# Subdivisions of Ring Dupin Cyclides Using Bézier Curves with Mass Points

Lionel Garnier, Lucie Druoton, Jean-Paul Becar, Laurent Fuchs, Géraldine Morin

► To cite this version:

Lionel Garnier, Lucie Druoton, Jean-Paul Becar, Laurent Fuchs, Géraldine Morin. Subdivisions of Ring Dupin Cyclides Using Bézier Curves with Mass Points. WSEAS Transactions on Mathematics, 2021, 20, pp.581-597. 10.37394/23206.2021.20.62. hal-03521704

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# Subdivisions of Ring Dupin Cyclides Using Bézier Curves with Mass Points

LIONEL GARNIER<fnref n="1" />, LUCIE DRUOTON², JEAN-PAUL BÉCAR³
LAURENT FUCHS⁴, GÉRALDINE MORIN⁵

¹L.I.B., University of Burgundy, B.P. 47870, 21078 Dijon Cedex, France;

²I.U.T. of Dijon, University of Burgundy-Franche-Comté, B.P. 47870, 21078 Dijon Cedex, France;

³U.P.H.F. - Campus Mont Houy - 59313 Valenciennes Cedex 9, France;

⁴X.L.I.M., U.M.R. 7252, University of Poitiers, France;

⁵Laboratoire IRIT, U.M.R. 5505, University Paul Sabatier, 31 000 Toulouse, France;

**Abstract:** The paper deals in the Computer-Aided Design or Computer-Aided Manufacturing domain with the Dupin cyclides as well as the Bézier curves. It shows that the same algorithms can be used either for subdivisions of ring Dupin cyclides or Bézier curves. The Bézier curves are described with mass points here. The Dupin cyclides are considered in the Minkowski-Lorentz space. This makes a Dupin cyclide as the union of two conics on the unit pseudo-hypersphere, called the space of spheres. And the conics are quadratic Bézier curves modelled by mass points. The subdivision of any Dupin cyclide, is equivalent to subdivide two curves of degree 2, independently, whereas in the 3D Euclidean space, the same work implies the subdivision of a rational quadratic Bézier surface and resolutions of systems of three linear equations. The first part of this work is to consider ring Dupin cyclides because the conics are bounded circles which look like ellipses.

**Key-Words:** Mass points, Rational quadratic Bézier curves, Conics, Subdivisions, Space of spheres, Minkowski-Lorentz space, Ring Dupin cyclides

Received: April 23, 2021. Revised: October 9, 2021. Accepted: October 23, 2021. Published: November 9, 2021.

## 1 Introduction

Dupin cyclides are algebraic surfaces introduced in 1822 by the french mathematician Pierre-Charles Dupin [1] and were introduced in Computer Aided Design (C.A.D.) by R. Martin in 1982 [2]. They have a low algebraic degree: at most 4. These Dupin cyclides have a parametric equation and two equivalent implicit equations [3, 4] and they have been studied by a lot of mathematicians [4, 5, 6]. Since a score of years, much of authors used them in Computer Aided Geometric Design (C.A.G.D.). In previous studies [7, 8, 9, 10], Dupin cyclides are represented by Rational Biquadratic Bézier Surfaces (R.B.B.S.) in the usual 3D Euclidean affine space $\mathcal{E}_3$. Four patches are necessary to model the whole cyclide. Then, the conditions of the control points of a R.B.B.S. that can represent a Dupin cyclide patch have been given [11, 12, 13, 14, 15, 16]. A similar construction is

possible in the space of spheres [17].

In C.A.D., rational Bézier curves are the basis for the standard Non-Uniform Rational Bézier Splines (N.U.R.B.S.) representation. In particular, second order rational Bézier curves model conic arcs [18, 19, 14, 20]. However, since point on Bézier curve are expressed as barycenters of a set of the control points, they are, in the classical setting, limited to modeling bounded arcs. Unbounded conics arcs, like parabola or hyperbola branches, may be modeled by considering the joint space of weighted points and vectors [21]. In this space, vectors correspond to points associated the weight zero. Here, we propose to further investigate this general setting to propose an original subdivision algorithm for bounded and unbounded conic arcs<fnref n="1" /> and the subdivision is independent of the rational parameterization. The subdivision

<footnote n="1">¹The work will be presented in a future paper.</footnote>
theorems in Paragraph 4, lay on Theorem 5 which offers to change mass points. The Theorem 5 in B.2.3 and his Corollary 1 give a way to build a conic arc that eases the calculation of the conic invariants [20, 22, 23].

The space of spheres was introduced in various ways. For example, M. Berger [6] works in the projective space of the quadratic forms on the affine Euclidean space, M. Paluszny [24] works in a four dimensional projective space using the hypersphere of Moebius. While U. Hertrich-Jeromin [25], T. Cecil [26], R. Langevin and P. Walczak [27] use a four dimensional quadric $\Lambda^4$ in the five dimensional Minkowski-Lorentz space which is endowed with a non-degenerate indefinite quadratic form of signature (4, 1). That space is a generalization of the space-time used in Einstein's theory, equipped of the non-degenerate indefinite quadratic form

$$Q_M(\vec{u}) = x^2 + y^2 + z^2 - c^2 t^2$$

where $(x, y, z)$ are the spacial components of the vector $\vec{u}$ and $t$ is the time component of $\vec{u}$ and $c$ is the constant of the speed of light. The Minkowski-Lorentz space permits to solve the three contact condition problem [28] whereas the other space leads to nothing: with the projective spaces, the orientations of the spheres are lost. In these projective space, from two spheres of centers $O_0$ and $O_1$ and of non-null radius $r$, we can not distinguish a cone and a cylinder. In the Minkowski-Lorentz spheres, the two spheres of centers $O_0$ and $O_1$ and of non-null radius $r$ lead to a circular cylinder whereas the sphere of center $O_0$ and of radius $r$ and a sphere of center $O_1$ and radius $-r$ lead to a circular cone.

