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 Bernstein-Bézier patches. An analysis of these patches is given and some numerical experiments with quartic and quintic

A technique is presented for deforming solid geometric models in a free-form manner. The technique can be used with any solid modeling system, such as CSG or B-rep. It can deform surface primitives of any type or degree: planes, quadrics, parametric surface patches, or implicitly defined surfaces

using degenerate polynomial patches are described in [3,5,1] and also in an article in these proceedings [2]. No...

‘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 1-parameter deformation of the Harish-Chandra homomorphism from D(g) g, the algebra of invariant polynomial differential operators on the Lie algebra g = gl n

Gröbner degeneration. Interpreting the Hilbert series of the flat limit in equivariant K-theory, another corollary of the proof is that Grothendieck polynomials represent the classes of Schubert varieties in K-theory of the flag manifold. An inductive procedure for listing the limit coordinate subspaces

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.

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

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 single-sheeted algebraic surface patch, We call this an A-patch. We present algorithms to build a mesh of cubic

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

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 Stanley-Reisner scheme. Similar results had been known for Schubert patches in various types of Grassmannians. We maintain reducedness using