Smooth interpolation in triangles

Abstract

Boolean sum smooth interpolation to boundary data on a triangle is described. Sufficient conditions are given so that the functions when pieced together form a C N-1 (Ω) function over a triangular subdivision of a polygonal region Ω and the precision sets of the interpolation functions are derived. The interpolants are modified so that the compatability conditions on the function which is interpolated can be removed and a C 1 interpolant is used to illustrate the theory. The generation of interpolation schemes for discrete boundary data is also discussed.

Extension of Some Positive and Linear Operators to Domains with Curved Sides

“…Gordon, there have been constructed interpolation operators of Lagrange, Hermite and Birkhoff type, that interpolate the values of a given function or the values of the function and of certain of its derivatives on the boundary of a triangle with straight sides. These operators were applied in computer aided geometric design (see, e.g., [1]- [3], [5]]) and in finite element analysis (see, e.g., [1], [7], [8], [13], [17]- [19], [21], [22]). …”

Constrained Visualization Using the Shepard Interpolation Family

“…Smoother interpolants over triangulations will typically fail to remain within the bounds of the data, but several have been modified to incorporate constraints. For example, Asim [Asi00] modifies the Barnhill et al [BBG73] blending method (constrained cubics as suggested by Asim and Brodlie [AB03] are created along triangle edges and blended in the interior); Ong and Wong [OW96] create a C 1 interpolant by blending constrained rational cubics along triangle edges using the Nielson [Nie79] side-vertex method. Mulansky and Schmidt [MS94] construct a constrained C 1 interpolant using quadratic splines on a Powell-Sabin refinement of the original triangulation.…”

C1 Smooth Triangular Surface Patch Constructed by C-curves

“…This paper gives a method of constructing triangular patches by C-curves. The first smooth interpolation triangular patch to boundary curves of a triangle was proposed by Barnhill, Birkhoff and Gordon (7) . The triangular patch is constructed using the Boolean sum scheme.…”