Curvature continuous curves and surfaces

A simple methods is given for constructing the Bézier points of curvature continuous cubic spline curves and surfaces from their B-spline control points. The method is similar to the well-known construction of Bézier points of C² splines from their B-spline control points. The new construction allows the use of all results of the powerful Bernstein-Bézier technique in the realm of geometric splines.

A simple introduction to nu- and beta-splines is also derived, as well as some simple geometric properties of beta-splines.     

Geometric construction of spline curves with tension properties

In this paper a class of C²∩FC³ spline curves possessing tension properties is described. These curves can be constructed using a simple modification of the well-known geometric construction of C⁴ quintic splines; therefore their shape can be easily controlled using the control net. Their applications in approximation and interpolation of spatial data will be discussed.     

High accuracy geometric Hermite interpolation

We describe a parametric cubic spline interpolation scheme for planar curves which is based on an idea of Sabin for the construction of C¹ bicubic parametric spline surfaces. The method is a natural generalization of [standard] Hermite interpolation. In addition to position and tangent, the curvature is prescribed at each knot. This ensures that the resulting interpolating piecewise cubic curve is twice continuously differentiable with respect to arclength and can be constructed locally. Moreover, under appropriate assumptions, the interpolant preserves convexity and is 6-th order accurate.     

Convexity-preserving fairing

This paper develops a two-stage automatic algorithm for fairing C²-continuous cubic parametric B-splines under convexity, tolerance and end constraints. The first stage is a global procedure, yielding a C² cubic B-spline which satisfies the local-convexity, local-tolerance and end constraints imposed by the designer. The second stage is a local finefairing procedure employing an iterative knot-removal knotreinsertion technique, which adopts the curvature-slope discontinuity as the fairness measure of a C² spline. This procedure preserves the convexity and end properties of the output of the first stage and, moreover, it embodies a globaltolerance constraint. The performance of the algorithm is discussed for four data sets.     

Biquartic C-surface splines over irregular meshes

C¹-surface splines define tangent continuous surfaces from control points in the manner of tensor-product (B-)splines, but allow a wider class of control meshes capable of outlining arbitrary free-form surfaces with or without boundary. In particular, irregular meshes with non-quadrilateral cells and more or fewer than four cells meeting at a point can be input and are treated in the same conceptual frame work as tensor-product B-splines; that is, the mesh points serve as control points of a smooth piecewise polynomial surface representation that is local and evaluates by averaging. Biquartic surface splines extend and complement the definition of C¹-surface splines in a previous paper (Peters, J SLAM J. Numer. Anal. Vol 32 No 2 (1993) 645–666) improving continuity and shape properties in the case where the user chooses to model entirely with four-sided patches. While tangent continuity is guaranteed, it is shown that no polynomial, symmetry-preserving construction with adjustable blends can guarantee its surfaces to lie in the local convex hull of the control mesh for very sharp blends where three patches join. Biquartic C¹-surface splines do as well as possible by guaranteeing the property whenever more than three patches join and whenever the blend exceeds a certain small threshold.     

Rectangular v-Splines

This article describes and presents examples of some techniques for the representation and interactive design of surfaces based on a parametric surface representation that user v-spline curves. These v-spline curves, similar in mathematical structure to v-splines, were developed as a more computationally efficient alternative to splines in tension. Although splines in tension can be modified to allow tension to be applied at each control point, the procedure is computationally expensive. The v-spline curve, however, uses more computationally tractable piecewise cubic curves segments, resulting in curves that are just as smoothly joined as those of a standard cubic spline. After presenting a review of v-splines and some new properties, this article extends their application to a rectangular grid of control points. Three techniques and some application examples are presented.     

Cardinal interpolation: A bivariate polynomial example

The general interpolation problem over a linear space is solved by providing explicit formulas for the cardinal basis of the space. As an example of this technique, the cardinal form of a bivariate degree-nine polynomial interpolating to function and derivative values through order four at various points on a triangle is derived. The piecewise polynomial interpolant over an arbitrary triangulated domain in has C² continuity.     

A class of general quartic spline curves with shape parameters

With a support on four consecutive subintervals, a class of general quartic splines are presented for a non-uniform knot vector. The splines have C² continuity at simple knots and include the cubic non-uniform B-spline as a special case. Based on the given splines, piecewise quartic spline curves with three local shape parameters are given. The given spline curves can be C²∩G³ continuous by fixing some values of the curve?s parameters. Without solving a linear system, the spline curves can also be used to interpolate sets of points with C² continuity. The effects of varying the three shape parameters on the shape of the quartic spline curves are determined and illustrated.