A rational parametrization of B\'ezier like curves

In this paper, we construct a family of Bernstein functions using a class of rational parametrization. The new family of rational Bernstein basis on an index $\alpha \in {\left(-\infty \, , \, 0 \right)}\cup {\left(1 \, , \, +\infty\right)}$, and for a given degree $k\in \mathbb{N}^*$, these basis functions are rational with a numerator and a denominator are polynomials of degree k. All of the classical properties as positivity, partition of unity are hold for these rational Bernstein basis and they constitute approximation basis functions for continuous functions spaces. The B\'ezier curves obtained verify the classical properties and we have the classical computational algorithms like the deCasteljau Algorithm and the algorithm of subdivision with the similar accuracy. Given a degree k and a control polygon points all of these algorithms converge to the same B\'ezier curve as the classical case. That means the B\'ezier curve is independent of the index $\alpha$. The classical polynomial Bernstein basis seems a asymptotic case of our new class of rational Bernstein basis.


Introduction
We know that, for a Bézier curve B of degree n ∈ I N * in I R d with d ∈ I N * and 1 ≤ d ≤ 3, the polynomial Bernstein basis (B n i ) n i=0 can be define for a < b ∈ I R and a parametrization x ∈ [a , b], by : where C i n = n! i!(n − i)! and 0! = 1.
Let (d i ) n i=0 be the control points of B, d i ∈ I R d for all i then we have In the same way we define the rational Bernstein basis (R i ) n i=0 of degree n ∈ I N * on [a , b] by where ω i > 0, ∀i = 0, . . . , n.
We can then define the rational Bézier curves by substiting the polynomial basis by the rational one.
We observe the that, by putting w(x) = Therefore, This function w we defined is increasing on [a , b] with w(a) = 0 and w(b) = 1.
The goal here is to retain these properties while requiring that w is homographic, to get a basis Bernstein naturally formed, of rational functions of degree (k, k), ie a numerator of degree k and a denominator of degree k.
2 New class of rational Bézier like basis 2

.1 A class of rational parametrizations
We start with a lemma who fixed the foundation of our new class rational Bezier curves and can be expressed as follow : Let α, β ∈] − ∞ , 0[∪]1 , +∞[ and f α , f β ∈ H([a , b]) the associate homographic functions Thus obtaining the standard case as a asymptotic one : b]) and n ∈ I N * . A rational Bernstein basis of α index and degree n on [a , b], is a family

A new class of rational Bernstein functions
It was agreed that α B 0 0 (x) = 1, ∀x ∈ [a , b] and [a , b] the parametrization space ( or parameter space). Definition 2.2 Let n, d ∈ I N * and a, b ∈ I R such that a < b.
is the rational Bernstein basis of α index and degree n as defined by the equation 2 is the classical polynomial Bernstein basis defined in 1.  A flash exploration of these figures shows that the parameter α has a notable effect on the functions of the Bernstein basis for any degree. We now refine this first reading by the check of the classical properties of Bernstein functions.

Properties of the new Bernstein basis
the rational Bernstein basis of α index and degree n defined by the equation 2.
The following properties hold :

(The positivity)
By remark 2.2, for all  Using the binomial formula of Newton's, we have , the rational Bernstein basis of α index and degree n on the parameter space [a , b]. satisfies : Proposition 2.2 Let n ∈ I N * and a, b ∈ I R such that a < b. For all x ∈ [a , b] and w(x) = f α (x), we have : By a direct computation.
The rational Bernstein basis ( α B n i ) n i=0 of α index and degree n on the parameter space [a , b], is formed of rational functions of degree (n , n) and infinitely differentiable on [a, b]. Moreover, we have : 1. First order derivative : For all x ∈ [a , b] and i = 0, . . . , n we have , it is the same for α B n i . We complete the proof by simple computation.
Remark 2.5 For all n ∈ I N * and a, b ∈ I R such that a < b and α ∈] − ∞ , 0[∪]1 , +∞[, using the relation 4, we observe that the derivatives of elements of the rational Bernstein basis of α index and degree n on the parameter space [a , b], ( α B n i ) n i=0 satisty the following relations : and and be the rational Bernstein basis of α index and degree n on the parameter space [a , b], 1. Uniqueness of the maximun : For i = 0, . . . , n, α B n i has a unique maximun in The value of this maximun is independant of α. More precisely, we have and it can be obtained by simple computation 2. Maximun symmetrical property : By simple computation, we have The rational Bernstein basis of α index and degree n on is a linear independent family.

