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Fair webs
, 2007
"... Fair webs are energyminimizing curve networks. Obtained via an extension of cubic splines or splines in tension to networks of curves, they are efficiently computable and possess a variety of interesting applications. We present properties of fair webs and their discrete counterparts, i.e., fair ..."
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Fair webs are energyminimizing curve networks. Obtained via an extension of cubic splines or splines in tension to networks of curves, they are efficiently computable and possess a variety of interesting applications. We present properties of fair webs and their discrete counterparts, i.e., fair polygon networks. Applications of fair curve and polygon networks include fair surface design and approximation under constraints such as obstacle avoidance or guaranteed error bounds, aesthetic remeshing, parameterization and texture mapping, and surface restoration in geometric models.
Smoothing imprecise 1dimensional terrains
"... An imprecise 1dimensional terrain is an xmonotone polyline where the ycoordinate of each vertex is not fixed but only constrained to a given interval. In this paper we study four different optimization measures for imprecise 1dimensional terrains, related to obtaining smooth terrains. In particu ..."
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An imprecise 1dimensional terrain is an xmonotone polyline where the ycoordinate of each vertex is not fixed but only constrained to a given interval. In this paper we study four different optimization measures for imprecise 1dimensional terrains, related to obtaining smooth terrains. In particular, we present algorithms to minimize the largest and total turning angle, and to maximize the smallest and total turning angle.
Constrained Optimization with EnergyMinimizing Curves and Curve Networks — A Survey
"... We survey recent research results in constrained optimization with curves and curve networks. The addressed topics include constrained variational curve and curve network design, variational motion design, and guaranteed error bound approximation of point cloud data with curve networks. The main the ..."
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We survey recent research results in constrained optimization with curves and curve networks. The addressed topics include constrained variational curve and curve network design, variational motion design, and guaranteed error bound approximation of point cloud data with curve networks. The main theoretic results are summarized with a focus on geometric solutions of the studied problems. A variety of applications is outlined including obstacle avoiding rigid body motion design and smoothing of digital terrain elevation data.
NOTE ON CURVE AND SURFACE ENERGIES
"... ABSTRACT. Energies of curves and surfaces together with their discrete variants play a prominent role as fairness functionals in geometric modeling and computer aided geometric design. This paper deals with a particular discrete surface energy which is expressible in terms of curve energies, and whi ..."
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ABSTRACT. Energies of curves and surfaces together with their discrete variants play a prominent role as fairness functionals in geometric modeling and computer aided geometric design. This paper deals with a particular discrete surface energy which is expressible in terms of curve energies, and which occurs naturally in the problem of smoothing digital elevation data with tolerance zone constraints. We also discuss geometrically meaningful surface energies in general from the viewpoint of invariant theory, and the role of the GaussBonnet theorem. 1. A FAIRING PROBLEM WITH HARD CONSTRAINTS For a the polyline p with vertices pi we consider the discrete linearized bending energy (1) E2(p) = ∑‖ ∆ 2 pi ‖ 2, where the forward difference operator ∆ is defined by ∆pi = pi+1 − pi, which implies ∆ 2 pi = pi+2 − 2pi+1 + pi. This energy functional occurs e.g. in the context of energyminimizing curves and curve networks (cf. Hofer and Pottmann (2004) and Wallner et al. (2005)). Hofer et al. (2005) apply energyminimizing curve networks to the problem of smoothing digital elevation data, as described below. Suppose that a terrain is modeled by a rectangular grid of points pi j = (xi j,yi j,zi j), with i =