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796
Degenerate Polynomial Patches of Degree 4 and 5 Used for Geometrically Smooth Interpolation in R³
 IN IR , COMPUTER AIDED GEOMETRIC DESIGN 11
"... The problem of interpolating scattered 3D data by a geometrically smooth surface is considered. A completely local method is proposed, based on employing degenerate triangular BernsteinBézier patches. An analysis of these patches is given and some numerical experiments with quartic and quintic patc ..."
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Cited by 14 (1 self)
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The problem of interpolating scattered 3D data by a geometrically smooth surface is considered. A completely local method is proposed, based on employing degenerate triangular BernsteinBézier patches. An analysis of these patches is given and some numerical experiments with quartic and quintic
Freeform deformation of solid geometric models
 IN PROC. SIGGRAPH 86
, 1986
"... A technique is presented for deforming solid geometric models in a freeform manner. The technique can be used with any solid modeling system, such as CSG or Brep. It can deform surface primitives of any type or degree: planes, quadrics, parametric surface patches, or implicitly defined surfaces, f ..."
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Cited by 701 (1 self)
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A technique is presented for deforming solid geometric models in a freeform manner. The technique can be used with any solid modeling system, such as CSG or Brep. It can deform surface primitives of any type or degree: planes, quadrics, parametric surface patches, or implicitly defined surfaces
On Degenerate Surface Patches
, 1992
"... A local construction of a GC 1 interpolating surface to given scattered data in R 3 can give rise to degenerate BernsteinB'ezier patches. That means the parametrization at vertices is not regular in the sense that the length of the tangent vector to any curve passing through a vertex is z ..."
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Cited by 1 (1 self)
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using degenerate polynomial patches are described in [3,5,1] and also in an article in these proceedings [2]. No...
Symplectic reflection algebras, CalogeroMoser space, and deformed HarishChandra homomorphism
 Invent. Math
"... To any finite group Γ ⊂ Sp(V) of automorphisms of a symplectic vector space V we associate a new multiparameter deformation, Hκ, of the algebra C[V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter κ runs over points of CP r, where r = number of conjugacy classes of symplectic ..."
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Cited by 280 (39 self)
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‘rational ’ degenerations of the double affine Hecke algebra introduced earlier by Cherednik. Let Γ = Sn, the Weyl group of g = gl n. We construct a 1parameter deformation of the HarishChandra homomorphism from D(g) g, the algebra of invariant polynomial differential operators on the Lie algebra g = gl n
Gröbner geometry of Schubert polynomials
 Ann. Math
"... Schubert polynomials, which a priori represent cohomology classes of Schubert varieties in the flag manifold, also represent torusequivariant cohomology classes of certain determinantal loci in the vector space of n ×n complex matrices. Our central result is that the minors defining these “matrix S ..."
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Cited by 99 (15 self)
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Gröbner degeneration. Interpreting the Hilbert series of the flat limit in equivariant Ktheory, another corollary of the proof is that Grothendieck polynomials represent the classes of Schubert varieties in Ktheory of the flag manifold. An inductive procedure for listing the limit coordinate subspaces
Toric degenerations of Bézier patches
, 2011
"... Parametric Bézier curves and surfaces are widely used to represent geometric objects in a computer for design and manufacturing. While classical, they were rediscovered in 1959/62 by De Casteljau (Citroën) and Bézier (Renault) for automotive design. Used in animation software such as Adobe flash. Yo ..."
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Cited by 2 (2 self)
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Parametric Bézier curves and surfaces are widely used to represent geometric objects in a computer for design and manufacturing. While classical, they were rediscovered in 1959/62 by De Casteljau (Citroën) and Bézier (Renault) for automotive design. Used in animation software such as Adobe flash. You were able to fly here because of design, testing, and manufacture of airplane wings based on Bézier surfaces. You are looking at several thousand Bézier curves: Many fonts (L ATEX, and type 1 fonts) are defined in terms of cubic Bézier curves. Frank Sottile (TAMU) 3 June 2011 2 / 17Rational Bézier Curves For i ∈ [0, d] ∩ Z, and s ∈ [0, d], we have βi;d(s): = s i (1 − s) d−i. Weights wi> 0 and control points bi ∈ R n (typically, n = 2, 3), define a rational Bézier curve of degree d, which is the image of [0, d] ∋ s ↦−→ ∑ di=0 wiβi;d(s)bi i wiβi;d(s) The control polygons of these rational Bézier curves connect the control points, and large weights pull the curve towards the control points.
POLYNOMIAL PATCHES THROUGH GEODESICS
, 2007
"... We construct low degree patches that contain any given 3D polynomial curve as a pregeodesic (i.e. geodesic up to reparametrization). A curve is a pregeodesic if and only if its rectifying plane coincides with the tangent plane to the surface, we use this fact to construct patches through pregeodesic ..."
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Cited by 3 (0 self)
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We construct low degree patches that contain any given 3D polynomial curve as a pregeodesic (i.e. geodesic up to reparametrization). A curve is a pregeodesic if and only if its rectifying plane coincides with the tangent plane to the surface, we use this fact to construct patches through
Modeling with Cubic APatches
, 1995
"... We present a sufficient criterion for the Bernstein Bezier (BB) form of a trivariate polynomial within a tetrahedron, such that the real zero contour of the polynomial defines a smoothand singlesheeted algebraic surface patch, We call this an Apatch. We present algorithms to build a mesh of cubic ..."
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Cited by 68 (37 self)
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We present a sufficient criterion for the Bernstein Bezier (BB) form of a trivariate polynomial within a tetrahedron, such that the real zero contour of the polynomial defines a smoothand singlesheeted algebraic surface patch, We call this an Apatch. We present algorithms to build a mesh of cubic
From Degenerate Patches to Triangular and Trimmed Patches
 CURVES AND SURFACES
, 1997
"... CAD systems are usually based on a tensor product representation of free form surfaces. In this case, trimmed patches are used for modeling non rectangular zones. Trimmed patches provide a reasonable solution for the representation of general topologies, provided that the gap between equivalent trim ..."
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Cited by 1 (1 self)
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trimming curves in the euclidean space is small enough. Several commercial CAD systems, however, represent certain non rectangular surface regions through degenerate rectangular patches. Degenerate patches produce rendering artifacts and can lead to malfunctions in the subsequent geometric operations
SCHUBERT PATCHES DEGENERATE TO SUBWORD COMPLEXES
, 2008
"... We study the intersections of general Schubert varieties Xw with permuted big cells, and give an inductive degeneration of each such “Schubert patch” to a StanleyReisner scheme. Similar results had been known for Schubert patches in various types of Grassmannians. We maintain reducedness using th ..."
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Cited by 9 (1 self)
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We study the intersections of general Schubert varieties Xw with permuted big cells, and give an inductive degeneration of each such “Schubert patch” to a StanleyReisner scheme. Similar results had been known for Schubert patches in various types of Grassmannians. We maintain reducedness using
Results 1  10
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796