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 
:
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):
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.