In this paper, we investigate subdivision tree representations of arbitrary triangle meshes. By subdivision, we mean the recursive topological partitioning of a triangle into subtriangles. Such a process can be represented by a subdivision tree. We identify the class of regular triangle meshes, those that can be losslessly represented by a nontrivial subdivision tree, i.e. by a subdivision tree...

Traditionally, modelling using spline curves and surfaces is facilitated by control points. We propose to enhance the modelling process by the use of control vectors. This improves upon existing spline representations by providing such facilities as modelling with local (semi-sharp) creases, vanishing and diagonal features, and hierarchical editing. While our prime interest is in surfaces, most...

The Bezier formulation for parametric curves has many quali ties, among them the intuitive relationship between the shape of the control polygon and the shape of the curve , and the case of computation and subdivision. Other formulations , however, have become more popular because they offer local control, or because they are interpolating, or even more recently because they provide the added f...

Algorithms for drawing smooth curves and surfaces are well-known but may be too expensive in CPU time to use for certain high speed interactive graphics applications. This paper looks at some new geometrically-based smoothing algorithms that arrive at the final smooth curve or surface after the application of an infinite number of iterations of the algorithm using subdivision refinement. When C...

In this paper, we construct a family of Bernstein functions using class rational parametrization. The new basis on an index $\alpha \in {\left(-\infty \, , 0 \right)}\cup {\left(1 +\infty\right)}$, and for given degree $k\in \mathbb{N}^*$, these are with numerator denominator polynomials k. All the classical properties as positivity, partition unity hold they constitute approximation continuous...

We present an approach to finding the implicit equation of a planar rational parametric cubic curve, by defining a new basis for the representation. The basis, which contains only four cubic bivariate polynomials, is defined in terms of the Bézier control points of the curve. An explicit formula for the coefficients of the implicit curve is given. Moreover, these coefficients lead to simple exp...

A new technique and an algorithm have been designed which are meant to automatically capture the outlines of fonts. A more flexible class of B! ezier cubic (generalized Hermite cubic) functions are the basis of this technique. This class of functions involves two families of control parameters, to produce more flexible shapes than ordinary Hermite or B! ezier cubics, in each segment. These func...

This paper presents two self-similar models that allow the control of curves and surfaces. The first model is based on IFS (Iterated Function Systems) theory and the second on subdivision curve and surface theory. Both of these methods employ the detail concept as in the wavelet transform and allow the multiresolution control of objects with control points at any resolution level. In the first ...

Here we show how reversible computation processes, like Margolus diffusion[1], can be envisioned as physical turning operations on a 2dimensional rigid surface that is cut by a regular pattern of intersecting circles. We then briefly explore the design-space of these patterns, and report on the discovery of an interesting fractal subdivision of space by iterative circle packings. We devise two ...