Proof
We proceed by induction on n. Let In particular : i=0 is a linear independent family Assumes that the property holds up to an order n ∈ I N * , and show that Using the assumption : ( α B n i ) n i=0 is a linear independent system we have Otherwise The linear system from the equation 6 completed by equation 7 is inversible triangular upper since the terms of the diagonal are nonzero. We deduce that λ i = 0, ∀i = 0, . . . , n + 1. Then we can conclude that α B n+1 i n+1 i=0 is a linear independent system.
is a linear independent system then there is a basis of an approximation space of continuous real functions.

Properties of the new class of Bézier curves
Let n ∈ I N * and a, b ∈ I R such that a < b.
be the rational Bernstein basis of α index and degree n on Consider the rational Bézier curve B α of α index and degree n with control

Geometric properties
The curves of this new class check the properties of the classical Bézier curves . The proposition below lists the most important of these properties. Extremities tangents properties : The Bézier curve B α is tangent to its control polygon to the extremities. More precisely, we have 3. Convex hull property : B α is in the convex hull of its control points (d i ) Consider the rational Bézier curve B α of α index and degree n with control polygon points (d i ) n i=0 on the parameter space [a, b].

Using the remark (3), we have
Using the remark (5), we have Using a unity partition property we obtain 4. Let T be an affine transformation in I R d . There is a square matrix M of order d and a point C ∈ I R d such that for all X ∈ I R d , we have This proves the property. More precisely we have : Using the proposition 2.2, we have : This proves the result. A comparative analysis of figures 6, 7, 8, and 9 suggest that the Bézier curves are independent in index α, but each curve is function of its control polygon.

Algorithms for computing the Bézier curve
These algorithms show that it is possible to calculate a point of the Bézier curve or all without the need to build the basis of Bernstein. The most fondemental is deCasteljau algorithm which can be formulate by the following proposition : As in [1], we proceed by induction to prove this algorithm. Let and we deduce the result. The proof of this proposition uses the following three lemmas : The points d j k (c) defined by the algorithm of the proposition 3.4, satisty for all 0 ≤ k ≤ j ≤ n, the relation Let c ∈]a , b[ and w = f α (c). From the proposition 3.4, for all 0 ≤ k ≤ j ≤ n, . we proceed by induction to prove that for all 1 ≤ r ≤ j, d j k (c) = r i=0 d j−r k+i (c) α B r i (c) and we deduce the result for This completes the proof.  H([a , b]). There exists a bijective mapping u from [a , b] to [a , c] such that f α • u = f α (c)f α Moreover, for all n ∈ I N * and i ∈ I N with i ≤ n we have Step 1 First of all let us prove the existence of u Using the fact that f α is a bijection we have an unique u(t) ∈ [a , c] such that This implies the mapping t ∈ We deduce By an iterative use of the proposition 2.2 we have We deduce that Since for all i > j we have α B j i (c) = 0, the expected result follows.
Moreover, for all n ∈ I N * and i ∈ I N with i ≤ n Step 1  Since f α is increasing strictly we deduce that, for all t ∈ [a , b], we have an unique Using the fact that f α is a bijection we have an unique v(t) ∈ [a , c] such that f α (v(t)) = w(t).
This implies a mapping Then v is a bijection from [a , b] to [c , b] and satisfies, for all t ∈ [a , b] Step 2 Let t ∈ [a , b], w(t) = f α (t) and w(c) = f α (c) we havē Then we have An iterative use of the proposition 2.2 implies We deduce that because that, for all i < j we have α B i−j n−j (c) = 0 Proof Proof of the proposition 3.4 For all t ∈ [a , b], using the lemma 3.2 we have u(t) ∈ [a , c] and That is the expected result by using the lemma 3.1 With the lemma 3.3, for all t ∈ [a , b] we havê using the lemma 3.1. The point B α (c) = d n 0 computed by the relation The points equirepartition in the figure 13 confirms the self symmetry shown in the standard case which corresponds to α = ∞.  The following proposition shows that the curves B α are completely defined by their points of control. Which reinforces this observation. In this proof we only used the fact that f α is a diffeomorphism of the class C 2 .  dy dx 3 We deduce that κ α (x) and κ β (y) the respective curvatures of B α and B β at the common point B α (x) = B β (y) satisty This complete the proof.

Conclusion
We have an approximation basis of rational functions which unfortunately can not create the functions t → t 1 + t 2 and t → 1 − t 2 1 + t 2 . Thus, this new family of Bézier curves do not resolve the primary motivation of rational Bézier curves, which consists of generating planar conics exactly. This new class, however, gives another alternative construction of Bézier curves using algorithms as soon as accurate than the standard one with a large conservation of usual properties of Bernstein functions and Bézier curves.
The analysis of the approximation properties of this new class of Bernstein functions can be of some interest.