Intrinsic parametrization for approximation

Hoschek, Josef

doi:10.1016/0167-8396(88)90017-9

“…In such cases the solution given in Equation (14) will also be finite. However, for complex trim curves this is often not possible, in which case the solution will be the infinite series given in Equation (14). For the purpose of meshing the interior region of the trimmed surface an approximate solution of Equation (13) is obtained which satisfy exactly at the given boundary conditions.…”

Section: Fast Solution Methods For Laplace Equationmentioning

confidence: 99%

“…In the case of polynomial surface patches the problem of trimming was most probably discussed by Hoscheck [13][14][15] at first. These early methods involved the division of the parameter space of a given polynomial surface patch into a set of subsets whose specific forms are defined by cubic Bézier surface patches.…”

Section: Introductionmentioning

confidence: 99%

Method of trimming PDE surfaces

Ugail

¹

2006

Computers & Graphics

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A method for trimming surfaces generated as solutions to Partial Differential Equations (PDEs) is presented. The work we present here utilises the 2D parameter space on which the trim curves are defined whose projection on the parametrically represented PDE surface is then trimmed out. To do this we define the trim curves to be a set of boundary conditions which enable us to solve a low order elliptic PDE on the parameter space. The chosen elliptic PDE is solved analytically, even in the case of a very general complex trim, allowing the design process to be carried out interactively in real time. To demonstrate the capability for this technique we discuss a series of examples where trimmed PDE surfaces may be applicable.

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“…The quality of the approximation and/or interpolation of a set of points P j ∈ R d , (d = 2, 3) by a parametric curve C(s) depends on the choice of parameters s. Additional degrees of freedom in the case of spline curves are given by the knot vector t. The optimal parametrization problem has been extensively studied, see for instance Hoschek [13] and Speer et. al.…”

Section: Previous Workmentioning

confidence: 99%

Spline Curve Approximation and Design by Optimal Control Over the Knots

Goldenthal

¹

,

Bercovier

²

2004

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In [1] Optimal Control methods over re-parametrization for curve and surface design were introduced. The advantage of Optimal Control over Global Minimization such as in [17] is that it can handle both approximation and interpolation. Moreover a cost function is introduced to implement a design objective (shortest curve, smoothest one etc...). The present work introduces the Optimal Control over the knot vectors of non-uniform B-Splines.An interesting aspect is that the interpolation or the approximation matrix might become singular due to invalid knot vector values with respect to the current parametrization (violation of Schoenberg-Whitney condition). This situation is dealt naturally within the Optimal Control framework. A geometric description of the resulting null space is provided as well.

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“…The optimal parametrization problem has been extensively studied, see for instance Hoschek [13] and Speer et. al.…”

Section: Previous Workmentioning

confidence: 99%

Spline Curve Approximation and Design by Optimal Control Over the Knots

Goldenthal

¹

,

Bercovier

²

2004

Geometric Modelling

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In [1] Optimal Control methods over re-parametrization for curve and surface design were introduced. The advantage of Optimal Control over Global Minimization such as in [17] is that it can handle both approximation and interpolation. Moreover a cost function is introduced to implement a design objective (shortest curve, smoothest one etc...). The present work introduces the Optimal Control over the knot vectors of non-uniform B-Splines.An interesting aspect is that the interpolation or the approximation matrix might become singular due to invalid knot vector values with respect to the current parametrization (violation of Schoenberg-Whitney condition). This situation is dealt naturally within the Optimal Control framework. A geometric description of the resulting null space is provided as well.

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Intrinsic parametrization for approximation

Cited by 141 publications

References 3 publications

Method of trimming PDE surfaces

Method of trimming PDE surfaces

Spline Curve Approximation and Design by Optimal Control Over the Knots

Spline Curve Approximation and Design by Optimal Control Over the Knots

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