1,911 research outputs found
Exponential Splines and Pseudo-Splines: Generation versus reproduction of exponential polynomials
Subdivision schemes are iterative methods for the design of smooth curves and
surfaces. Any linear subdivision scheme can be identified by a sequence of
Laurent polynomials, also called subdivision symbols, which describe the linear
rules determining successive refinements of coarse initial meshes. One
important property of subdivision schemes is their capability of exactly
reproducing in the limit specific types of functions from which the data is
sampled. Indeed, this property is linked to the approximation order of the
scheme and to its regularity. When the capability of reproducing polynomials is
required, it is possible to define a family of subdivision schemes that allows
to meet various demands for balancing approximation order, regularity and
support size. The members of this family are known in the literature with the
name of pseudo-splines. In case reproduction of exponential polynomials instead
of polynomials is requested, the resulting family turns out to be the
non-stationary counterpart of the one of pseudo-splines, that we here call the
family of exponential pseudo-splines. The goal of this work is to derive the
explicit expressions of the subdivision symbols of exponential pseudo-splines
and to study their symmetry properties as well as their convergence and
regularity.Comment: 25 page
Subdivision Directional Fields
We present a novel linear subdivision scheme for face-based tangent
directional fields on triangle meshes. Our subdivision scheme is based on a
novel coordinate-free representation of directional fields as halfedge-based
scalar quantities, bridging the finite-element representation with discrete
exterior calculus. By commuting with differential operators, our subdivision is
structure-preserving: it reproduces curl-free fields precisely, and reproduces
divergence-free fields in the weak sense. Moreover, our subdivision scheme
directly extends to directional fields with several vectors per face by working
on the branched covering space. Finally, we demonstrate how our scheme can be
applied to directional-field design, advection, and robust earth mover's
distance computation, for efficient and robust computation
L-systems in Geometric Modeling
We show that parametric context-sensitive L-systems with affine geometry
interpretation provide a succinct description of some of the most fundamental
algorithms of geometric modeling of curves. Examples include the
Lane-Riesenfeld algorithm for generating B-splines, the de Casteljau algorithm
for generating Bezier curves, and their extensions to rational curves. Our
results generalize the previously reported geometric-modeling applications of
L-systems, which were limited to subdivision curves.Comment: In Proceedings DCFS 2010, arXiv:1008.127
A survey of partial differential equations in geometric design
YesComputer aided geometric design is an area
where the improvement of surface generation techniques
is an everlasting demand since faster and more accurate
geometric models are required. Traditional methods
for generating surfaces were initially mainly based
upon interpolation algorithms. Recently, partial differential
equations (PDE) were introduced as a valuable
tool for geometric modelling since they offer a number
of features from which these areas can benefit. This work
summarises the uses given to PDE surfaces as a surface
generation technique togethe
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