Construction of 3D Triangles on Dupin Cyclides

Construction of 3D Triangles on Dupin Cyclides

Bertrand Belbis (Université de Bourgogne, France), Lionel Garnier (Université de Bourgogne, France) and Sebti Foufou (Université de Bourgogne, France and Qatar University, Qatar)
Copyright: © 2011 |Pages: 16
DOI: 10.4018/ijcvip.2011040104
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Abstract

This paper considers the conversion of the parametric Bézier surfaces, classically used in CAD-CAM, into patched of a class of non-spherical degree 4 algebraic surfaces called Dupin cyclides, and the definition of 3D triangle with circular edges on Dupin cyclides. Dupin cyclides was discovered by the French mathematician Pierre-Charles Dupin at the beginning of the 19th century. A Dupin cyclide has one parametric equation, two implicit equations, and a set of circular lines of curvature. The authors use the properties of these surfaces to prove that three families of circles (meridian arcs, parallel arcs, and Villarceau circles) can be computed on every Dupin cyclide. A geometric algorithm to compute these circles so that they define the edges of a 3D triangle on the Dupin cyclide is presented. Examples of conversions and 3D triangles are also presented to illustrate the proposed algorithms.
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2. Background

2.1. Rational Bézier Curves and are Quadratic Bernstein Polynomials Defined As:

, ,

and , , are weights associated with the control points . For a standard rational quadratic Bézier curve, and are equal to 1, while can be used to control the type of the conic defined by the curve (Farin, 1993, 1999; Garnier, 2007). As we will model circular arcs using rational quadratic Bézier curve in our algorithm, let us first recall a theorem to compute the weight :

  • Theorem 1.Circle defined by two points and the tangents at these points.

Let , and be three non-collinear points. and are the tangents to the circle C at and . The circle C has center and radius R. Let be the middle of . Let P be the perpendicular bisector plane of . Let be the plane containing the Bézier curve. Let and . Let , where bar is an abbreviation for barycentre.

The rational quadratic Bézier curve y of weighted control points and is an arc of circle if and only if:

(2) which is equivalent to the equation , where and are given by:

The positive, , and the negative, , solutions of the previous equation are:

(3)

Points and decompose the circle C in two adjacent arcs: the small arc, , and the big arc, . Let us note that if then the computation of weight is simplified to:

(4)

A rational biquadratic Bézier surfaces is defined by parameters , a set of control points and the associated weights as:

(5)

The following theorem for the computation of the barycentric middle curve on a Bézier surface will be used in Section 4. The proof of this theorem can be found in (Garnier, Belbis, & Foufou, 2009):

  • Theorem 2.Barycentric middle curve

Let us consider a Bézier surface defined by control points and weights .

Let and where

Let and where .

If , the barycentric middle curve has control points .

If , the barycentric middle curve has control points .

More details on Bézier surfaces can be found in (Farin, 1993; 1999; Garnier, 2007). In the remaining of this paper, we only consider rational Bézier curves and surfaces of degree two to which we refer, for short, by Bézier curves and Bézier surfaces.

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