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,
{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:
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) 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
=0, where
and
are given by:
The positive, and the negative, solutions of the previous equation are:
(3)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:
(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.