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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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 ijcvip.2011040104.m01 are Quadratic Bernstein Polynomials Defined As:

ijcvip.2011040104.m02, ijcvip.2011040104.m03, ijcvip.2011040104.m04

and ijcvip.2011040104.m05, ijcvip.2011040104.m06, are weights associated with the control points ijcvip.2011040104.m07. For a standard rational quadratic Bézier curve, ijcvip.2011040104.m08 and ijcvip.2011040104.m09 are equal to 1, while ijcvip.2011040104.m10 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 ijcvip.2011040104.m11:

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

Let ijcvip.2011040104.m12, ijcvip.2011040104.m13 and ijcvip.2011040104.m14 be three non-collinear points. ijcvip.2011040104.m15 and ijcvip.2011040104.m16 are the tangents to the circle C at ijcvip.2011040104.m17 and ijcvip.2011040104.m18. The circle C has center ijcvip.2011040104.m19 and radius R. Let ijcvip.2011040104.m20 be the middle of ijcvip.2011040104.m21. Let P be the perpendicular bisector plane of ijcvip.2011040104.m22. Let ijcvip.2011040104.m23 be the plane containing the Bézier curve. Let ijcvip.2011040104.m24 and ijcvip.2011040104.m25. Let ijcvip.2011040104.m26, where bar is an abbreviation for barycentre.

The rational quadratic Bézier curve y of weighted control points ijcvip.2011040104.m27 and ijcvip.2011040104.m28 is an arc of circle if and only if:

(2) which is equivalent to the equation ijcvip.2011040104.m30, where ijcvip.2011040104.m31 and ijcvip.2011040104.m32 are given by:


The positive, ijcvip.2011040104.m34, and the negative, ijcvip.2011040104.m35, solutions of the previous equation are:


Points ijcvip.2011040104.m37 and ijcvip.2011040104.m38 decompose the circle C in two adjacent arcs: the small arc, ijcvip.2011040104.m39, and the big arc, ijcvip.2011040104.m40. Let us note that if ijcvip.2011040104.m41 then the computation of weight ijcvip.2011040104.m42 is simplified to:


A rational biquadratic Bézier surfaces ijcvip.2011040104.m45 is defined by parameters ijcvip.2011040104.m46, a set of control points ijcvip.2011040104.m47 and the associated weights ijcvip.2011040104.m48 as:


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 ijcvip.2011040104.m50 and weights ijcvip.2011040104.m51.

Let ijcvip.2011040104.m52 and ijcvip.2011040104.m53 where ijcvip.2011040104.m54

Let ijcvip.2011040104.m55 and ijcvip.2011040104.m56 where ijcvip.2011040104.m57.

If ijcvip.2011040104.m58, the barycentric middle curve ijcvip.2011040104.m59 has control points ijcvip.2011040104.m60.

If ijcvip.2011040104.m61, the barycentric middle curve ijcvip.2011040104.m62 has control points ijcvip.2011040104.m63.

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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