Non-self-intersection is both a topological and a geometric property. Recent theorems show how non-self-intersecting Bézier curves also have non-self-intersecting control polygons, after sufficiently many uniform subdivisions. As a partial generalization, a sufficient condition is given within R for a non-self-intersecting, C cubic Bézier curve to be ambient isotopic to its control polygon form...

We present a new 3D surface modeling approach that enables curve-based creation and modification of smooth surfaces by sketching. The key feature of the proposed methods is a two-way communication between the user-designed curve networks and the generated surfaces. A user-drawn curve network serves as a control cage, from which a subdivision surface is generated. The subdivision surface is upda...

This paper presents an algorithm dealing with the data reduction and the approximation of 3D polygonal curves. Our method is able to approximate efficiently a set of straight 3D segments or points with a piecewise smooth subdivision curve, in a near optimal way in terms of control point number. Our algorithm is a generalization for subdivision rules, including sharp vertex processing, of the Ac...

ABSTRACT. A multiresolution analysis of a curve is normal if each wavelet detail vector with respect to a certain subdivision scheme lies in the local normal direction. In this paper we study properties such as regularity, convergence, and stability of a normal multiresolution analysis. In particular we show that these properties critically depend on the underlying subdivision scheme and that i...

In recent years, subdivision has emerged as a major geometric modeling technique. Algorithms for generating subdivision curves are often specified in terms of iterated matrix multiplication. Each multiplication maps a globally indexed sequence of points that represents a coarser approximation of the curve onto a longer sequence that represents a finer approximation. Unfortunately, this use of m...

2. MAIN RESULTS In this paper, subdivision methods for rectangular Bézier A rectangular Bézier surface of degree n 3 m can be surfaces are generalized to subdivide a rectangular Bézier surface patch of degree n 3 m into two rectangular Bézier surrepresented by face patches of degree n 3 (m 1 n), while the parameter domain of the Bézier surface is decomposed into two trapezoids. As an applicatio...

A Bézier curve segment is generally rendered by subdividing it into a series of disjoint curve subsegments, and then approximating each subsegment by joining its endpoints by a line segment (chord). The maximum transverse deviation of each curve subsegment from the corresponding chord (the achieved flatness) should be no greater than a minimum error value, , called the flatness. The standard te...

This paper presents an interpolating ternary butterfly subdivision scheme for triangular meshes based on a 1-9 splitting operator. The regular rules are derived from a C2 interpolating subdivision curve, and the irregular rules are established through the Fourier analysis of the regular case. By analyzing the eigen-structures and characteristic maps, we show that the subdivision surfaces genera...

A non-uniform 3-point ternary interpolatory subdivision scheme with variable subdivision weights is introduced. Its support is computed. The C and C convergence analysis are presented. To elevate its controllability, a modified edition is proposed. For every initial control point on the initial control polygon a shape weight is introduced. These weights can be used to control the shape of the c...

For a wide class of stationary subdivision methods, we derive necessary and suucient conditions for these schemes to produce C k continuous limit curves. These stationary schemes include those arising from midpoint subdivision of irregularly-spaced knot sequences. We also describe a matrix method for computing various derivative schemes associated with such stationary schemes. x1. Introduction ...