Bézout coefficients of coprime numbers approximate quadratic Bézier curves

Bézier curves are parametrized curves extensively used in computer aided geometric design (CAGD) [2]. Bézier curves are related to Berenstein polynomials, they have very interesting mathematical properties and the literature contains many proposals of algorithms for their algebraic construction and their visualization [3, 5]. In this work, we will focus on quadratic Bézier curves. Since a quadratic Bézier curve may be described as the envelope of a family of segments whose endpoints move along two straight lines, then it is conceivable to replace this family of lines with another suitable family of lines whose corresponding endpoints are close to each other. When the quadratic Bézier curve is relative to the points $(p,q)$, $(0,0)$ and $(q,p)$, it turns out that such replacement lines may be characterized as having as endpoints pairs of Bézout coefficients of coprime numbers close to the points $(p,q)$ and $(q,p)$, respectively. The main observation that leads to the verification of this statement is fairly simple: Denote $P_{0}=(p,q)$, $P_{1}=(0,0)$ and $P_{2}=(q,p)$ and suppose that the coprime pair of positive numbers $(r,s)$ is close to the pair $(p,q)$. Then, the Bézout coefficients $Q_{1}$ and $Q_{2}$ of the two coprime pairs $(r,s)$ and $(s,r)$, respectively, are close to their projections $R_{1}$ and $R_{2}$ on the lines $P_{0}P_{1}$ and $P_{1}P_{2}$, respectively. Moreover, the distance from $R_{1}$ to $P_{0}$ is approximately the same as the distance from $R_{2}$ to $P_{1}$. Hence, the line with endpoints $Q_{1}$ and $Q_{2}$ is the candidate to replace a line in the envelope of the quadratic Bézier curve associated to $P_{0}$, $P_{1}$ and $P_{2}$. When the norm of $(p,q)$ is sufficiently large, there will be many coprime pairs in a neighbourhood of $(p,q)$. Thus, when plotting all such lines with endpoints $Q_{1}$ and $Q_{2}$, they will approximate the quadratic Bézier curve corresponding to $P_{0}$, $P_{1}$ and $P_{2}$.

This work is divided in two sections. In Section 1 we establish the notation and state the elementary results needed in the rest of the paper and Section 2 contains the main results.

The linear Bézier curve for two points ${P}_{0}$ and ${P}_{1}$ in the plane, is defined as just the parametrized line $t\mapsto(1-t){P}_{0}+t{P}_{1}$ from $P_{0}$ to $P_{1}$. Given three different points ${P}_{0}$, ${P}_{1}$ and ${P}_{2}$ in the plane, the quadratic Bézier curve is a parametrized curve in terms of the linear Bézier curves from ${P}_{0}$ to ${P}_{1}$ and from ${P}_{1}$ to ${P}_{2}$, that is, the quadratic Bézier curve for the three points ${P}_{0}$, ${P}_{1}$ and ${P}_{2}$ in the Cartesian plane, is the parametric curve:

For each $0<t_{0}<1$, it is well known that a parametrization of the straight line tangent to $c_{p,q}(t)$ at the point $c_{p,q}(t_{0})$, when $t=t_{0}$, is the line $\gamma_{t_{0}}$. Indeed, a parametrization of such a tangent line is:

In fact, $c_{p,q}$ may be described as the envelope of the family of lines $\gamma_{s}$.

The following notion was introduced as Definition 1 in [1].

The following proposition gives a couple of formulas relating the Bézout coefficients of a coprime pair and the Bézout coefficients of the flipped pair.

We now prove a result that allows to argue why the points $\mathcal{B}(r,s)$ and $\mathcal{B}(s,r)$ are suitable substitutes for the endpoints of the envelope of the family of lines $\gamma_{s}$ of the quadratic Bézier curve $c_{p,q}$ relative to $(p,q)$, $(0,0)$ and $(q,p)$, when $(r,s)$ is close to $(p,q)$. More precisely, we prove that the distances from the projections of $\mathcal{B}(p,q)$ and $\mathcal{B}(q,p)$ on the lines $y=\frac{q}{p}x$ and $y=\frac{p}{q}x$ to $(p,q)$ and $(0,0)$, respectively, are equal. Furthermore, the distances from $\mathcal{B}(p,q)$ and $\mathcal{B}(q,p)$ to the lines $y=\dfrac{q}{p}x$ and $y=\dfrac{p}{q}x$ are both equal to the inverse of the norm of $(p,q)$. It will be proved in the next section why such segments with endpoints $\mathcal{B}(r,s)$ and $\mathcal{B}(s,r)$ plays the role of a segment $\gamma_{s}$ in defining the quadratic Bézier curve $c_{p,q}$.

Since given a coprime pair $(p,q)$ the line with endpoints $\mathcal{B}(p,q)$ and $\mathcal{B}(q,p)$ will play an important role in the sequel, we introduce a notation for it next.

Finally, we will define what we mean for two segments to be “close”.

We have the following straightforward result.

In this section we prove our main result, namely, that a quadratic Bézier curve $c_{p,q}$ as in Definition 1.1 can be approximated through a family of Bézier-Bézout line segments given in Definition 1.5.

For integers $p>3$ and $0\leq q<p$, our next proposition will show how for a given $1\leq\epsilon\leq\frac{1}{2}\|(p,q)\|$ and any pair of relatively prime numbers $(r,s)$ in a neighborhood of radius $\epsilon$ and center $(p,q)$ one gets a real number $0<t_{0}<1$ such that both the distances from $\mathcal{B}(r,s)$ to $\alpha_{p,q}(t_{0})$ and from $\mathcal{B}(s,r)$ to $\beta_{p,q}(t_{0})$ are less to $\epsilon+1$. Since $\mathcal{B}(r,s)$ and $\mathcal{B}(s,r)$ are the endpoints of the Bézier-Bézout line $L_{r,s}$, $\alpha_{p,q}(t_{0})$ and $\beta_{p,q}(t_{0})$ are the endpoints of $\gamma_{t_{0}}$, and the Bézout quadratic $c_{p,q}$ is the envelope of the family of lines $\gamma_{s}$, this proposition will prove to be crucial for our main result.

Notice that the smaller the $\epsilon$ is, the fewer the pairs $(r,s)$ of relatively prime numbers satisfying the condition of Proposition 2.1. For example, for $p=300$, $q=21$ and $\epsilon=1$, then $(299,21)$ is the only such pair. Therefore the size of $\epsilon$ has to increase to obtain more coprime pairs within $\epsilon$ of $(p,q)$; however, increasing the size of $\epsilon$ will worsen our approximation of the Bézier quadratic.

The following theorem is our main result. It establishes the approximation of the Bézier quadratic curve $c_{p,q}$ in Definition 1.1 through segments having as endpoints pairs of Bézout coefficients. More precisely, it establishes conditions for a point in the Bézier-Bézout segment $L_{r,s}$ to be close to a point of the quadratic Bézier curve $c_{p,q}$.

In Figure 1 we show some examples of approximately quadratic Bézier curves using $\epsilon=10$.