Rational surfaces with linear normals and their convolutions with rational surfaces

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Rational surfaces with linear normals and their
convolutions with rational surfaces

• Maria Lucia Sampoli
• Martin Peternell
• Bert Jüttler

Reporter Xingwang Zhang
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Authors

• Maria Lucia Sampoli
• Università di Siena, Dipartimento di Scienze
  Matematiche ed Informatiche, Italy
• Martin Peternell
• Technische Universität Wien, Inst. für Diskrete
  Mathematik und Geometrie, Austria
• Bert Jüttler
• J. Kepler Universität Linz, Institut für
  Angewandte Geometrie, Austria

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Topics

• Polynomial (or rational) parametric surfaces with
  a linear field of normal vectors
• dual to graphs bivariate polynomials (or rational
  functions)
• the convolution with general rational surfaces
  yields again rational surfaces

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Minkowski sums

• Given two sets and
• the Minkowski sums
• the Minkowski sum is the union of all
  translations of by a point defined within
  

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Minkowski sums
Minkowski sum of two planar domains and the
convolution of their boundaries
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Minkowski sums
Minkowski sum of a ball and a cube
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Convolution surfaces
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Kinematic generation of convolutions
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Preliminaries

• A polynomial(or rational) surface
  
• is said to LN surface if

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Preliminaries
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LN surface equation

• The tangent surface of an LN surface
• On the other hand, if satisfies
• the envelop surface
• is a LN surface

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LN surface equation

• Normal vector
• Singular points
• Gaussian curvature
• The quotient of the determinants of fundamental
  forms

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LN surface equation

• If hyperbolic points
• If elliptic points
• Singular curve composed of singular
• points
• in the parameter domain

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Dual representation

• is dual to the graph surface
• are elliptic, parabolic or
  hyperbolic, if the sign of
  is 1, 0, -1, respectively
• Corollary There is a correspondence between
  and

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Example
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Construction of LN surfaces
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Convolution of LN surfaces and rational surfaces
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Conclusion

• A class of free form surfaces
• Rational convolution surfaces with general
  rational surfaces
• This is first result on rational convolution
  surfaces
• Computation of Minkowski sum
• Generalization to LN-curves and rational curves

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• Thank you, everybody!