Triangular Bernstein-Bézier patches

Farin, Gerald

doi:10.1016/0167-8396(86)90016-6

Cited by 509 publications

(259 citation statements)

References 21 publications

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“…As a conséquence (7.4) p n < 0 and p 02 < 0 since otherwise one could slightly perturb A into a new triangle A * such that <</J 12 >0» (or «^02=>0» respectively) and still have «/» 0i >0» which contradicts (7,3). On introducing the notation B(p \ T) for the Bézier net of p with respect to the triangle T and (r l9 ..., r m ) for the convex huil of the points r h ...,r m one has (d o ,d l9 c)).…”

Section: Constructevg a Convex Bézier Net For A Convex Quadraticmentioning

confidence: 84%

On convex Bézier triangles

Prautzsch¹

1992

ESAIM: M2AN

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Section: Constructevg a Convex Bézier Net For A Convex Quadraticmentioning

confidence: 84%

On convex Bézier triangles

Prautzsch¹

1992

ESAIM: M2AN

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“…Wachspress extended barycentric coordinates to convex polygons [16,13]. The Wachspress coordinates satisfies properties (1)(2)(3)(4)(5) and recently Floater and Kosinka proved that property (6) also holds [5]. Note that property (7) only makes sense in a simplicial setting.…”

Section: Generalized Barycentric Coordinatesmentioning

confidence: 99%

“…The construction of a simplicial diffeomorphism can be divided in two stages: the first stage consists in to define a K-invariant mapping X in a given simplicial complex K. Initially, we define the functions x i (p) in a single simplex σ in such a way that the properties (1), (3)(4)(5) and (7) be satisfied, which is generally easy. These functions depend of control parameters associated to faces of σ.…”

Section: Construction Of Simplicial Diffeomorphismsmentioning

confidence: 99%

Simplicial diffeomorphisms

Mello

Velho

2010

Computer Aided Geometric Design

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In this paper we will present a theory for simplicial diffeomorphims, that is, diffeomorphims that preserve the incidence relations of a simplicial complex. Simplicial diffeomorphisms can be regarded as curvilinear barycentric coordinates. Using the combination of piecewise linear functions on complexes with simplicial diffeomorphisms, we also propose a new representation of curves and surfaces (and hypersurfaces, in general) that is simultaneously implicit and parametric.

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“…The piecewise linear interpolant of the Bézier control points, defined as b ijk = (ξ ijk , b ijk ), is called the Bézier control net. This control net is tangent to the polynomial surface at the three vertices [7].…”

Section: Bivariate Polynomials In Bernstein-bézier Representationmentioning

confidence: 99%

“…In this Bernstein-Bézier representation the RCT-splines can be easily represented, evaluated, manipulated and displayed using the de Casteljau algorithm [7].…”

Section: A Bernstein-bézier Representation For Rct-splinesmentioning

confidence: 99%

A normalized basis for reduced Clough–Tocher splines

Speleers

2010

Computer Aided Geometric Design

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We present the construction of a suitable normalized B-spline representation for reduced Clough-Tocher splines. The basis functions have a local support, they are nonnegative, and they form a partition of unity. Geometrically, the problem can be interpreted as the determination of a set of triangles that must contain a specific set of points. This leads to a natural definition of tangent control triangles. We also consider a stable computation of the Bézier control net of the spline surface.Keywords : Clough-Tocher splines, normalized B-splines, control points, control triangles, Bézier control net MSC : Primary : 65D07, Secondary : 65D17, 68U07A normalized basis for reduced Clough-Tocher splines Hendrik SpeleersDepartment of Computer Science, Katholieke Universiteit Leuven Celestijnenlaan 200A, B-3001 Leuven, Belgium AbstractWe present the construction of a suitable normalized B-spline representation for reduced Clough-Tocher splines. The basis functions have a local support, they are nonnegative, and they form a partition of unity. Geometrically, the problem can be interpreted as the determination of a set of triangles that must contain a specific set of points. This leads to a natural definition of tangent control triangles. We also consider a stable computation of the Bézier control net of the spline surface.

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Triangular Bernstein-Bézier patches

Cited by 509 publications

References 21 publications

On convex Bézier triangles

On convex Bézier triangles

Simplicial diffeomorphisms

A normalized basis for reduced Clough–Tocher splines

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