Monotone cubic interpolation

About: Monotone cubic interpolation is a research topic. Over the lifetime, 1740 publications have been published within this topic receiving 38111 citations.

A Parametric Cubic Element with Tension Properties

Abstract: In this paper we present the construction and study the properties of a new parametric cubic element having tension properties. The element is based on the Clough--Tocher split of a given triangle. Due to its tension properties the element can be used to obtain a local interpolation scheme for scattered data.

An Animation Engine with the Cubic Spline Interpolation

Abstract: We present a novel and efficient local spline interpolation algorithm and apply it into our application of key frame based 2D animation. Unlike global algorithms which need to solve a linear system every time a vertex is moved, our method performs a constant number of iteration that affects only a small number of control points over time. Therefore the cost of the interpolation does not depend on the total control point number.

Hermite interpolation by triangular cubic patches with small Willmore energy

Abstract: In this paper, a construction of a cubic Bezier spline surface that interpolates prescribed spatial points and the corresponding normal directions of tangent planes is proposed. Boundary curves of each triangular patch minimize the approximated strain energy. A comparison of optimal boundary curves is given. The interpolant minimizes Willmore energy functional. Some numerical examples and applications of the interpolation scheme are presented: surface approximation, hole filling and condensation of parameters.

Loop subdivision surfaces interpolating B-spline curves

Abstract: This paper presents a novel method for defining a Loop subdivision surface interpolating a set of popularly-used cubic B-spline curves. Although any curve on a Loop surface corresponding to a regular edge path is usually a piecewise quartic polynomial curve, it is found that the curve can be reduced to a single cubic B-spline curve under certain constraints of the local control vertices. Given a set of cubic B-spline curves, it is therefore possible to define a Loop surface interpolating the input curves by enforcing the interpolation constraints. In order to produce a surface of local or global fair effect, an energy-based optimization scheme is used to update the control vertices of the Loop surface subjecting to curve interpolation constraints, and the resulting surface will exactly interpolate the given curves. In addition to curve interpolation, other linear constraints can also be conveniently incorporated. Because both Loop subdivision surfaces and cubic B-spline curves are popularly used in engineering applications, the curve interpolation method proposed in this paper offers an attractive and essential modeling tool for computer-aided design.

On cubic Hermite coalescence hidden variable fractal interpolation functions

Abstract: Hermite interpolation is a very important tool in approximation theory and numerical analysis, and provides a popular method for modeling in the area of computer aided geometric design. However, the classical Hermite interpolant is unique for a prescribed data set, and hence lacks freedom for the choice of an interpolating curve, which is a crucial requirement in design environment. Even though there is a rather well developed fractal theory for Hermite interpolation that offers a large flexibility in the choice of interpolants, it also has the shortcoming that the functions that can be well approximated are highly restricted to the class of self-affine functions. The primary objective of this paper is to suggest a $$\mathcal{C}^1$$ -cubic Hermite interpolation scheme using a fractal methodology, namely, the coalescence hidden variable fractal interpolation, which works equally well for the approximation of a self-affine and non-self-affine data generating functions. The uniform error bound for the proposed fractal interpolant is established to demonstrate that the convergence properties are similar to that of the classical Hermite interpolant. For the Hermite interpolation problem, if the derivative values are not actually prescribed at the knots, then we assign these values so that the interpolant gains global $$\mathcal{C}^2$$ -continuity. Consequently, the procedure culminates with the construction of cubic spline coalescence hidden variable fractal interpolants. Thus, the present article also provides an alternative to the construction of cubic spline coalescence hidden variable fractal interpolation functions through moments proposed by Chand and Kapoor [Fractals, 15(1) (2007), pp. 41–53].