In the Minkowski-Lorentz space, a Dupin cyclide is the union of two conics, see Definition 1 and Table 1. We can choose a vector using perpendicular conditions and pseudo-metric conditions to determine a Bézier curve which models a circle for the non-degenerate indefinite quadratic form and this circle looks like an ellipse or hyperbola with an Euclidean point of view. Some recalls about the Minkowski-Lorentz spaces $L_{4,1}$ and $L_{4,1}$ can be found in [29, 30, 31, 32, 33, 34, 17, 28]. Some formulae are given in Appendix A. The use of Minkowski-Lorentz space permits to simplify the subdivision algorithms developed by L. Garnier and C. Gentil [15]. One can note that an inversion is not an affine, transformation and so, the control points of the image of a Bézier curve are not the images of the control points of the original Bézier curve. Then, the general Dupin cyclide case and torus case must be distinguish. Some recalls

about mass points and Bézier curves can be found in [21, 35, 36, 29, 30], to facilitate the read of this paper, some formulae are given in Appendix B.

The paper is organized as follows: Section 2 presents Dupin cyclides in the 3-dimensional Euclidean affine space $\mathcal{E}_3$ and in the Minkowskis-Lorentz space. In section 3, the authors present the adaptation of De Casteljau algorithm to Bézier curves with mass points. Section 4 presents methods to subdivide Bézier curves which model ellipse arcs and then Ring Dupin Cyclide. Before the conclusion and the perspectives, in section 5, the authors present the method to subdivide Ring Dupin cyclides patches. The Appendix A (resp. B) presents some fundamental recalls about Minkowski-Lorentz space (resp. Bézier curves with mass points) to clarify this paper.

## 2 Dupin cyclide in the Minkowskis-Lorentz space

An Euclidean sphere $S$ of the 3-dimensional usual affine Euclidean space $\mathcal{E}_3$ of center $\Omega$ and of radius $r$ defines two oriented spheres $S^+$ and $S^-$ of center $\Omega$ and of radius $\rho = r$ and $\rho = -r$ respectively. For any point $M$ belonging to the sphere $S^+$ or $S^-$, we have

$$\vec{\Omega M} = \rho \vec{N} \qquad (1)$$

where $\vec{N}$ is the unit normal vector to the considered sphere at the point $M$.

The space of spheres $\Lambda^4$ is the 4-dimensional pseudo-unit hypersphere of the Minkowskis-Lorentz space, see Appendix A.

Dupin cyclides can be defined in different ways [4, 3]. Using the space of spheres in the Minkowski-Lorentz space, we use the following definition:

**Definition 1:**

A Dupin cyclide is, in two different ways, the envelope of an one-parameter family of oriented spheres. Each family of spheres can be seen as a conic in the space of spheres $\Lambda^4$. These two conics are called brother circles.

We can distinguish five kinds of Dupin cyclides and the type of the conic in the Minkowskis-Lorentz space depends of the number of singular points of these surfaces [2, 7, 37, 38], see Table 1. A Ring Dupin cyclide is a Dupin cyclide without singular points, see Fig. 1.

The representation of a Ring Dupin cyclide in the space of spheres $\Lambda^4$ is the union of two circles which look like ellipses (with an Euclidean point of view), Fig. 2. In a second paper, we will deal
<table><caption>Table 1: Kinds of Dupin cyclides and their representations in the Minkowski-Lorentz space.</caption><thead><tr><th>Name of Dupin cyclide</th><th>Number of singular point(s)</th><th>Lorentz property</th><th>Euclidean point of view</th></tr></thead><tbody><tr><td>Ring</td><td>0</td><td>Two circles</td><td>Two ellipses</td></tr><tr><td>Horned</td><td rowspan="2">2</td><td rowspan="2">Two circles</td><td rowspan="2">An ellipse and a hyperbola</td></tr><tr><td>Spindle</td></tr><tr><td>One-singularity spindle</td><td rowspan="2">1</td><td rowspan="2">A circle and a parabola</td><td rowspan="2">An ellipse and a parabola</td></tr><tr><td>Singly horned</td></tr></tbody></table>

<figure><img src="image_2.png" /><figcaption>Figure 1: Three ring Dupin cyclides in $\mathcal{E}_3$. (a): torus. (b): general quartic Dupin cyclide. (c): cubic Dupin cyclide.</figcaption></figure>

with the other Dupin cyclides represented by a circle which looks like an ellipse and an other conic: a circle which looks like a hyperbola or an affine parabola isometric to a line [39]. For $i$ in $\{1, 2\}$, let $\Omega_i$ be the center of the circle $C_i$ which is contained into the affine 2-plane $\mathcal{P}_i$. Then, the planes $\mathcal{P}_i$ and $\mathcal{P}_{3-i}$ are perpendicular, the line $(\Omega_1\Omega_2)$ is perpendicular to $\mathcal{P}_i$.

From the Minkowski-Lorentz spaces, the set $\tilde{\mathcal{L}}_{4,1}$ of mass points $(A, a)$ are defined, see Appendix B, where

* $a = 0$ implies that $A$ is a vector of $\overrightarrow{\mathcal{L}}_{4,1}$;

* $a \neq 0$ implies that $A$ is a point of $\mathcal{L}_{4,1}$.

