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Mini-Workshops Accepted
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On
Geometry and Optimization I
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Organizers:
Francisco Tovar & Marco Paluszny
Speakers:
Adriana Padron, Marianela Lentini,
& Mauricio Londao
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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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Jonnathan
Otero
Distribution of Circles Along a Cyclide |
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Adriana
Padron
Modeling of the Van Der Waals Surface
Using Triangular Patches |
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In the field of geometric
modeling the approximation of surfaces
with triangular meshes is a very popular
technique for the visualization and
the simulation of three dimensional
objects. This allows for the object's
3D display using standard graphical
interfaces systems such as those in
Matlab® and Maple®. However,
this representation does not provide
for the interactive deformation of
the object.
A possible solution to this problem
is the representation of objects by
means of patches defined in terms
of control
points, whose positions in space can
be modified interactively.
The patches can be polynomial or rational,
and triangular or rectangular. Usually,
the triangular patches are more
convenient because they adapt naturally
to any surface in space.
A necessary tool for building this
type of patch are barycentric coordinates,
which allow for the generalization
of
the univariate Bernstein polynomials
and the definition of Bezier triangular
patches. Working with patches has
the
advantage of reducing the number of
elements that describe the surface
and also facilitates its deformation
by manipulating its control points.
There are several models for the description
of molecular surfaces, the Van der
Waals being the simplest way to describe
a molecule, namely as a union of spheres
whose radii are chosen according to
the type of atom and the centers of
these spheres, which represent the
atomic nuclei are arranged according
to molecular geometry. The outer border
of the union all the spheres of S
is called the van der Waals surface.
Therefore the description of the Van
der Waals surface is reduced to the
list of the spherical polygons that
compose these outer borders and the
decomposition of each polygon into
triangular Bezier patches.
This paper proposes a way of describing
the Van der Waals surface through
a set of rational Bezier triangular
patches. The patches are constructed
from the decomposition of spherical
polygons, for which it is required:
To calculate intersections of spheres
of S, by pairs, to obtain a set of
circles in 3D. Given all the circles
that lie on a sphere of S to consider
all intersections in pairs. This subdivides
each circle into arcs. Determination
of spherical polygons from the arcs.
Every spherical polygon is mapped
to the plane by stereographic projection,
obtaining a planar polygon of curved
sides and we determine its representation
as a planar rational Bezier triangular
patch. This presents the Van der Waals
surface as a union of triangular rational
Bezier patches.
Keywords. Stereographic projection.
Molecular surface. Parametric surfaces.
Rational Bezier triangular patches.
Van
der Waals surface. |
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Marianela
Lentini
Grid Generation Using Lemniscates
of Two Foci |
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SDSU: Computational Science
and Engineering Gateway to Latin America
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Last
updated:
June 2, 2010 4:00 PM
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