rational surfaces with linear normals and their convolutions with rational surfaces maria lucia...
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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(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
Computation
given a system of tangent planes
Then,the envelope surface
is a LN surface. The normal vector
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
Computer Graphics Image Processing Computational Geometry NC tool path generation Robot Motion Planning 何青 , 仝明磊 , 刘允才 . 用卷积曲面生成脸部皱纹的
方法 , Computer Applications, June 2006
Convolution surfaces of general rational surfaces
Two surfaces A=a(u,v) , B=b(s,t) parameter domains ΩA, ΩB.
unit normal vectors , .
Then,
is a parametric representation of the convolution surface of
Convolution surfaces of general 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
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.