Curvature and the fairness of curves and surfaces

Farin, Gerald; Sapidis, Nickolas S.

doi:10.1109/38.19051

Cited by 126 publications

(35 citation statements)

References 10 publications

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“…Is it possible to erase unwanted curvature extrema, and how should we proceed? Among B-spline standard algorithms, knot removal, which amounts to reducing the number of control points may sometime be an effective method for removing curvature extrema and smoothing a curve [1]. However, for demanding applications, reducing the number of control points is not a solution.…”

Section: Introductionmentioning

confidence: 99%

A selective eraser of curvature extrema for B-spline curves

Demers

Tribes

Guibault

2015

Computers & Graphics

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Section: Introductionmentioning

confidence: 99%

A selective eraser of curvature extrema for B-spline curves

Demers

Tribes

Guibault

2015

Computers & Graphics

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“…Contrary to what might be the case in the field of computer graphics, where often the goal is to design aesthetically pleasing curves (Farin and Sapidis, 1989), in motion control scenarios it is preferable to assign paths that consist of straight lines and arc segments rather than splines, mainly because the latter implies that at least one of the control surfaces (a ship's rudder, for instance) will always be active, due to the relentlessly varying curvature. In addition, using straight lines and arc segments makes it more likely and easier to ensure that the path will not include wiggles or zigzags between two waypoints.…”

Section: Introductionmentioning

confidence: 99%

Continuous-Curvature Path Generation Using Fermat's Spiral

Lekkas¹,

Dahl²,

Breivik³

et al. 2013

MIC

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This paper proposes a novel methodology, based on Fermat's spiral (FS), for constructing curvaturecontinuous parametric paths in a plane. FS has a zero curvature at its origin, a property that allows it to be connected with a straight line smoothly, that is, without the curvature discontinuity which occurs at the transition point between a line and a circular arc when constructing Dubins paths. Furthermore, contrary to the computationally expensive clothoids, FS is described by very simple parametric equations that are trivial to compute. On the downside, computing the length of an FS arc involves a Gaussian hypergeometric function. However, this function is absolutely convergent and it is also shown that it poses no restrictions to the domain within which the length can be calculated. In addition, we present an alternative parametrization of FS which eliminates the parametric speed singularity at the origin, hence making the spiral suitable for path-tracking applications. A detailed description of how to construct curvature-continuous paths with FS is given.

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“…The osculating circle at a point x on a curve is the circle that approximates the curve by matching its tangent and curvature [11]. Instead of solving a linear system, we also propose to use the corresponding control points from a circle template.…”

Section: Circle and Sphere Templatesmentioning

confidence: 99%

Volumetric parametrization from a level set boundary representation with PHT-splines

Chan

Anitescu

Rabczuk

2017

Computer-Aided Design

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A challenge in isogeometric analysis is constructing analysis-suitable volumetric meshes which can accurately represent the geometry of a given physical domain. In this paper, we propose a method to derive a splinebased representation of a domain of interest from voxel-based data. We show an efficient way to obtain a boundary representation of the domain by a level-set function. Then, we use the geometric information from the boundary (the normal vectors and curvature) to construct a matching C 1 representation with hierarchical cubic splines. The approximation is done by a single template and linear transformations (scaling, translations and rotations) without the need for solving an optimization problem. We illustrate our method with several examples in two and three dimensions, and show good performance on some standard benchmark test problems.

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Curvature and the fairness of curves and surfaces

Cited by 126 publications

References 10 publications

A selective eraser of curvature extrema for B-spline curves

A selective eraser of curvature extrema for B-spline curves

Continuous-Curvature Path Generation Using Fermat's Spiral

Volumetric parametrization from a level set boundary representation with PHT-splines

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