41 research outputs found
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Variational skinning of an ordered set of discrete 2D balls
This paper considers the problem of computing an interpolating
envelope of an ordered set of 2D balls. By construction, the envelope
is constrained to be C1 continuous, and for each ball, it touches the
ball at a point and is tangent to the ball at the point of contact. Using
an energy formulation, we derive differential equations that are designed
to minimize the envelope’s arc length and/or curvature subject to these
constraints. Given an initial envelope, we update the envelope’s parameters
using the differential equations until convergence occurs. We demonstrate
the method’s usefulness in generating interpolating envelopes of
balls of different sizes and in various configurations
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3D ball skinning using PDEs for generation of smooth tubular surfaces
We present an approach to compute a smooth, interpolating skin of an ordered set of 3D balls. By construction, the skin is constrained to be C-1 continuous, and for each ball, it is tangent to the ball along a circle of contact. Using an energy formulation, we derive differential equations that are designed to minimize the skin's surface area, mean curvature, or convex combination of both. Given an initial skin, we update the skin's parametric representation using the differential equations until convergence occurs. We demonstrate the method's usefulness in generating interpolating skins of balls of different sizes and in various configurations
3D ball skinning using PDEs for generation of smooth tubular surfaces
We present an approach to compute a smooth, interpolating skin of an ordered set of
3D balls. By construction, the skin is constrained to be C1 continuous, and for each
ball, it is tangent to the ball along a circle of contact. Using an energy formulation,
we derive differential equations that are designed to minimize the skin’s surface area,
mean curvature, or convex combination of both. Given an initial skin, we update the
skin’s parametric representation using the differential equations until convergence
occurs. We demonstrate the method’s usefulness in generating interpolating skins
of balls of different sizes and in various configurations
How round is a protein? Exploring protein structures for globularity using conformal mapping.
We present a new algorithm that automatically computes a measure of the geometric difference between the surface of a protein and a round sphere. The algorithm takes as input two triangulated genus zero surfaces representing the protein and the round sphere, respectively, and constructs a discrete conformal map f between these surfaces. The conformal map is chosen to minimize a symmetric elastic energy E S (f) that measures the distance of f from an isometry. We illustrate our approach on a set of basic sample problems and then on a dataset of diverse protein structures. We show first that E S (f) is able to quantify the roundness of the Platonic solids and that for these surfaces it replicates well traditional measures of roundness such as the sphericity. We then demonstrate that the symmetric elastic energy E S (f) captures both global and local differences between two surfaces, showing that our method identifies the presence of protruding regions in protein structures and quantifies how these regions make the shape of a protein deviate from globularity. Based on these results, we show that E S (f) serves as a probe of the limits of the application of conformal mapping to parametrize protein shapes. We identify limitations of the method and discuss its extension to achieving automatic registration of protein structures based on their surface geometry
3-Webs generated by confocal conics and circles
We consider families of confocal conics and two pencils of Apollonian circles having the same foci. We will show that these families of curves generate trivial 3-webs and find the exact formulas describing them
3-Webs generated by confocal conics and circles
We consider families of confocal conics and two pencils of Apollonian circles having the same foci. We will show that these families of curves generate trivial 3-webs and find the exact formulas describing them
Dense point sets have sparse Delaunay triangulations
The spread of a finite set of points is the ratio between the longest and
shortest pairwise distances. We prove that the Delaunay triangulation of any
set of n points in R^3 with spread D has complexity O(D^3). This bound is tight
in the worst case for all D = O(sqrt{n}). In particular, the Delaunay
triangulation of any dense point set has linear complexity. We also generalize
this upper bound to regular triangulations of k-ply systems of balls, unions of
several dense point sets, and uniform samples of smooth surfaces. On the other
hand, for any n and D=O(n), we construct a regular triangulation of complexity
Omega(nD) whose n vertices have spread D.Comment: 31 pages, 11 figures. Full version of SODA 2002 paper. Also available
at http://www.cs.uiuc.edu/~jeffe/pubs/screw.htm