### 3 The De Casteljau algorithm adapted to $\tilde{\mathcal{L}}_{4,1}$

A recall about the rational quadratic Bézier curve with mass points of control $(P_0; \omega_0)$, $(P_1; \omega_1)$ and $(P_2; \omega_2)$ is done in Appendix B.2. To simplify the rest of this paper, we introduce the following notation:

**Notation 1:**

$BR\{(P_0; \omega_0); (P_1; \omega_1); (P_2; \omega_2)\}$ denotes a rational quadratic Bézier curve with the following control mass points $(P_0; \omega_0)$, $(P_1; \omega_1)$ and $(P_2; \omega_2)$.

<figure><img src="image_3.png" /><figcaption>Figure 2: Representation of a ring Dupin cyclide on $\Lambda^4$: the two brother circles $C_1$ and $C_2$ look like ellipses.</figcaption></figure>

Given that the law $\oplus$, defined in Appendix B, is associative, the De Casteljau algorithm can be generalized to the Bézier curve in the space of mass points, Algorithm 1.
Algorithm 1 The De Casteljau algorithm in the space of mass points.

Input : Three mass points $(P_0; \omega_0)$, $(P_1; \omega_1)$ and $(P_2; \omega_2)$ not collinear.

1. Choice of $t$ in $]0; 1[$

2. Calculation of:

$$(N_1; \varpi_1) = (1 - t) \odot (P_0; \omega_0) \oplus t \odot (P_1; \omega_1)$$

3. Calculation of:

$$(N_2; \varpi_2) = (1 - t) \odot (P_1; \omega_1) \oplus t \odot (P_2; \omega_2)$$

4. Calculation of:

$$(N_3; \varpi_3) = (1 - t) \odot (N_1; \varpi_1) \oplus t \odot (N_2; \varpi_2)$$

Output : A mass point $(N_3; \varpi_3)$ belonging to the conic arc modeled by $BR\{(P_0; \omega_0); (P_1; \omega_1); (P_2; \omega_2)\}$

From Algorithm 1, it yields:

$$ \begin{aligned} (N_3; \varpi_3) &= B_0(t) \odot (P_0; \omega_0) \\ &\oplus B_1(t) \odot (P_1; \omega_1) \\ &\oplus B_2(t) \odot (P_2; \omega_2) \end{aligned} \quad (2) $$

In the following, we point out regular subdivisions (e.g. $t_0 = \frac{1}{2}$ in Algorithm 1) and distinguish weighted points and vectors. It is not possible to obtain directly this result with the Algorithm 1 : given that for each step of iteration the weights added implies a disturbance, see [40]. The use of Theorem 5 or the Corollary 1, see B.2.3 or [20, 23], provides a regular subdivision (the weighted point $N_3$ belongs to the $L_{4,1}$-perpendicular bisector from $P_1$ in the triangle $P_0P_1P_2$). We impose that:

* if the endpoint is a weighted point, its weight equals 1;

* if the endpoint is a vector, its first component is 1, see Table 2 in Appendix A.2. The representation of a point $M_0(x_0, y_0, z_0) \in \mathcal{E}_3$ in the Minkowski-Lorentz space is the light-like vector $\overrightarrow{m_0} \cdot (1, x_0, y_0, z_0, \frac{x_0^2 + y_0^2 + z_0^2}{2}) \in \overrightarrow{L_{4,1}}$.

Moreover if $(N_3; \varpi_3)$ is a weighted point the straight line defined by the mass points $(N_3; \varpi_3)$ and $(N_1; \varpi_1)$ or $(N_2; \varpi_2)$ represent the tangent line to the BR curve at $N_3$.

# 4 Regular iterative subdivision of Dupin cyclides or Bézier curves

Using the Theorem 5 or the Corollary 1, we define two homographies $h_0$ and $h_1$ from $[0; 1]$ into

$[0; \frac{1}{2}]$ and $[\frac{1}{2}; 1]$ respectively. For $i$ in $[0; 1]$, $h_i$ is defined by four real numbers and we have two degrees of freedom. Let us give more details about these homographies.

## 4.1 Homography $h_0$

We have

$$ h_0(u) = \frac{a_0 (1 - u) + b_0 u}{c_0 (1 - u) + d_0 u} \quad (3) $$

and we have to solve

$$ \left\{ \begin{array}{l} h_0(0) = 0 \\ h_0(1) = \frac{1}{2} \end{array} \right. \quad (4) $$

which leads to

$$ \left\{ \begin{array}{l} a_0 = 0 \\ d_0 = 2 b_0 \end{array} \right. \quad (5) $$

and the homography becomes

$$ h_0(u) = \frac{b_0 u}{c_0 (1 - u) + 2 b_0 u} \quad (6) $$

with $(b_0, c_0) \in (\mathbb{R}^+)^2$.

