Minimum Distance between curved surfaces Li Yajuan Oct.25,2006

Computation of the minimum distance between two objects is very important   Collision detection   Physical simulation in computer graphics   Animation   Virtual prototyping in haptic rendering   Robot motion planning and path modification   Computer games

Computation of the minimum distance between two polyhedra   A polygonal representation is not considered a true restriction since real objects can be approximated arbitrarily precisely by a polyhedron.   The basic algorithms and predicates can be implemented robustly and very efficiently on polygons.  Time: the number of polyhedral faces approximating the given curved surfaces is usually very large

Computation of the minimum distance between two curved surfaces   Ellipsoids and degenerate quadrics, such as cylinders and cones, are important primitives for solid modeling systems.   Ellipsoids can be used for efficiently bounding more general solids.   Bischoff and Kobbelt (02) developped techniques for approximating general objects by ellipsoids.

References   [1]Kim K-J. Minimum distance between a canal surface and a simple surface. CAD, 2003;35(10):871–9.   [2]Lennerz C, Schomer E. Efficient distance computation for quadratic curves and surfaces. In: Proceeding of Geometric Modeling and Processing p. 60–9.   [3]Sohn K-A, Juttler B, Kim M-S, Wang W. Computing distances between surfaces using line geometry. In: Pacific conference on computer graphics and applications p. 236–45.   [4]Chen Xiaodiao, etc.Computing minimum distance between two implicit algebraic surfaces. CAD, 2006; –1061.

Minimum distance between a canal surface and a simple surface [1] The minimum distance between two parametric surfaces F(u,v) and G(s,t) are described by Piegl(1995):

A Canal  A canal surface is defined by the trajectory of the center C(t) and the function determining the radius r(t).

A Canal  A canal surface is defined by the trajectory of the center C(t) and the function determining the radius r(t).

A general solution  Finding roots of a function of a single parameter of a necessary condition:

Distance between a canal surface and a plane

Distance between a canal surface and a sphere

Distance between a canal surface and a cylinder

Distance between a canal surface and a cone

Distance between a canal surface and a torus

Efficient distance computation for quadratic curves and surfaces [2].

Efficient distance computation for quadratic curves and surfaces.

Computing Distances Between Surfaces Using Line Geometry [3]  Using line geometry, the distance computation is reformulated as a simple instance of a surface-surface intersection problem, which leads to lowdimensional root finding in a system of equations.

Line Coordinates(Plucker,1846)

The normal congruence of a surface:

 Parameter representation

The normal congruence of a surface:  Implicit representation

The normal congruence of a surface:  Implicit representation

The normal congruence of a surface:

Distance Computation

 Distance between two ellipsoids;  Distance between an ellipsoid and a cylinder;  Distance between an ellipsoid and a cone;  Distance between an ellipsoid and a torus.

Experimental Results

Computing minimum distance between two implicit algebraic surfaces [4].

Computing minimum distance between two implicit algebraic surfaces.

Resultant method

Computing minimum distance between two implicit algebraic surfaces. Eliminate λandμ If S 1 is an implicit surface If S 1 is a parameter surface

Computing minimum distance between two implicit algebraic surfaces.

Algorithm

Minimum distance between a quadric surface and an implicit algebraic surface.  A cylinder and an implicit algebraic surface.

Minimum distance between a quadric surface and an implicit algebraic surface.  A cone and an implicit algebraic surface.

Minimum distance between a quadric surface and an implicit algebraic surface.  An elliptic paraboloid and an implicit algebraic surface.

Minimum distance between a quadric surface and an implicit algebraic surface.  An ellipsoid and an implicit algebraic surface.

Minimum distance between a quadric surface and an implicit algebraic surface.  A torus and an implicit algebraic surface.

Minimum distance between a canal and an implicit algebraic surface.

Minimum distance between two canal surfaces.

Minimum distance between two implicit surfaces.

comparison  [1]Kim K-J. Minimum distance between a canal surface and a simple surface. CAD, 2003;35(10):871–9.  [2]Lennerz C, Schomer E. Efficient distance computation for quadratic curves and surfaces. In: Proceeding of Geometric Modeling and Processing p. 60–9.  [3]Sohn K-A, Juttler B, Kim M-S, Wang W. Computing distances between surfaces using line geometry. In: Pacific conference on computer graphics and applications p. 236–45.  [4]Chen Xiaodiao, etc.Computing minimum distance between two implicit algebraic surfaces. CAD, 2006; –1061.

comparison

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