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On
Geometry and Optimization I
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Organizers:
Francisco Tovar & Marco Paluszny
Speakers:
Marco Paluszny, Jonnathan Otero,
& Miguel Martin-Landrove
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| Abstract
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Since early 2008, the
Laboratorio de Computacion Grafica
y Geometria Aplicada (CGGA) and the
Computacion Cientfica research group
of the Universidad Nacional de Colombia,
Sede Medellin have been collaborating
in a series of themes concerning geometry
and optimization with a strong emphasis
on applications. In this minisymposium
we will be presenting results concerning
the following topics:
- Using lemniscates to construct orthogonal
grids on meander-like regions
- Using envelopes of 1-parameter families
to construct path splines
- The Prony method in the detection
of tissue types in brain
- Construction of developable low
degree patches
- Models of molecular surfaces
- Distribution of circles along a
cyclide. |
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Marco
Paluszny
Path Splines with Envelope Conics |
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Given a sequence of
point-line pairs on the plane we produce
a G1 path spline that passes
through the sequence of points and
has the prescribed contacts by the
given lines at these points. Each
segment of the spline is a section
of the envelope of a one parameter
family of conics and it is composed
of points which are the intersections
of infinitesimally near conics belonging
to the one parameter family. The conics
comprise the real projective space
P5. Lines P5
correspond pencils and these are characterized
by their base points i.e., the common
intersection points of all the conics
of a pencil. A curve in P5
corresponds to a one parameter family
of conics in the plane and conics
in correspond to quadratic families
of conics in the plane. We consider
conditions on a quadratic Bézier
curve for its envelope to consist
of pairs of points and satisfy the
initial contact conditions by the
point-line pairs at the endpoints.
A quadratic spline in P5
produces a G1 path spline
on the plane which consists of sectors
of envelopes which are joined smoothly.
Each envelope has algebraic degree
four. Our main tool is the classification
of pencils of conics for conditions
on these guarantee that the envelope
consists of pairs of points. |
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Jonnathan
Otero
Distribution of Circles Along a Cyclide
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