Rational surfaces with linear normals and their convolutionswith rational surfaces

Maria Lucia Sampoli, Martin Peternell, Bert Jüttler Computer Aided Geometric Design 23 (2006) 179–192

Reporter: Wei WangThursday, Dec 21, 2006

About the authors

Marai Lucia Sampoli, Italy Università degli Studi di Siena Dipartimento di Scienze Matematic

he ed Informatiche http://

www.mat.unisi.it/newsito/docente.php?id=32

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 Lin

z 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. 431–446.

Jüttler, B., Sampoli, M.L., 2000. Hermite interpolation by piecewise polynomial surfaces with rational offsets. CAGD 17, 361–385.

Peternell, M., Manhart, F., 2003. The convolution of a paraboloid and a parametrized surface. J. Geometry Graph. 7, 157–171.

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 Sam

poli 2000] Main advantageous LN surfaces posse

ss exact rational offsets.

Definition

LN surface a polynomial surface p(u,v) with Linea

r 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 > 0 elliptic points,

K < 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)>0 elliptic points, det(H)=0 parabolic points, det(H)<0 hyperbolic points.

2

2 2det

1uu vv uv

u v

f f fH

f f

The dual representation

Graph surface LN surface q(u,v) p(u,v) elliptic elliptic hyperbolic hyperbolic parabolic singular points

dual to

Convolution surfaces and Minkowski sums Application

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方法 , Computer Applications, June 2006

Definition

Given two objects P,Q in , then

Minkowski sum

3R

Definition

Given two surfaces A,B in ,then

Convolution surface

3R

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 A=a(u,v) , B=b(s,t) parameter domains ΩA, ΩB.

unit normal vectors , .

Reparameterization

such that

Where, .

Convolution of generalrational surfaces

∥

Then,

is a parametric representation of the convolution surface of

Convolution surfaces of general 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 s

urface B has an explicit parametric representation.

If A and B are rational surfaces, their convolution A★B is rational, too.

Convolution of LN surfaces and rational surfaces

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 !