Shape-preserving interpolation in R3

Abstract

Contents ABSTRACT 1. Introduction 2. A notion of shape-preserving interpolation in IR 3 3. The family I?(K) of curvature-and torsion-continuous polynomial splines of non-uniform degree 4. The asymptotic properties of the family I?(K) for large degrees 5. Shape-preserving interpolation in IR 3 with the aid of I?(K) 5.1.

A biarc based subdivision scheme for space curve interpolation

“…Convexity preserving criteria of space curves have been suggested by Goodman and Ong (1997), Kaklis and Karavelas (1997) and Costantini and Pelosi (2001). Because the data processed in our biarc based subdivision scheme are points as well as their tangent vectors, different with other definitions of convexity, we define the convexity property of edges and points using points and their tangent vectors.…”

Shape-preserving interpolation of spatial data by Pythagorean-hodograph quintic spline curves

“…, N − 1. ∆ p j is the displacement vector for the spline segment u ∈ [ u j , u j+1 ]; N j,i and N j, f are the discrete binormals 1 at its end-points; and τ j has the same sign as the discrete torsion for that segment (Kaklis & Karavelas (1997)). Now if |S ′ (u)| = 0 and |S ′ (u) × S ′′ (u)| = 0 , the curvature vector and torsion are defined by…”

“…In the second example, the interpolation points are from a benchmark test for shape-preserving spline interpolation, proposed in Kaklis & Karavelas (1997) -see Remark 2 below for the unit tangents. The data specify a symmetric closed curve, with three collinear points and also several coplanar points.…”

“…Right: the torsion of the PH quintic spline curve (green), and the sign of the discrete torsion (blue). The "chair" data from Kaklis & Karavelas (1997). The PH quintic spline is constructed by defining α, β through the CC criterion and setting s = 1 in each segment.…”

“…Among such methods, the well-known tension schemes, originally introduced for univariate functional interpolation, employ free parameters that can be used to make a smooth interpolant converge toward the piecewise-linear curve connecting the data points (which is inherently shape-preserving). While shape-preserving planar interpolation can be traced to the introduction of the exponential tension spline in Schweikert (1966), the development of shape-preserving interpolation algorithms for spatial data is more recent -see, for example, Kaklis & Karavelas (1997); Goodman et al (1998); Karavelas & Kaklis (2000); Asaturyan et al (2001); Costantini et al (2001); Costantini & Manni (2003); Renka (2005).…”