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Rational surfaces with linear normals and their convolutions with rational surfaces

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Title: Rational surfaces with linear normals and their convolutions with rational surfaces


1
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

2
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

3
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

4
About the authors
  • Bert Jüttler, Austria
  • J. Kepler Universität Linz
  • Research Interests
  • Computer Aided Geometric Design (CAGD)
  • Applied Geometry
  • Kinematics, Robotics
  • Differential Geometry

5
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.

6
Introduction(1)
  • LN surfaces
  • Some geometric properties
  • Its dual representation

7
Introduction(2)
  • Convolution surfaces
  • Computation of convolution surfaces
  • Convolution of LN surfaces and rational surfaces

8
LN surface
  • Linear normal vector field
  • Model free-form surfaces Juttler and Sampoli
    2000
  • Main advantageous LN surfaces possess exact
    rational offsets.

9
Definition
  • LN surface
  • a polynomial surface p(u,v) with Linear
  • Normal vector field
  • certain constant coefficient vectors

10
Properties(1)
  • Obviously
  • Assume
  • That is

11
Properties(2)
  • Tangent plane of LN surface p(u, v)
  • where

12
Computation
  • given a system of tangent planes
  • Then,the envelope surface
  • is a LN surface.
  • The normal vector

13
Geometric property
  • Gaussian curvature of the envelope

14
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.

15
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.

16
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.

17
The dual representation
  • Graph surface LN surface
  • q(u,v) p(u,v)
  • elliptic
    elliptic
  • hyperbolic hyperbolic
  • parabolic singular points

18
Convolution surfaces and Minkowski sums
  • Application
  • Computer Graphics
  • Image Processing
  • Computational Geometry
  • NC tool path generation
  • Robot Motion Planning
  • ??,???,???.??????????????, Computer
    Applications, June 2006

19
Definition
  • Given two objects P,Q in , then
  • Minkowski sum

20
Definition
  • Given two surfaces A,B in ,then
  • Convolution surface

21
Relations between them
  • In general,
  • In particular, if P and Q are convex sets
  • Where,

22
Kinematic generation(1)
23
Kinematic generation(1)
24
Kinematic generation(1)
25
Kinematic generation(1)
26
Kinematic generation(1)
27
Kinematic generation(2)
28
Kinematic generation(2)
29
Kinematic generation(2)
30
Convolution surfaces of general rational surfaces
  • Two surfaces Aa(u,v) , Bb(s,t)
  • parameter domains OA, OB.
  • unit normal vectors , .

31
Convolution of general rational surfaces
  • Reparameterization
  • such that
  • Where, .

32
Convolution surfaces of general rational surfaces
  • Then,
  • is a parametric representation of the
    convolution surface of

33
Convolution of LN surfaces and rational surfaces
  • Assumed
  • LN-surface A
  • rational surface B

34
Convolution of LN surfaces and rational surfaces
  • If correspond, that is
  • Then,

35
Convolution of LN surfaces and rational surfaces
  • So,
  • That is
  • Where

36
Convolution of LN surfaces and rational surfaces
  • The parametric representation c(s, t) of the
    convolution C A?B

37
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.

38
Example
39
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.

40
  • Thank you !
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