Sweeping of three-dimensional objects
Computer-Aided Design
Curves and surfaces for computer aided geometric design (3rd ed.): a practical guide
Curves and surfaces for computer aided geometric design (3rd ed.): a practical guide
Computation of the solutions of nonlinear polynomial systems
Computer Aided Geometric Design
Fundamentals of computer aided geometric design
Fundamentals of computer aided geometric design
The NURBS book
Polynomial/rational approximation of Minkowski sum boundary curves
Graphical Models and Image Processing
Geometric constraint solver using multivariate rational spline functions
Proceedings of the sixth ACM symposium on Solid modeling and applications
Geometric modeling with splines: an introduction
Geometric modeling with splines: an introduction
The convex Hull of Rational Plane Curves
Graphical Models
An algebraic condition for the separation of two ellipsoids
Computer Aided Geometric Design
The intersection of two ringed surfaces and some related problems
Graphical Models
Shape Interrogation for Computer Aided Design and Manufacturing
Shape Interrogation for Computer Aided Design and Manufacturing
Computing Point/Curve and Curve/Curve Bisectors
Proceedings of the 5th IMA Conference on the Mathematics of Surfaces
Problem Reduction to Parameter Space
Proceedings of the 9th IMA Conference on the Mathematics of Surfaces
Precise Voronoi cell extraction of free-form rational planar closed curves
Proceedings of the 2005 ACM symposium on Solid and physical modeling
Two-Dimensional Visibility Charts for Continuous Curves
SMI '05 Proceedings of the International Conference on Shape Modeling and Applications 2005
Perspective silhouette of a general swept volume
The Visual Computer: International Journal of Computer Graphics
Intersecting a freeform surface with a general swept surface
Computer-Aided Design
Computing roots of polynomials by quadratic clipping
Computer Aided Geometric Design
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We review a family of related techniques for geometric computations in the parameter space of freeform curves and surfaces. Geometric constraint equations for freeform curves and surfaces have low degrees (often linear or quadratic) in x,y,z and considerably higher degrees in the curve or surface parameters. We eliminate x,y, and z, so that the constraints are expressed in terms of the curve or surface parameters, while making the variables x,y,z the functions of these parameters under those same constraints. It is relatively straightforward to compute the differential geometric properties of many constructs using this representation. We have successfully addressed the following classes of computation for freeform curves and surfaces: Minkowski sums, bisectors and α-sectors, surface-surface intersections, collision detection, offset trimming, swept volume computation, constructing Voronoi diagrams, convex hulls and kernels, silhouette, and visibility computations. We provide a few simple examples to demonstrate how to apply this technique to a variety of problems in geometric computation.