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Rational surfaces with linear normals and their

convolutions with rational surfaces

- Maria Lucia Sampoli, Martin Peternell, Bert

Jüttler - Computer Aided Geometric Design 23 (2006)

179192 - Reporter Wei Wang
- Thursday, Dec 21, 2006

About the authors

- Marai Lucia Sampoli, Italy
- Università degli Studi di Siena
- Dipartimento di Scienze Matematiche ed

Informatiche - http//www.mat.unisi.it/newsito/docente.php?id32

About the authors

- Martin Peternell, Austria
- Vienna University of Technology
- Research Interests
- Classical Geometry
- Computer Aided Geometric Design
- Reconstruction of geometric objects from dense 3D

data - Geometric Modeling and Industrial Geometry

About the authors

- Bert Jüttler, Austria
- J. Kepler Universität Linz
- Research Interests
- Computer Aided Geometric Design (CAGD)
- Applied Geometry
- Kinematics, Robotics
- Differential Geometry

Previous related work

- Jüttler, B., 1998. Triangular Bézier surface

patches with a linear normal vector field. In

The Mathematics of Surfaces VIII. Information

Geometers, pp. 431446. - Jüttler, B., Sampoli, M.L., 2000. Hermite

interpolation by piecewise polynomial surfaces

with rational offsets. CAGD 17, 361385. - Peternell, M., Manhart, F., 2003. The convolution

of a paraboloid and a parametrized surface. J.

Geometry Graph. 7, 157171. - Sampoli, M.L., 2005. Computing the convolution

and the Minkowski sum of surfaces. In

Proceedings of the Spring Conference on Computer

Graphics, Comenius University, Bratislava. ACM

Siggraph, in press.

Introduction(1)

- LN surfaces
- Some geometric properties
- Its dual representation

Introduction(2)

- Convolution surfaces
- Computation of convolution surfaces
- Convolution of LN surfaces and rational surfaces

LN surface

- Linear normal vector field
- Model free-form surfaces Juttler and Sampoli

2000 - Main advantageous LN surfaces possess exact

rational offsets.

Definition

- LN surface
- a polynomial surface p(u,v) with Linear
- Normal vector field
- certain constant coefficient vectors

Properties(1)

- Obviously
- Assume
- That is

Properties(2)

- Tangent plane of LN surface p(u, v)
- where

Computation

- given a system of tangent planes
- Then,the envelope surface
- is a LN surface.
- The normal vector

Geometric property

- Gaussian curvature of the envelope

Geometric property

- K gt 0 elliptic points,
- K lt 0 hyperbolic points,
- If the envelope possesses both, the corresponding

domains are separated by the singular curve C.

The dual representation

- A polynomial or rational function f
- the LN-surfaces p (u,v)
- the associated graph surface
- q(u,v) is dual to LN-surface in the sense of

projective geometry.

The dual representation

- Since det(H) of q(u,v)
- So,
- det(H)gt0 elliptic points,
- det(H)0 parabolic points,
- det(H)lt0 hyperbolic points.

The dual representation

- Graph surface LN surface
- q(u,v) p(u,v)
- elliptic

elliptic - hyperbolic hyperbolic
- parabolic singular points

Convolution surfaces and Minkowski sums

- Application
- Computer Graphics
- Image Processing
- Computational Geometry
- NC tool path generation
- Robot Motion Planning
- ??,???,???.??????????????, Computer

Applications, June 2006

Definition

- Given two objects P,Q in , then
- Minkowski sum

Definition

- Given two surfaces A,B in ,then
- Convolution surface

Relations between them

- In general,
- In particular, if P and Q are convex sets
- Where,

Kinematic generation(1)

Kinematic generation(1)

Kinematic generation(1)

Kinematic generation(1)

Kinematic generation(1)

Kinematic generation(2)

Kinematic generation(2)

Kinematic generation(2)

Convolution surfaces of general rational surfaces

- Two surfaces Aa(u,v) , Bb(s,t)
- parameter domains OA, OB.
- unit normal vectors , .

Convolution of general rational surfaces

- Reparameterization
- such that
- Where, .

Convolution surfaces of general rational surfaces

- Then,
- is a parametric representation of the

convolution surface of

Convolution of LN surfaces and rational surfaces

- Assumed
- LN-surface A
- rational surface B

Convolution of LN surfaces and rational surfaces

- If correspond, that is
- Then,

Convolution of LN surfaces and rational surfaces

- So,
- That is
- Where

Convolution of LN surfaces and rational surfaces

- The parametric representation c(s, t) of the

convolution C A?B

Convolution of LN surfaces and rational surfaces

- The convolution surface A?B of an
- LN-surface A and a parameterized surface B has

an explicit parametric representation. - If A and B are rational surfaces, their

convolution A?B is rational, too.

Example

Conclusion and further work

- To our knowledge, this is the first result on

rational convolution surfaces of surfaces which

are capable of modeling general free-form

geometries. - This result may serve as the starting point for

research on computing Minkowski sums of general

free-form objects.

- Thank you !