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Adapted and directed rotation-minimizing frames on
space curves: theory, algorithms, and applications |
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Abstract: |
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Two types of
rotation-minimizing orthonormal frames are defined for twisted space
curves: the adapted frame, in which one frame vector coincides with
the unit tangent vector of the curve, and the directed frame, in
which one frame vector coincides with the unit polar vector from the
origin to each curve point. These orthonormal frames are
rotation-minimizing in the sense that their angular velocity vectors
have vanishing components along the curve tangent and polar vectors,
respectively. They are also intimately connected: the directed frame
coincides with the adapted frame for the anti-hodograph (indefinite
integral) of the specified curve. After presenting the basic theory
of these frames, we describe algorithms for their computation, and
applications to path planning, animation, swept surface
construction, and camera motion control. |
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Title: |
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Think globally, act locally:
Recent trends in geometry processing |
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Abstract: |
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Many problems in discrete geometry processing can be posed as global
optimization problems, frequently non-linear. Seemingly hopeless to
solve, they typically involve ony local interactions because of the
differential nature of the cost function. Thus efficient algorithms
may be designed to take advantage of this by solving a large number
of small local problems, and then merging these into one coherent
global solution.
The speaker will provide a overview of this general approach, and
demonstrate how it may be applied to solve a number of seemingly
unrelated problems in geometry processing. |
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Title: |
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An evaluation based approach to deal with
intersection problems including topology determination and offset
manipulation |
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Abstract: |
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Many algebraic techniques potentially
useful when dealing with intersection problems for curves and
surfaces in Computer-Aided Geometric Design suffer a very
serious lack of efficiency and stability due to its strong
dependence on the presentation of the curves and surfaces to
intersect in terms of the monomial basis.
In this talk, by using a new formulation
of the classical Bezout matrix, we construct matrix polynomials
expressed in a tensor-product Lagrange or Hermite basis to be
used to solve some common tasks in Computer-Aided Geometric
Design: these matrix polynomials will serve as stable and
efficient implicit representations for curves and surfaces for a
variety of intersection problems including topology
determination and offset manipulation.
The algebraic framework in this approach is based on matrix
polynomials whose computation depends only on the evaluation of
the curve and surface representation at some points (and its
derivatives if needed or available) and not on the availability
of the representation of the involved polynomials in the
monomial basis. The polynomial system solving tasks needed in
this approach depend only on well stablished numerical linear
algebra techniques like solving generalized eigenvalue problems
or determining singular value decompositions.
This is a joint work with D. A. Aruliah,
R. M. Corless, I. F. Rua and A. Shakoori.
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GMP 2008 : : Geometric Modeling and Processing 2008 April 23-25, 2008, Hangzhou, China
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