Interactive Procedural Computer-Aided Design

Séquin, Carlo H.

doi:10.1109/cad-cg.2005.52

Cited by 2 publications

(2 citation statements)

References 12 publications

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“…These studies have focused on different aspects of Be ´zier curves and surfaces, including their properties, applications, and computational methods. This includes the investigation of variational improvement in Be ´zier surfaces using quasi-minimal functions [13,14], the applications of Be ´zier curves and surfaces in computer-aided geometric design and modeling [15][16][17]. Furthermore, Be ´zier surfaces have been employed to solve the Plateau-Be ´zier problem, which involves finding a Be ´zier surface of minimal area among all possible surfaces spanned by the same boundary [2,18,19].…”

Section: Introductionmentioning

confidence: 99%

Geometric analysis of non-degenerate shifted-knots Bézier surfaces in Minkowski space

Bashir,

Ahmad

2024

PLoS ONE

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In this paper, we investigate the properties of timelike and spacelike shifted-knots Bézier surfaces in Minkowski space-E 1 3. These surfaces are commonly used in mathematical models for surface formation in computer science for computer-aided geometric design and computer graphics, as well as in other fields of mathematics. Our objective is to analyze the characteristics of timelike and spacelike shifted-knots Bézier surfaces in Minkowski space-E 1 3. To achieve this, we compute the fundamental coefficients of shifted-knots Bézier surfaces, including the Gauss-curvature, mean-curvature, and shape-operator of the surface. Furthermore, we present numerical examples of timelike and spacelike bi-quadratic (m = n = 2) and bi-cubic (m = n = 3) shifted-knots Bézier surfaces in Minkowski space-E 1 3 to demonstrate the applicability of the technique in Minkowski space.

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Section: Introductionmentioning

confidence: 99%

Geometric analysis of non-degenerate shifted-knots Bézier surfaces in Minkowski space

Bashir,

Ahmad

2024

PLoS ONE

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“…The applications of minimal surfaces are found in the emerging disciplines of the computer-aided geometric design, computer graphics, computer-aided design, and geometric modeling. Séquin's studies [20][21][22] are based on a variety of applications of these minimal surfaces in constructing, for example, for motor vehicles, the interactive CAD tools can be used for engineering purpose. The minimal surfaces can be built through a variational technique by finding solution of vanishing condition of gradient of a constraint usually in the form of an integral having the property of minimizing something, for example, minimizing an energy integral.…”

Section: Introductionmentioning

confidence: 99%

Variationally Improved Bézier Surfaces with Shifted Knots

Ahmad

Hassan

Mahmood

et al. 2021

Advances in Mathematical Physics

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The Plateau-Bézier problem with shifted knots is to find the surface of minimal area amongst all the Bézier surfaces with shifted knots spanned by the admitted boundary. Instead of variational minimization of usual area functional, the quasi-minimal Bézier surface with shifted knots is obtained as the solution of variational minimization of Dirichlet functional that turns up as the sum of two integrals and the vanishing condition gives us the system of linear algebraic constraints on the control points. The coefficients of these control points bear symmetry for the pair of summation indices as well as for the pair of free indices. These linear constraints are then solved for unknown interior control points in terms of given boundary control points to get quasi-minimal Bézier surface with shifted knots. The functional gradient of the surface gives possible candidate functions as the minimizers of the aforementioned Dirichlet functional; when solved for unknown interior control points, it results in a surface of minimal area called quasi-minimal Bézier surface. In particular, it is implemented on a biquadratic Bézier surface by expressing the unknown control point P 11 as the linear combination of the known control points in this case. This can be implemented to Bézier surfaces with shifted knots of higher degree, as well if desired.

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Interactive Procedural Computer-Aided Design

Abstract: Abstract

Cited by 2 publications

References 12 publications

Geometric analysis of non-degenerate shifted-knots Bézier surfaces in Minkowski space

Geometric analysis of non-degenerate shifted-knots Bézier surfaces in Minkowski space

Variationally Improved Bézier Surfaces with Shifted Knots

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