7. On the Problem of Determining the Distance between Parametric Curves

Fritsch, Frederick N.; Nielson, Gregory M.

doi:10.1137/1.9781611971651.ch7

Cited by 5 publications

(2 citation statements)

References 2 publications

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“…Computing the true Hausdorff distance between / and 5 is no simple matter [6]. For the experiments reported below, we first took a sample of points along /, getting higher density in areas of high curvature by using a fixed number of samples per segment in the piecewise linear /.…”

Section: Numerical Results For Free-knot Splinesmentioning

confidence: 99%

Improved Rounding for Spline Coefficients and Knots

Grosse¹,

Hobby²

1994

Mathematics of Computation

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Abstract. When representing the coefficients and knots of a spline using only small integers, independently rounding each infinite-precision value is not the best strategy. We show how to build an affine model for the error expanded about the optimal full-precision free-knot or parameterized spline, then use the Lovász basis reduction algorithm to select a better rounding. The technique could be used for other situations in which a quadratic error model can be computed.

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Section: Numerical Results For Free-knot Splinesmentioning

confidence: 99%

Improved Rounding for Spline Coefficients and Knots

Grosse¹,

Hobby²

1994

Mathematics of Computation

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“…While it is realistic to assume a certain degree for the resulting curve, the number of control points and the knot vector need to be determined based on user-specified error bounds on the positions and derivatives. This error specified is generally different from the least square error E (l) , since most users are more interested in an estimate of the set theoretic distance measure: the Hausdorff distance e (l) between the ideal and approximated curves [15]. Thus, a high-level iterative procedure for finding the number of control points and the knot vector based on repeatedly using the core procedure is outlined.…”

Section: The Core Proceduresmentioning

confidence: 99%

The continuous non-linear approximation of procedurally defined curves using integral B-splines

Sarma

2004

Engineering with Computers

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This paper outlines an algorithm for the continuous non-linear approximation of procedurally defined curves. Unlike conventional approximation methods using the discrete L_2 form metric with sampling points, this algorithm uses the continuous L_2 form metric based on minimizing the integral of the least square error metric between the original and approximate curves. Expressions for the optimality criteria are derived based on exact B-spline integration. Although numerical integration may be necessary for some complicated curves, the use of numerical integration is minimized by a priori explicit evaluations. Plane or space curves with high curvatures and/or discontinuities can also be handled by means of an adaptive knot placement strategy. It has been found that the proposed scheme is more efficient and accurate compared to currently existing interpolation and approximation methods.

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Choosing nodes in parametric blending function interpolation

Conti

Sestini

1996

Computer-Aided Design

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7. On the Problem of Determining the Distance between Parametric Curves

Cited by 5 publications

References 2 publications

Improved Rounding for Spline Coefficients and Knots

Improved Rounding for Spline Coefficients and Knots

The continuous non-linear approximation of procedurally defined curves using integral B-splines

Choosing nodes in parametric blending function interpolation

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