Tangent Plane Continuity Research Articles

Algebraic surface design with Hermite interpolation

This paper presents an efficient algorithm called Hermite interpolation, for constructing low-degree algebraic surfaces, which contain, with C 1 or tangent plane continuity, any given collection of points and algebraic space curves having derivative information. Positional as well as derivative constraints on an implicitly defined algebraic surface are translated into a homogeneous linear system, where the unknowns are the coefficients of the polynomial defining the algebraic surface. Computaional details of the Hermite interpolation algorithm are presented along with several illustrative applications of the interpolation technique to construction of joining or blending surfaces for solid models as well as fleshing surfaces for curved wire frame models. A heuristic approach to interactive shape control of implicit algebraic surfaces is also given, and open problems in algebraic surface design are discussed.

An investigation of curvature variations over recursively generated B-spline surfaces

The continuity properties of recursively generated B-spline surfaces over an arbitrary topology have been related to the eigenproperties of the local subdivision tranformation, and conditions have been established on the subdivision weightings for tangent plane continuity at extraordinary points. In this paper, curves through an extradordinary point, which align in both the tagent and binormal direction, are identified, and their curvatures are compared either side of the point. Further restrictions on the subdivision weightings are derived to optimize the curvature properties of the surface. In general continuity of curvature is not attained.

Necessary and sufficient conditions for tangent plane continuity of Bézier surfaces

Sets of conditions are derived that are necessary and sufficient for tangent plane continuity between two polynomial or rational Bézier surfaces. The patches may be given in either triangular or rectangular form, and no assumptions are made concerning the relative degrees of the patches; the only assumption is that the patches share common boundary control points (and weights in the case of rational surfaces). The conditions are shown to be minimal in the sense that they are, in general, independent.

Smooth mesh interpolation with cubic patches

An algorithm for the local interpolation of a mesh of cubic curves with 3- and 4-sided facets by a piecewise cubic C 1 surface is stated and illustrated by an implementation. Precise necessary and sufficient conditions for oriented tangent-plane continuity between adjacent patches are derived, and the explicit constructions are characterized by the degree of the three scalar weight functions that relate the versal to the two transversal derivatives. The algorithm fully exploits the possibility of reparametrization by choosing all three weight functions nonconstant and not just degree-raising polynomials. The construction is local and consists mainly of averaging. The only systems to be solved are linear and of size 2 × 2. The algorithm guarantees interpolating surfaces without cusps and has a simple, implemented extension to n-sided facets.

Design of an n-sided surface patch from Hermite boundary data

The design of surface patches of other than four sides has traditionally been a time-consuming problem. In this paper it is demonstrated that a B-spline subdivision patch can be fitted into a general n-sided area of a bicubic surface with at least tangent plane continuity on the boundary. One degree of freedom is identified and related to shape control.

Nonrectangular surface patches with curvature continuity

An n-sided region can be decomposed into n rectangles, all meeting at a common vertex. For odd n, derivative data can be defined for each rectangular patch at the common vertex that allows continuity of tangent planes and, optionally, normal curvatures between all n rectangles. For tangent plane continuity, the resulting patches are biquintic; for curvature continuity, biseptic.

Conditions for tangent plane continuity over recursively generated B-spline surfaces

The continuity properties of recursively generated B-spline surfaces over an arbitrary topology have been related to the eigenproperties of the local subdivision transformation. In this paper a discrete Fourier transform technique is employed to derive these eigenproperties for a general choice of subdivision weightings. Conditions on these weightings are identified for tangent plane continuity at the extraordinary points and a geometric interpretation is given.

Recursively generated B-spline surfaces

The paper reports the progress of an SERC sponsored project on recursive subdivision over arbitrary topologies. The limit surface is generally a standard B-spline surface except at the so called ‘extraordinary points’ where, it is reported, tangent plane continuity can be guaranteed and discrepancies in the curvature can be confined to 10% or less. The optimal subdivision procedure is currently being implemented in a designer-oriented form in collaboration with CADCentre Ltd Cambridge.

A construction for visual C1 continuity of polynomial surface patches

A geometric construction is described which ensures tangent plane (visual C 1) continuity between adjacent polynomial surface patches. The algorithm works for both three- and four-sided patches.