Let $(P_0; \omega_0)$, $(P_1; \omega_1)$ and $(P_2; \omega_2)$ be the control mass points of the Bézier curve $\gamma$. Using the Theorem 5, the control mass points of the Bézier curve $\gamma \circ h_0$ are $(Q_0; \varpi_0)$, $(Q_1; \varpi_1)$ and $(Q_2; \varpi_2)$ with

$$ \left\{ \begin{array}{l} (Q_0; \varpi_0) = c_0^2 \odot (P_0; \omega_0) \\ (Q_1; \varpi_1) = c_0 b_0 \odot (P_0; \omega_0) \\ \qquad \oplus b_0 c_0 \odot (P_1; \omega_1) \\ (Q_2; \varpi_2) = b_0^2 \odot (P_0; \omega_0) \\ \qquad \oplus 2 b_0^2 \odot (P_1; \omega_1) \\ \qquad \oplus b_0^2 \odot (P_2; \omega_2) \end{array} \right. \quad (7) $$

Since we want that the first control mass point of the two curves $\gamma$ and $\gamma \circ h_0$ is the same, we have $c_0 = 1$. If $\omega_0 + 2\omega_1 + \omega_2 \neq 0$, the last control mass point is a weighted point, in order to have $\varpi_2 = 1$, we choose

$$ b_0 = \frac{1}{\sqrt{\omega_0 + 2\omega_1 + \omega_2}} \quad (8) $$

else the computation of $b_0$ depends on the vector $\overrightarrow{Q_2}$: either $\overrightarrow{Q_2}$ is $\overrightarrow{e_\infty}$ or its first component equals 1.
## 4.2 Homography $h_1$

In the same way, we have

$$h_1(u) = \frac{a_1 (1-u) + b_1 u}{c_1 (1-u) + d_1 u} \quad (9)$$

and we have to solve

$$\left\{ \begin{array}{l} h_1(0) = \frac{1}{2} \\ h_1(1) = 1 \end{array} \right. \qquad (10)$$

which leads to

$$\left\{ \begin{array}{l} c_1 = 2a_1 \\ d_1 = b_1 \end{array} \right. \qquad (11)$$

and the homography becomes

$$h_1(u) = \frac{a_1 (1-u) + b_1 u}{2a_1 (1-u) + b_1 u} \quad (12)$$

with $(a_1, b_1) \in (\mathbb{R}^+)^2$.

Let $(P_0; \omega_0)$, $(P_1; \omega_1)$ and $(P_2; \omega_2)$ be the control mass points of the Bézier curve $\gamma$. Using the Theorem 5, the control mass points of the Bézier curve $\gamma \circ h_1$ are $(Q_0; \varpi_0)$, $(Q_1; \varpi_1)$ and $(Q_2; \varpi_2)$ with

$$\left\{ \begin{array}{l} (Q_0; \varpi_0) = a_1^2 \odot (P_0; \omega_0) \\ \oplus 2 a_1^2 \odot (P_1; \omega_1) \\ \oplus a_1^2 \odot (P_2; \omega_2) \\ (Q_1; \varpi_1) = a_1 b_1 \odot (P_1; \omega_1) \\ \oplus a_1 b_1 \odot (P_2; \omega_2) \\ (Q_2; \varpi_2) = b_1^2 \odot (P_2; \omega_2) \end{array} \right. \qquad (13)$$

Since we want that the last control mass point of the two curves $\gamma$ and $\gamma \circ h_1$ is the same, we have $b_1 = 1$. If $\omega_0 + 2\omega_1 + \omega_2 \neq 0$, the first control mass point is a weighted point, in order to have $\varpi_0 = 1$, we choose

$$a_1 = \frac{1}{\sqrt{\omega_0 + 2\omega_1 + \omega_2}} \qquad (14)$$

else the computation of $a_1$ depends on the vector $\overrightarrow{Q_0}$: either $\overrightarrow{Q_0}$ is $\overrightarrow{e_\infty}$ or its first component is 1.

From Table 4, one can see that Bézier curves are ellipse arcs if the weights $(\omega_0; \omega_1; \omega_2)$ belong to $\{1\} \times ]-1, 1[ \times \{1\}$.

## 4.3 Subdivision methods

We distinct two cases:

1. the segment $[P_0P_2]$ is not a diameter of the ellipse<fnref n="2" /> and the intermediate control mass point is a weighted point;

2. the Bézier curve is a semi-ellipse<fnref n="3" /> and the intermediate control mass point is a vector.

Since the rational quadratic Bézier curve with mass points of control $(P_0; \omega_0)$, $(P_1; \omega_1)$ and $(P_2; \omega_2)$ is a circle arc in the Minkowski-Lorentz space and an ellipse arc in the usual Euclidean affine plane, our methods do not depend on the metric of the space.

### 4.3.1 Case with 3 weighted points :

$$(\omega_0; \omega_1; \omega_2) \in \{1\} \times ]-1, 1[ - \{0\} \times \{1\}$$

Let us recall that

$$\mathrm{Bar}\{(A_k; \omega_k)\}$$

designates the barycentre of the weighted points $\{(A_k; \omega_k)\}$.

Theorem 1 Let $\gamma$ be a Bézier curve with control mass points $(\sigma_0; 1)$, $(P_1; \omega_1)$ and $(\sigma_2; 1)$ laying on the conic C.

Let $h_0: [0, 1] \to [0, \frac{1}{2}]$ defined by :

$$h_0: u \mapsto \frac{\frac{1}{\sqrt{2 + 2\omega_1}} u}{(1 - u) + 2 \frac{1}{\sqrt{2 + 2\omega_1}} u} \qquad (15)$$

then $\gamma \circ h_0$ equals a Bézier curve with control mass points $(\sigma_{00}; 1)$, $(P_{10}; \varpi_{10})$ and $(\sigma_{20}; 1)$ laying on the conic C with :

$$\left\{ \begin{array}{l} (\sigma_{00}; 1) = (\sigma_0; 1) \\ P_{10} = \mathrm{Bar}\{(\sigma_0; 1); (P_1; \omega_1)\} \\ \varpi_{10} = \sqrt{\frac{1 + \omega_1}{2}} \qquad (16) \\ \sigma_{20} = \mathrm{Bar}\{(\sigma_0; 1); (P_1; 2\omega_1); (\sigma_2; 1)\} \\ \varpi_{10} = 1 \end{array} \right.$$

**Proof:** by the use of Formula (7) with

$$b_0 = \frac{1}{\sqrt{2 + 2\omega_1}}$$

and $c_0 = 1$.

