### 2.1. Rational Bézier Curves and Surfaces

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

*(1)* where

*B*_{i}(t) are quadratic Bernstein polynomials defined as:

*B*_{o}(t)= (1-t)^{2}*B*_{1}(t)= 2t(1-t)*B*_{2}(t)=t^{2}and

*w*_{i},

{0,1,2}, are weights associated with the control points

*P*_{i}. For a standard rational quadratic Bézier curve,

*w*_{o} and

*w*_{1} are equal to 1, while

*w*_{1} 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

*w*_{1}:

Let *P*_{o}, *P*_{1}, and *P*_{2} be three non-collinear points. (*P*_{0}P_{1}) and (*P*_{2}P_{1}) are the tangents to the circle C at *P*_{o} and *P*_{2}. The circle *C* has center *O*_{o} and radius *R*. Let *I*_{2} be the middle of (*P*_{0}P_{2}). Let *P* be the perpendicular bisector plane of (*P*_{0}P_{2}). Let *P*_{c} be the plane containing the Bézier curve. Let (*w*_{0},*w*_{2}) and *w*= *w*_{0}+ *w*_{2}. Let *G*= *bar*{(*P*_{0},*w*_{0}),(*P*_{2},*w*_{2})}, where *bar* is an abbreviation for barycentre.

The rational quadratic Bézier curve *y* of weighted control points (*P*_{0},*w*_{0}), (*P*_{1},*w*_{1}) and (*P*_{2},*w*_{2}) is an arc of circle if and only if:

*(2)* which is equivalent to the equation

=0, where

and

are given by:

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

*(3)*Points *P*_{0} and *P*_{2} decompose the circle *C* in two adjacent arcs: the small arc, *P*_{0}*P*_{2}, and the big arc, *P*_{2}*P*_{0}. Let us note that if *w*_{0}= *w*_{2} =1 then the computation of weight *w*_{1} is simplified to:

*(4)*A rational biquadratic Bézier surfaces *S*(*u,v*) is defined by parameters (*u,v*) [0,1]^{2}, 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):

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 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.