Chongyang Deng

A unified interpolatory subdivision scheme for quadrilateral meshes

For approximating subdivision schemes, there are several unified frameworks for effectively constructing subdivision surfaces generalizing splines of an arbitrary degree. In this article, we present a similar unified framework for interpolatory subdivision schemes. We first decompose the 2n-point interpolatory curve subdivision scheme into repeated local operations. By extending the repeated local operations to quadrilateral meshes, an efficient algorithm can be further derived for interpolatory surface subdivision. Depending on the number n of repeated local operations, the continuity of the limit curve or surface can be of an arbitrary order CL, except in the surface case at a limited number of extraordinary vertices where C1 continuity with bounded curvature is obtained. Boundary rules built upon repeated local operations are also presented.

Improved bounds on the magnitude of the derivative of rational Bezier curves

Abstract In this paper we derive some new derivative bounds of rational Bezier curves according to some existing identities and inequalities. The comparison of the new bounds with some existing ones is also presented.

On the bounds of the derivative of rational Bézier curves

According to some identities and inequalities about the intermediate weights and control points of the de Casteljau algorithm for rational Bezier curve, we derive a new bound of the derivative of rational Bezier curves. The comparison of the new bounds with some existing ones is also presented. Based on some previous bounds, the bound presented in this paper and many numerical examples, we propose a conjecture of the bound for the derivative of rational Bezier curves.

Conditions for the coincidence of two quartic Bézier curves

This paper presents efficient and necessary coincidence conditions for two quartic Bezier curves, i.e., either their two control polygons are coincident or the two curves can be reparameterized or degenerated into the same quadratic Bezier curve.

C-shaped G2 Hermite interpolation by rational cubic Bézier curve with conic precision

Abstract We present a simple method for C-shaped G 2 Hermite interpolation by a rational cubic Bezier curve with conic precision. For the interpolating rational cubic Bezier curve, we derive its control points according to two conic Bezier curves, both matching the G 1 Hermite data and one end curvature of the given G 2 Hermite data, and the weights are obtained by the two given end curvatures. The conic precision property is based on the fact that the two conic Bezier curves are the same when the given G 2 Hermite data are sampled from a conic. Both the control points and weights of the resulting rational cubic Bezier curve are expressed in explicit form.

Characteristic conic of rational bilinear map

Abstract Bilinear and rational bilinear maps defined by planar quadrilaterals play important roles in computer graphics and geometric design. We show that the inverse of a rational bilinear map has a close relationship with a conic, which is named as ”characteristic conic” in this paper. We show that the characteristic conic of a rational bilinear map can be expressed as the rational quadratic Bezier curve, with its control points and weights in a closed form. Furthermore, we show that the characteristic conic is an envelope of two one-parameter families of straight lines, and all characteristic conics of a fixed quadrilateral form a one-parameter family.

Repeated local operations and associated interpolation properties of dual 2n-point subdivision schemes

Abstract In this paper we first derive a recursive relation of the generating functions of a family of dual 2 n -point subdivision schemes. Based on the recursive relation we design repeated local operations for implementing the 2 n -point subdivision schemes. Associated interpolation properties of the limit curve sequence of the dual 2 n -point subdivision schemes when n tends to infinity are then investigated. Based on the repeated local operations, we further prove that the limit curves of the family of the dual 2 n -point subdivision scheme sequence approach a circle that interpolates all initial control points as n approaches infinity, provided that the initial control points form a regular control polygon. Other interpolation properties show that the limit curve interpolates all closed initial control points with odd points or with even points but satisfying an extra condition, and interpolates all newly inserted vertices of an original closed polygon, when n approaches infinity. Some numerical examples are provided to illustrate the validity of our theoretic analyses.

The monotonicity of a family of barycentric coordinates for quadrilaterals

Abstract Recently, Floater (2016) proved that four well-known kinds of generalized barycentric coordinates in convex polygons share a simple monotonicity property. In this note we proved that a family of barycentric coordinates for quadrilaterals ( Floater, 2015a ) are monotonic, too.

A formula for estimating the deviation of a binary interpolatory subdivision curve from its data polygon

Abstract This paper introduces a new formula to evaluate the deviation of a binary interpolatory subdivision curve from its data polygon. We first bound the deviation of the new control points of each subdivision step from its data polygon by accumulating the distances between the new control points and the midpoints of their corresponding edges. Then, by finding the maximum deviation of each subdivision step, a formula for estimating the deviation of the limit curve from its data polygon can be deduced. As the applications of the formula, we evaluate the deviations of the uniform, centripetal and chord parametrization four-point interpolatory subdivision scheme, and find that the bounds derived by our method are sharper than bounds by [3]. Of course, we also deduce the new deviations of the six-point interpolatory subdivision scheme, Dyn et al’s four- and six-point subdivision schemes with tension parameters, and Deslauriers–Dubucs eight- and ten-point subdivision schemes.

Efficient evaluation of subdivision schemes with polynomial reproduction property

In this paper we present an efficient framework for the evaluation of subdivision schemes with polynomial reproduction property. For all interested rational parameters between 0 and 1 with the same denominator, their exact limit positions on the subdivision curve can be obtained by solving a system of linear equations. When the framework is applied to binary and ternary 4-point interpolatory subdivision schemes, we find that the corresponding coefficient matrices are strictly diagonally dominant, and so the evaluation processes are robust. For any individual irrational parameters between 0 and 1, its approximate value is computed by a recursive algorithm which can attain an arbitrary error bound. For surface schemes generalizing univariate subdivision schemes with polynomial reproduction property, exact evaluation methods can also be derived by combining Stams method with that of this paper. The method is applicable to all subdivision schemes with polynomial reproduction.It performs exact evaluation at rational parameters and approximate evaluation at other arbitrary parameters with tolerance control.It is efficient and robust for the presented schemes with corresponding coefficient matrix being strictly and diagonally dominant.It can also evaluate derivatives under the same framework.Extension of the method to surface cases is straightforward.