Hierarchical B-spline networks consist of multiple B-spline networks assembled in different level or cascade architecture. To identify the hierarchical B-spline networks and select important input features for each sub-B-spline network automatically, a predefined instruction/operator set was used. The structures of hierarchical B-spline networks were created and evolved by using Probabilistic I...

A computationally inexpensive algorithm for signal interpolation using B-spline functions is presented in this work. Speciically, the convolution between the B-spline coeecients and the B-spline function itself, widely acknowledged as the most costly aspect of B-spline interpolation, is reformulated such that it is directly amenable to implementation at a much lower computational cost (16.25% o...

The density conjecture of Bers, Sullivan and Thurston predicts that each complete hyperbolic 3-manifold M with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We prove that the conjecture obtains for each complete hyperbolic 3manifold with no cusps and incompressible ends.

We initiate a triangle geometry in the projective metrical setting, based on the purely algebraic approach of universal geometry, and yielding in particular a new form of hyperbolic triangle geometry. There are three main strands: the Orthocenter, Incenter and Circumcenter hierarchies, with the last two dual. Formulas using ortholinear coordinates are a main objective. Prominent are five partic...

Surface blending is a useful operation in geometric design for rounding sharp edges or corners. Meanwhile, NURBS has already become the de facto industrial standard in existing CAD/CAM systems. Therefore, it is required to study how to blend two B-spline surfaces. However, two arbitrary B-spline surfaces (called base surfaces) are hard to be blended with a B-spline surface (called blending surf...

i=0 Pn n=0 bijB m i (u)B n j (v) of bi-degree (m;n) has total degree m + n, however, the class of Gmn(u; v) is a subset of polynomials of total degree m + n. G (resp G) continuity implies curve segments share the same tangent(curvature) at join points(knots). In each of the G and G constructions, we develop a spline curve family whose member satis es given interpolation conditions. Each family ...

A tangle is a pair (B, T ), where B is a 3-ball, T is a pair of properly embedded arcs. When there is no ambiguity we will simply say that T is a tangle. Let E(T ) = B − IntN(T ) be the exterior of T , usually called the tangle space. T is simple if E(T ) is a simple manifold, that is, it is irreducible, ∂-irreducible, atoroidal, and anannular. By Thurston’s geometrization theorem, simple tangl...