<figure><img src="image_4.png" /></figure>

Note that $\varpi_1 = \sqrt{\frac{1+\omega_1}{2}}$ is the well known the recurrence equation in the Euclidean case. For symmetry reasons, we can formulate,

<footnote n="2">²The curve is a circle for the Lorentz metric in $L_{4,1}$</footnote>

<footnote n="3">³The curve is a semi-circle for the Lorentz metric in $L_{4,1}$</footnote>
Theorem 2 Let $\gamma$ be a Bézier curve with control mass points $(\sigma_0; 1)$, $(P_1; \omega_1)$ and $(\sigma_2; 1)$ laying on the conic C.

Let $h_1 : [0, 1] \to [\frac{1}{2}, 1]$ defined by

$$h_1 : u \mapsto \frac{\frac{1}{\sqrt{2+2\omega_1}}(1-u) + u}{2\frac{1}{\sqrt{2+2\omega_1}}(1-u) + u} \quad (17)$$

then $\gamma \circ h_1$ equals a Bézier curve with control mass points $(\sigma_{01}; 1)$, $(P_{11}; \varpi_{11})$ et $(\sigma_{21}; 1)$ laying on the conic C with :

$$\left\{
\begin{array}{l}
\sigma_{01} = \operatorname{Bar}\{(\sigma_0; 1); (P_1; 2\omega_1); (\sigma_2; 1)\} \\
\qquad \varpi_{01} = 1 \\
P_{11} = \operatorname{Bar}\{(P_1; \omega_1); (\sigma_2; 1)\} \\
\qquad \varpi_{11} = \sqrt{\frac{1+\omega_1}{2}} \\
\qquad (\sigma_{21}; 1) = (\sigma_2; 1)
\end{array}
\right.
\quad (18)$$

The Fig. 3 shows an iteration of the subdivision algorithm based on theorems 1 and 2 thus we have $\sigma_{20} = \sigma_{01}$, $\sigma_0 = \sigma_{00}$ and $\sigma_2 = \sigma_{21}$. The Bézier curve of control mass points $(\sigma_0; 1)$, $(P_1; \omega_1)$ and $(\sigma_2; 1)$ is subdivided into two Bézier curves $\gamma \circ h_0$ and $\gamma \circ h_1$. The control mass points of $\gamma \circ h_0$ are $(\sigma_{00}; 1)$, $(P_{10}; \omega_1)$ and $(\sigma_{20}; 1)$. The control mass points of $\gamma \circ h_1$ are $(\sigma_{01}; 1)$, $(P_{01}; \omega_1)$ and $(\sigma_{21}; 1)$.

### 4.3.2 Case of two endpoints and a intermediate vector: $(\omega_0; \omega_1; \omega_2) = (1; 0; 1)$

The curve in the Minkowski-Lorentz space is a semi-circle which looks like an Euclidean semi-ellipse. The endpoints of the Bézier curve are the weighted points $(\sigma_0; 1)$ and $(\sigma_2; 1)$, the intermediate control mass point is the vector $(\vec{P}_1; 0)$ and we have

$$\left\{
\begin{array}{l}
4 \vec{P}_1^2 = \overline{\sigma_2 \sigma_0}^2 \\
\vec{P}_1 \cdot \overline{\sigma_2 \sigma_0} = 0
\end{array}
\right.
\quad (19)$$

Theorem 3 Let a Bézier curve $\gamma$ with control mass points $(\sigma_0; 1)$, $(\vec{P}_1; 0)$ and $(\sigma_2; 1)$ laying on the conic C.

Let $\Omega_1$ be the midpoint of the segment $[\sigma_0; \sigma_2]$. Let $h_0$ defined by

$$h_0 : u \mapsto \frac{\frac{\sqrt{2}}{2} u}{(1-u) + \sqrt{2} u} \quad (20)$$

<figure><img src="image_5.png" /><figcaption>Figure 3: One step of an iterative subdivision of a conic arc $\gamma$ on $\Lambda^4$. The endpoints are $\sigma_0$ and $\sigma_2$. The tangent lines at the endpoints are the straight lines $(\sigma_0 P_1)$ and $(\sigma_2 P_1)$ where the weighted point $(P_1; \omega_1)$ verifies $0 < |\omega_1| < 1$.</figcaption></figure>

then $\gamma \circ h_0$ is a Bézier curve with control mass points $(\sigma_{00}; 1)$, $(P_{10}; \varpi_{10})$ and $(\sigma_{20}; 1)$ laying on the conic C with:

$$\left\{
\begin{array}{l}
(\sigma_{00}; 1) = (\sigma_0; 1) \\
(P_{10}; \varpi_{10}) = \left(\mathcal{T}_{\vec{P}_1}(\sigma_0); \frac{\sqrt{2}}{2}\right) \\
(\sigma_{20}; 1) = \left(\mathcal{T}_{\vec{P}_1}(\Omega_1); 1\right)
\end{array}
\right.
\quad (21)$$

*Proof:* by the use of Formula (7) with $b_0 = \frac{\sqrt{2}}{2}$ and $c_0 = 1$.

