On the numerical condition of polynomials in Berstein form
Computer Aided Geometric Design
Algorithms for polynomials in Bernstein form
Computer Aided Geometric Design
Real rational curves are not “unit speed”
Computer Aided Geometric Design
Real-time CNC interpolators for Pythagorean-hodograph curves
Computer Aided Geometric Design
Computer Aided Geometric Design
A vegetarian approach to optimal parameterizations
Computer Aided Geometric Design
Convergent inversion approximations for polynomials in Bernstein form
Computer Aided Geometric Design
Performance analysis of CNC interpolators for time-dependent feedrates along PH curves
Computer Aided Geometric Design
Sampling points on regular parametric curves with control of their distribution
Computer Aided Geometric Design
Euclidean and Minkowski Pythagorean hodograph curves over planar cubics
Computer Aided Geometric Design
Point-based methods for estimating the length of a parametric curve
Journal of Computational and Applied Mathematics
Optimal parameterization of rational quadratic curves
Computer Aided Geometric Design
Euclidean and Minkowski Pythagorean hodograph curves over planar cubics
Computer Aided Geometric Design
A new method for approximating optimal parameterization of polynomial curves
ISVC'06 Proceedings of the Second international conference on Advances in Visual Computing - Volume Part II
Optimal parameterizations of bézier surfaces
ISVC'06 Proceedings of the Second international conference on Advances in Visual Computing - Volume Part I
Improving angular speed uniformity by optimal C0 piecewise reparameterization
CASC'12 Proceedings of the 14th international conference on Computer Algebra in Scientific Computing
Equiareal parameterizations of NURBS surfaces
Graphical Models
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Rational re-parameterizations of a polynomial curve that preserve the curve degree and [0,1] parameter domain are characterized by a single degree of freedom. The ''optimal'' re-parameterization in this family (that comes closest under the L"2 norm to arc-length parameterization) can be identified by solving a quadratic equation, but may exhibit too much residual parametric speed variation for motion control and other applications. Closer approximations to arc-length parameterizations require more flexible re-parameterization functions, such as piecewise-polynomial/rational forms. We show that, for fixed nodes, the optimal piecewise-rational parameterization of the same degree is defined by a simple recursion relation, and we analyze its convergence to the arc-length parameterization. With respect to the new curve parameter, this representation is only of C^0 continuity, although the smoothness and geometry of the curve are unchanged. A C^1 parameterization can be obtained by using continuity conditions, rather than optimization, to fix certain free parameters, but the objective function is then highly non-linear and does not admit a closed-form optimization. Empirical results from implementations of these methods are presented.