John A. Roulier

An Algorithm for Computing a Shape-Preserving Osculatory Quadratic Spline

An algorithm is presented for calculating an osculatory quadratm sphne that preserves the monotonmlty and convexity of the data when consmtent with the given derivatives at the data points A method is also presented for calculatmn of an mterpolatory quadratm sphne with monotonicity and convexity consistent with that of the data Included is a discussion of pathologms that can occur when these algorithms are maplemented. Examples are given that illustrate the dependence of the method on only local informatmn and its usefulness m geometrm design.

MONOTONE AND CONVEX SPLINE INTERPOLATION

For a set of monotone (and/or convex) data, we consider the possibility of finding a spline interpolant, of pre-determined smoothness, which is monotone (and/or convex). The investigation is carried out by constructing an auxiliary set of points and using the well-known monotonicity and convexity preserving properties of Bernstein polynomials. In § 3 we consider the problem of piecewise monotone interpolation.

Algorithms for Computing Shape Preserving Spline Interpolations to Data

Algorithms are presented for computing a smooth piecewise polynomial interpolation which preserves the monotonicity and/or convexity of the data.

Interpolation by convex quadratic splines

Algorithms are presented for computing a quadratic spline interpolant with variable knots which preserves the monotonicity and convexity of the data. It is also shown that such a spline may not exist for fixed knots.

Shape-Preserving Spline Interpolation for Specifying Bivariate Functions on Grids

Compared with other methods, this technique for smooth surface interpolation over grid data reduces the number of extraneous local optima and inflection points of the surface.

Lipschitz and strong unicity constants for changing dimension

;I Tf,}& with lim,,m /! f -h 11 = 0 that lim,+, A(h) = h(f). For fixed f and n, Henry and Roulier [5] investigate the behavior of X(f) for changing intervals.

Designing faired parametric surfaces

Abstract The paper presents the theoretical foundation for the development of systems for the automatic fairing of parametric surfaces. Three fairness metrics are derived; each metric is a geometric property of the designed surface, which can be expressed explicitly in terms of the shape properties of the designed surface. The metrics are based on the surface areas of derived surfaces constructed from the geometric invariants of the designed surface. Consequently, these metrics can be evaluated in the original parameterization of the surface.

A Convexity-Preserving Grid Refinement Algorithm for Interpolation of Bivariate Functions

This article presents an algorithms to refine bevariate grid data that is convex (and monotonic) along the grid lines so that the refined data exhibits the same convexity (and monotonicity). The algorithm is based on some observations about univariate data and an algorithm for shape-preserving quadratic splines for such data. It can be used as is or with standard surface-path techniques.

Algorithm 574: Shape-Preserving Osculatory Quadratic Splines [E1, E2]

DESCRIPTION This algorithm is a FORTRAN implementation for the procedure developed in [3]. Let n data points {(x,, y,)} n.1 and n first derivatives {rn,),-1 at these data points be given, with x, <_-X,+l, 1 __ i __. n-1. The algorithm constructs a smooth osculatory quadratic spline S, which satisfies (1) S(x,) = y,, 1 <_ i <_ n; (2) S(xi) = m,, 1 <_ i <_ n; (3) S preserves monotonicity and convexity in the case that the slopes rn, are consistent with the shape of the data; (4) the knots of the spiine S include the data points and at most two additional knots between adjacent data points. The spline S is a piecewise quadratic Bernstein polynomial with a continuous first derivative. Permission to copy without fee all or part of this material is granted provided that the copras are not made or distributed for direct commercial advantage, the ACM copyright notice and the htle of the publication and its date appear, and notice is given that copying Is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission.

Preserving computational topology by subdivision of quadratic and cubic Bézier curves

Non-self-intersection is both a topological and a geometric property. It is known that non-self-intersecting regular Bézier curves have non-self-intersecting control polygons, after sufficiently many uniform subdivisions. Here a sufficient condition is given within ℝ3 for a non-self-intersecting, regular C2 cubic Bézier curve to be ambient isotopic to its control polygon formed after sufficiently many subdivisions. The benefit of using the control polygon as an approximant for scientific visualization is presented in this paper.