<figure><img src="image_6.png" /></figure>

For symmetric reasons, we can formulate

Theorem 4 Let a Bézier curve $\gamma$ with control mass points $(\sigma_0; 1)$, $(\vec{P}_1; 0)$ and $(\sigma_2; 1)$ laying on the conic C.

Let $\Omega_1$ be the midpoint of the segment $[\sigma_0; \sigma_2]$. Let $h_1$ defined by

$$h_1 : u \mapsto \frac{\frac{\sqrt{2}}{2}(1-u) + u}{\sqrt{2}(1-u) + u} \quad (22)$$

then $\gamma \circ h_1$ is a Bézier curve with control mass points $(\sigma_{01}; 1)$, $(P_{11}; \varpi_{11})$ and $(\sigma_{21}; 1)$ laying on
the conic C with:

$$ \left\{ \begin{array}{l} (\sigma_{01}; 1) = \left( \overrightarrow{\mathcal{T}_{\vec{P}_1}(\Omega_1)} ; 1 \right) \\ (P_{11}; \varpi_{11}) = \left( \overrightarrow{\mathcal{T}_{\vec{P}_1}(\sigma_2)} ; \frac{\sqrt{2}}{2} \right) \\ (\sigma_{21}; 1) = (\sigma_2; 1) \end{array} \right. \qquad (23) $$

The Fig. 4 shows an iteration of the subdivision of a semi-circle based on the Theorems 3 and 4.

### 4.3.3 Synthesis

The Fig. 5 shows a graph which synthesizes the links between the theorems which permit the subdivisions of a connected circle (an ellipse with an Euclidean point of view).

## 5 Subdivision of a Dupin cyclide patch

In this section, the algorithms given in [15] are simplified using the representation of the brother circles on $\Lambda^4$. These curves are modeled using Bézier curves. The same method is applied to all ring Dupin cyclides (non-degenerate or torus). Let $\sigma_0$ and $\tau_0$ be two representations of spheres which define the Dupin cyclide, if they do not belong to the same brother circle on $\Lambda^4$, then the light-like vector $\overrightarrow{\sigma_0\tau_0}$ defines the Dupin cyclide point.

Fig. 6 shows a subdivision of a Dupin cyclide patch. The first (resp. second) family of spheres is defined by the Bézier curve with control mass points $(\sigma_0; 1)$, $(P_1; \omega_1)$ and $(\sigma_2; 1)$ (resp. $(\tau_0; 1)$, $(Q_1; \varpi_1)$ and $(\tau_2; 1)$). The vertex $P_{00}$, $P_{02}$, $P_{20}$ and $P_{22}$ of the Dupin cyclide patch are defined by the light-like vectors $\overrightarrow{\sigma_0\tau_0}$, $\overrightarrow{\sigma_0\tau_2}$, $\overrightarrow{\sigma_2\tau_0}$ and $\overrightarrow{\sigma_2\tau_2}$.

First, the Bézier curve with control mass points $(\sigma_0; 1)$, $(P_1; \omega_1)$ and $(\sigma_2; 1)$ is subdivided to obtain the two Bézier curves with control mass points $(\sigma_0; 1)$, $(P_{10}; \omega_{10})$ and $(\sigma_{01}; 1)$ on one hand and $(\sigma_{01}; 1)$, $(P_{11}; \omega_{11})$ and $(\sigma_2; 1)$ on the other hand. Two new points $P_{010}$ and $P_{012}$ are computed in $\mathcal{E}_3$ by using the light-like vectors $\overrightarrow{\sigma_0\tau_0}$ and $\overrightarrow{\sigma_0\tau_2}$.

In the same way, the Bézier curve with control mass points $(\tau_0; 1)$, $(Q_1; \varpi_1)$ and $(\tau_2; 1)$ is subdivided to obtain the two Bézier curve with control mass points $(\tau_0; 1)$, $(Q_{10}; \varpi_{10})$ and $(\tau_{01}; 1)$ on one hand and $(\tau_{01}; 1)$, $(Q_{11}; \varpi_{11})$ and $(\tau_2; 1)$ on the other hand. Two new points $P_{001}$ and $P_{201}$ are computed in $\mathcal{E}_3$ by using the light-like vectors $\overrightarrow{\sigma_0\tau_0}$ and $\overrightarrow{\sigma_2\tau_0}$.

Finally, the point $P_{0101}$ is computed in $\mathcal{E}_3$ by using the light-like vector $\overrightarrow{\sigma_0\tau_0}$. In Fig. 7, the patch of vertices $P_{00}P_{20}P_{22}P_{02}$ is replaced by the four patches of vertices

* $P_{00}$, $P_{010}$, $P_{0101}$ and $P_{001}$,

* $P_{010}$, $P_{20}$, $P_{201}$ and $P_{0101}$,

* $P_{0101}$, $P_{201}$, $P_{22}$ and $P_{012}$,

* $P_{012}$, $P_{02}$, $P_{001}$ and $P_{0101}$.

The original spheres $S_0$ and $S_2$ are defined by the points $\sigma_0$ and $\sigma_2$ whereas the sphere $S_{01}$ is defined by the construction of the point $\sigma_{01}$.

Using the same algorithm, the Fig. 8 shows the subdivision of a path of ring torus. Let us recall than in [15], the algorithms depend on the type of the surface (torus or non-degenerate Dupin cyclide) and then, the first work provides the determination of the Dupin cyclide. Moreover, there is two algorithms to subdivide a torus, one for the meridians and one for the parallels.

