185 research outputs found
Kink-free deformations of polygons
We consider a discrete version of the Whitney-Graustein theorem concerning regular equivalence of closed curves. Two regular polygons P and P’, i.e. polygons without overlapping adjacent edges, are called regularly equivalent if there is a continuous one-parameter family Ps, 0 ≤ s ≤ 1, of regular polygons with P0 = P and P1 = P’. Geometrically the one-parameter family is a kink-free deformation transforming P into P’. The winding number of a polygon is a complete invariant of its regular equivalence class. We develop a linear algorithm that determines a linear number of elementary steps to deform a regular polygon into any other regular polygon with the same winding number
Riemannian simplices and triangulations
We study a natural intrinsic definition of geometric simplices in Riemannian
manifolds of arbitrary dimension , and exploit these simplices to obtain
criteria for triangulating compact Riemannian manifolds. These geometric
simplices are defined using Karcher means. Given a finite set of vertices in a
convex set on the manifold, the point that minimises the weighted sum of
squared distances to the vertices is the Karcher mean relative to the weights.
Using barycentric coordinates as the weights, we obtain a smooth map from the
standard Euclidean simplex to the manifold. A Riemannian simplex is defined as
the image of this barycentric coordinate map. In this work we articulate
criteria that guarantee that the barycentric coordinate map is a smooth
embedding. If it is not, we say the Riemannian simplex is degenerate. Quality
measures for the "thickness" or "fatness" of Euclidean simplices can be adapted
to apply to these Riemannian simplices. For manifolds of dimension 2, the
simplex is non-degenerate if it has a positive quality measure, as in the
Euclidean case. However, when the dimension is greater than two, non-degeneracy
can be guaranteed only when the quality exceeds a positive bound that depends
on the size of the simplex and local bounds on the absolute values of the
sectional curvatures of the manifold. An analysis of the geometry of
non-degenerate Riemannian simplices leads to conditions which guarantee that a
simplicial complex is homeomorphic to the manifold
The apolar bilinear form in geometric modeling
Some recent methods of Computer Aided Geometric Design are related to the apolar bilinear form, an inner product on the space of homogeneous multivariate polynomials of a fixed degree, already known in 19th century invariant theory. Using a generalized version of this inner product, we derive in a straightforward way some of the recent results in CAGD, like Marsden's identity, the expression for the de Boor-Fix functionals, and recursion schemes for the computation of B-patches and their derivatives
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