Voronoi-based interpolation with higher continuity

1. I N T R O D U C T I O N This paper presents a series of infinitely many multidimensional interpolants based on the Voronoi diagram, most of which are new and higher-order continuous than the previous ones. In this sense, this paper opens a new door for the research on the Voronoi-based interpolation. Interpolation is an extremely important technique in the field of engineering: it is applicable to various problems such as differential equations and geometric modeling. The finite element method is one of the most practical approaches to the interpolation problem, and is well-established today. However, this method has disadvantage in the sense that the interpolant is not continuous in the displacement of the data sites. For example, if we use the Delaunay mesh, the mesh sometimes changes topologically as the data sites move, and hence the resulting interpolation change drastically. To overcome this disadvantage, we focus on an alternative approach, which utilizes the Voronoi diagram. In this paper, we call this approach the Voronoi-based approach. Permission to make digital or hard copies of all or pall of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the thll citation on the first page. To copy other~,ise, to republish, to post on selwers or to redistribute to lists, requires prior specific permission and/or a fee. Computational Geometry 2000 Hong Kong China Copyright ACM 2000 1-58113-224-7/00/6...$5.00 The interpolation problem is formulated in the following way. Let P1, . . . , P,~ be fixed points in R d, called data sites. Suppose that zl, . . . , zn are given reals. The (exact) interpolation problem is to find a good function q5 such that ~(P~) = z~ for i = 1, . . . , n. The meaning of the word "good" depends on the context. Thiessen first applied the Voronoi diagram to the interpolation problem [15]. In his method, the Voronoi diagram for the data sites is constructed. Let P be the target point the value on which is to be estimated. If the point P belongs to the Voronoi region of Pi, the value at P is estimated as zi. By the definition, Thiessen's interpolant is a piecewise constant function, and hence is not continuous. About twenty years ago, Sibson found another interpolation method [12, 13]. In his method, the Voronoi diagram for {P1, . . . ,Pn ,P} is constructed. If the Voronoi regions of P and Pi are adjacent via a (d 1)-dimensional facet, we call Pi a neighbor of P. The critical fact he found is that the position vector of P can be expressed as a convex combination of the position vectors of the neighbors of P with the coefficients computed from the second-order Voronoi diagram [12]. Hence we can interpret the coefficients of this convex combination as the coordinates of P. Sibson constructed C o and C 1 interpolants based on this coordinate system. Sibson's interpolation method was further researched by Farin [3] and Piper [10]. In this connection, Farin proposed another C 1 interpolant based on the Bernstein polynomials [3]. On the other hand, Hiyoshi and Sugihara found another coordinate system, and proposed a C O interpolant [5, 7, 14]. We call this system Laplace's coordinate system, because it can be obtained as the finite-volume approximation of Laplace's equation. Compared with the finite element interpolation, the Voronoibased interpolation has a lot of virtues [13]. For example, Voronoi-based interpolants behave continuously when the data sites move. However, the Voronoi-based approach is less flexible because even C 2 interpolants have not been known yet. In order to make the Voronoi-based approach more flexible, the authors proposed a framework for constructing Voronoi-based interpolants, including both Sibson's and Laplace's interpolants, and infinitely many other