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, & Qatar University, Qatar)
DOI: 10.4018/978-1-4666-3906-5.ch009
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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 Surfaces

Rational quadratic Bézier curves are second degree parametric curves defined by:

(1) where Bi(t) are quadratic Bernstein polynomials defined as:Bo(t)= (1-t)2B1(t)= 2t(1-t)B2(t)=t2and wi, 978-1-4666-3906-5.ch009.m02{0,1,2}, are weights associated with the control points Pi. For a standard rational quadratic Bézier curve, wo and w1 are equal to 1, while w1 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 w1:

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

Let Po, P1, and P2 be three non-collinear points. (P0P1) and (P2P1) are the tangents to the circle C at Po and P2. The circle C has center Oo and radius R. Let I2 be the middle of (P0P2). Let P be the perpendicular bisector plane of (P0P2). Let Pc be the plane containing the Bézier curve. Let (w0,w2)978-1-4666-3906-5.ch009.m03 and w= w0+ w2. Let G= bar{(P0,w0),(P2,w2)}, where bar is an abbreviation for barycentre.

The rational quadratic Bézier curve y of weighted control points (P0,w0), (P1,w1) and (P2,w2) is an arc of circle if and only if:

(2) which is equivalent to the equation 978-1-4666-3906-5.ch009.m05=0, where 978-1-4666-3906-5.ch009.m06 and 978-1-4666-3906-5.ch009.m07 are given by:


The positive, 978-1-4666-3906-5.ch009.m09 and the negative, 978-1-4666-3906-5.ch009.m10 solutions of the previous equation are:


Points P0 and P2 decompose the circle C in two adjacent arcs: the small arc, P0P2, and the big arc, P2P0. Let us note that if w0= w2 =1 then the computation of weight w1 is simplified to:


A rational biquadratic Bézier surfaces S(u,v) is defined by parameters (u,v)978-1-4666-3906-5.ch009.m13 [0,1]2, a set of control points 978-1-4666-3906-5.ch009.m14 and the associated weights 978-1-4666-3906-5.ch009.m15 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 978-1-4666-3906-5.ch009.m17 and weights 978-1-4666-3906-5.ch009.m18 Let

where 978-1-4666-3906-5.ch009.m21


where 978-1-4666-3906-5.ch009.m24


the barycentric middle curve
has control points 978-1-4666-3906-5.ch009.m27


the barycentric middle curve
has control points 978-1-4666-3906-5.ch009.m30

More details on Bézier surfaces can be found in (Farin, 1993; 1999; Garnier, 2007). In the remainder 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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