## 6 Conclusion and future works

In this paper, we have given methods to subdivide Bézier curves representing ellipse arc or semi-ellipse using mass points. These conics representing Dupin cyclides are circles on the space of spheres in the Minkowski-Lorentz space: one conic is a family of spheres which generates the Dupin cyclide i.e. a canal surface. Using the two circles, the same algorithms permit to subdivide Dupin cyclide patches too than can be used in patch surfaces.

In a second paper, we will give methods to subdivide Dupin cyclides having one or two singular points i.e. subdivide Bézier curves which represent parabolae or hyperbolae arcs in the usual Euclidean affine plane.
<figure><img src="image_7.png" /><figcaption>Figure 4: One iteration of a connected semi-circle defined by a BR curve of control mass points $(\sigma_0; 1)$, $(\vec{P}_1; 0)$ and $(\sigma_2; 1)$. __________</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 5: Graph which synthesizes the subdivisions of a connected circle (an ellipse with an Euclidean point of view). __________</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 6: One iteration of a subdivision of a Dupin cyclide patch. Each Bézier curves represents the spheres whose Dupin cyclide is the envelope.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 7: Subdivision of a Dupin cyclide patch: the patch of vertices $P_{00}P_{20}P_{22}P_{02}$ is replaced by the four patches of vertices $P_{00}P_{010}P_{0101}P_{001}$, $P_{010}P_{20}P_{201}P_{0101}$, $P_{0101}P_{201}P_{22}P_{012}$ and $P_{012}P_{02}P_{001}P_{0101}$.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 8: Subdivision of a path of ring torus.</figcaption></figure>
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Appendices

A Recalls of Minkowski-Lorentz space and space of spheres

The Minkowski-Lorentz space is a generalization of the Spacetime used in Einstein's relativity theory.

A.1 Construction of Minkowski-Lorentz space and space of spheres

The Minkowski-Lorentz space $\overrightarrow{L_{4,1}}$ is the real vector space of dimension 5. The symmetric bilinear form $\mathcal{L}_{4,1}$ denoted by a dot product, is defined on the canonical basis $(\overrightarrow{e_\perp}, \overrightarrow{e_1}, \overrightarrow{e_2}, \overrightarrow{e_3}, \overrightarrow{e_+})$ as follows

$$ \left\{ \begin{array}{l} \overrightarrow{e_i} \cdot \overrightarrow{e_j} = 0 \quad \text{if } i \neq j \\ \overrightarrow{e_\perp} \cdot \overrightarrow{e_\perp} = -1 \\ \overrightarrow{e_i} \cdot \overrightarrow{e_i} = 1 \quad \text{if } i \neq - \end{array} \right. \qquad (24) $$

with $i \in \{1, 2, 3, +\}$ and $j \in \{-, 1, 2, 3, +\}$.

The affine Minkowski-Lorentz space $L_{4,1}$ is defined by the point $O_5 = (0, 0, 0, 0, 0)$ and $\overrightarrow{L_{4,1}}$.

A new basis $(\overrightarrow{e_0}, \overrightarrow{e_1}, \overrightarrow{e_2}, \overrightarrow{e_3}, \overrightarrow{e_\infty})$ with

$$ \left\{ \begin{array}{l} \overrightarrow{e_0} = \overrightarrow{e_\perp} - \overrightarrow{e_+} \\ \overrightarrow{e_\infty} = \frac{1}{2} (\overrightarrow{e_\perp} + \overrightarrow{e_+}) \end{array} \right. $$

eases to embed the usual 3D Euclidean affine space $\mathcal{E}_3$ in the Minkowski-Lorentz space. The
reader will check that

$$ \overrightarrow{e_o} \cdot \overrightarrow{e_o} = \overrightarrow{e_\infty} \cdot \overrightarrow{e_\infty} = 0 $$

and

$$ \overrightarrow{e_o} \cdot \overrightarrow{e_\infty} = -1 $$

The origin point $O_3$ of $\mathcal{E}_3$ is obtained by
$\overrightarrow{e_o} = \overrightarrow{O_5O_3}$ and the vector $\overrightarrow{e_\infty}$ represents the
point at infinity of $\mathcal{E}_3$.

According to Minkowski definitions, any
vector $\overrightarrow{u} \in \overrightarrow{L_{4,1}}$ such that $\overrightarrow{u}^2$ is negative, pos-
itive or zero is qualified as a time-like, space-
like or light-like vector respectively. The light-
cone $C_l$ is the set of the points $M$ which verify

$$ \overrightarrow{O_5M}^2 = 0 \qquad (25) $$

i.e. $\overrightarrow{O_5M}$ is an isotropic vector, is perpendicular to itself, and the pseudo-distance between $O_5$ and $M$ is null.

## A.2 Equivalent representations of elements in $\mathcal{E}_3$ and Minkowski-Lorentz space

The Table 2 gives important Formulae to determine the representation of elements of $\mathcal{E}_3$. The space of spheres $\Lambda^4$ is the set of points $\sigma$ such as $\overrightarrow{O_5\sigma}^2 = 1$. This space represents the oriented spheres and oriented planes of $\mathcal{E}_3$.

## A.3 Properties of points and spheres

The Table 3 gives some properties of the representations of element of $\mathcal{E}_3$ in the Minkowski-Lorentz spaces. If the orientation of the tangent spheres $S_1$ and $S_2$ at $P$ is the same, then

$$ \overrightarrow{O_5\sigma_1} \cdot \overrightarrow{O_5\sigma_2} = 1 \qquad (26) $$

and

$$ \overrightarrow{\sigma_1\sigma_2}^2 = 0 \qquad (27) $$

and the vectors $\overrightarrow{\sigma_1\sigma_2}$ and $\overrightarrow{p}$ are parallel. Moreover, the direction of the vector $\overrightarrow{\sigma_1\sigma_2}$ in $\overrightarrow{L_{4,1}}$ defines, in $\mathcal{E}_3$, the point of tangency between the spheres $S_1$ and $S_2$.

