Geometric modeling
Curved surfaces and coherence for non-penetrating rigid body simulation
SIGGRAPH '90 Proceedings of the 17th annual conference on Computer graphics and interactive techniques
Interval analysis for computer graphics
SIGGRAPH '92 Proceedings of the 19th annual conference on Computer graphics and interactive techniques
Interval methods for multi-point collisions between time-dependent curved surfaces
SIGGRAPH '93 Proceedings of the 20th annual conference on Computer graphics and interactive techniques
Free form surface analysis using a hybrid of symbolic and numeric computation
Free form surface analysis using a hybrid of symbolic and numeric computation
Computation of the solutions of nonlinear polynomial systems
Computer Aided Geometric Design
Direct haptic rendering of sculptured models
Proceedings of the 1997 symposium on Interactive 3D graphics
3D scan conversion of CSG models into distance volumes
VVS '98 Proceedings of the 1998 IEEE symposium on Volume visualization
Adaptively sampled distance fields: a general representation of shape for computer graphics
Proceedings of the 27th annual conference on Computer graphics and interactive techniques
Spatialized normal come hierarchies
I3D '01 Proceedings of the 2001 symposium on Interactive 3D graphics
The convex Hull of Rational Plane Curves
Graphical Models
Shape Interrogation for Computer Aided Design and Manufacturing
Shape Interrogation for Computer Aided Design and Manufacturing
Problem Reduction to Parameter Space
Proceedings of the 9th IMA Conference on the Mathematics of Surfaces
Signed Distance Transform Using Graphics Hardware
Proceedings of the 14th IEEE Visualization 2003 (VIS'03)
Minimum distance queries for haptic rendering
Minimum distance queries for haptic rendering
Distance extrema for spline models using tangent cones
GI '05 Proceedings of Graphics Interface 2005
The convex hull of freeform surfaces
Computing - Geometric modelling dagstuhl 2002
Intersecting a freeform surface with a general swept surface
Computer-Aided Design
Computing surface offsets and bisectors using a sampled constraint solver
Proceedings of Graphics Interface 2009
Critical point analysis using domain lifting for fast geometry queries
Computer-Aided Design
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We present an efficient and robust algorithm for computing the minimum distance between a point and freeform curve or surface by lifting the problem into a higher dimension. This higher dimensional formulation solves for all query points in the domain simultaneously, therefore providing opportunities to speed computation by applying coherency techniques. In this framework, minimum distance between a point and planar curve is solved using a single polynomial equation in three variables (two variables for a position of the point and one for the curve). This formulation yields two-manifold surfaces as a zero-set in a 3D parameter space. Given a particular query point, the solution space's remaining degrees-of-freedom are fixed and we can numerically compute the minimum distance in a very efficient way. We further recast the problem of analyzing the topological structure of the solution space to that of solving two polynomial equations in three variables. This topological information provides an elegant way to efficiently find a global minimum distance solution for spatially coherent queries. Additionally, we extend this approach to a 3D case. We formulate the problem for the surface case using two polynomial equations in five variables. The effectiveness of our approach is demonstrated with several experimental results.