteeing that all branches of the curve of intersection of two parametric surfaces patches contain a point on at least one of the patch boundary curves. This is of value because it eliminates a robustness limitation which arises when computing surface intersections using the marching method, namely, assuring that all branches of the intersection curve have been found. The intersection curve of any two surface patches meeting the prescribed condition can be computed by locating points where patch boundary curves intersect the opposing patch, and then using those points as starting points for the marching algorithm.