Minimum Distance between curved surfaces

Li YajuanLi Yajuan

Oct.25,2006Oct.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. 2002. 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. 2002. p. 236–45.

[4]Chen Xiaodiao, etc.Computing minimum distance between two implicit algebraic surfaces. CAD, 2006; 38 1053–1061.

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

A CanalA 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 plane

Distance between a canal surface and a sphere

Distance between a canal surface and a sphere

Distance between a canal surface and a cylinder

Distance between a canal surface and a cylinder

Distance between a canal surface and a cone

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[2].

Efficient distance computation for quadratic curves and surfaces.

Efficient distance computation for quadratic curves and surfaces.

Efficient distance computation for quadratic curves and surfaces.

Efficient distance computation for quadratic curves and surfaces.

Efficient distance computation for quadratic curves and surfaces.

Efficient distance computation for quadratic curves and surfaces.

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 lowdimensionalroot finding in a system of equations.

Line Coordinates(Plucker,1846)

The normal congruence of a surface:

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 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

Experimental Results

Experimental Results

Experimental Results

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

Computing minimum distance between two implicit algebraic surfaces.

Computing minimum distance between two implicit algebraic surfaces.

Resultant method

Computing minimum distance between two implicit algebraic surfaces.

Eliminate λandμ

If S1 is an implicit surface

If S1 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.

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. 2002. 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. 2002. p. 236–45.

[4]Chen Xiaodiao, etc.Computing minimum distance between two implicit algebraic surfaces. CAD, 2006; 38 1053–1061.

comparison

The End!The End!