A one parameter family of oriented spheres is a curve on $\Lambda^4$. The derivative spheres can be defined as follows.

**Definition 2** : Derivative sphere

Let $\gamma$ be a $\mathcal{C}^1$ curve, defined on an interval $I$, on $\Lambda^4$. The parameterization of the curve satisfies the following conditions:

* the curve is at least $\mathcal{C}^1$;

* the tangent vectors to the curve are always space-like vectors.

The intersection between $\Lambda^4$ and the line defined by $O_5$ and $\overrightarrow{\frac{\partial\gamma}{\partial\theta}}(\theta_0)$ is a sphere $\gamma(\theta_0)$ which is orthogonal to the sphere $\gamma(\theta_0)$ e.g.

$$ \overrightarrow{O_5 \gamma(\theta_0)} \cdot \overrightarrow{O_5 \gamma(\theta_0)} = 0 \qquad (28) $$

Moreover, if $\gamma(\theta_0)$ (resp. $\gamma(\theta_0)$) represents the sphere $\dot{S}$ (resp. $S$), then $\dot{S} \cap S$ is a circle, called characteristic circle if $\gamma$ models a canal surface. The canal Surface and the sphere $\gamma(\theta_0)$ are tangent along the circle $\dot{S} \cap S$.

# B Mass points and Minkowski-Lorentz spaces

## B.1 The set of mass points

A mass point is a couple $(M, m)$ such that: if the mass $m$ is equal to 0, $\overrightarrow{M}$ is a vector belonging to $\overrightarrow{L_{4,1}}$ otherwise $M$ is a point belonging to the affine space $L_{4,1}$. So, a mass point is a weighted point or a vector and the set of these mass points is denoted $\widetilde{L_{4,1}}$ i.e.

$$ \widetilde{L_{4,1}} = \overrightarrow{L_{4,1}} \times \{0\} \cup L_{4,1} \times \mathbb{R}^* \qquad (29) $$

The notation $\text{Bar}\{(M; \omega); (N; \mu)\}$ denotes the barycenter of the weighted points $(M; \omega)$ and $(N; \mu)$ and for any points $A$ and $B$ in $L_{4,1}$ we have:

$$ \text{Bar}\{(A; -1); (B; 1)\} \text{ is } \overrightarrow{AB} \in \widetilde{L_{4,1}} \qquad (30) $$

We define a stable addition $\oplus$ in $\widetilde{L_{4,1}}$, such that $(\widetilde{L_{4,1}}, \oplus)$ is a commutative group:
<table><caption>Table 2: correspondence between elements of $\mathcal{E}_3$ and points or vectors in the Minkowski-Lorentz spaces</caption><thead><tr><th>Type</th><th>ε<sub>3</sub> ∪ {∞}</th><th>L<sub>4,1</sub> or L&#x20D7;<sub>4,1</sub></th><th>Property</th></tr></thead><tbody><tr><td rowspan="2">Point</td><td>P ∈ ε<sub>3</sub></td><td>p&#x20D7; = e&#x20D7;<sub>o</sub> + O<sub>3</sub>p&#x20D7> + <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 0.2em;">||O<sub>3</sub>p&#x20D7;||<sup>2</sup></div><div style="padding: 0 0.2em;">2</div></div> e&#x20D7;<sub>∞</sub></td><td>p&#x20D7;<sup>2</sup> = 0</td></tr><tr><td>∞</td><td>e&#x20D7;<sub>∞</sub></td><td>e&#x20D7;<sub>∞</sub><sup>2</sup> = 0</td></tr><tr><td rowspan="2">Sphere</td><td>Center Ω</td><td>O&#x20D7;<sub>5</sub>σ = <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 0.2em;">1</div><div style="padding: 0 0.2em;">ρ</div></div> (e&#x20D7;<sub>o</sub> + Ω&#x20D7; + <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 0.2em;">1</div><div style="padding: 0 0.2em;">2</div></div> (||Ω&#x20D7;||<sup>2</sup> - ρ<sup>2</sup>) e&#x20D7;<sub>∞</sub>)</td><td>O&#x20D7;<sub>5</sub>σ<sup>2</sup> = 1</td></tr><tr><td>Radius ρ</td><td></td><td></td></tr><tr><td rowspan="2">Plane L<sub>4,1</sub></td><td>Normal vector N&#x20D7;</td><td>O&#x20D7;<sub>5</sub>π = N&#x20D7; + (N&#x20D7; • P&#x20D7;)</td><td>O&#x20D7;<sub>5</sub>π<sup>2</sup> = 1</td></tr><tr><td>Point P ∈ L<sub>4,1</sub></td><td></td><td></td></tr></tbody></table>

<table><caption>Table 3: Properties of the representations of points, planes and spheres in Minkowski-Lorentz spaces.</caption><thead><tr><th>Type</th><th>ε<sub>3</sub> ∪ {∞}</th><th>L<sub>4,1</sub> or L&#x20D7;<sub>4,1</sub></th><th>Property</th></tr></thead><tbody><tr><td>Point</td><td>P</td><td>p&#x20D7;</td><td>P ∈ S ⇐⇒ p&#x20D7> • O&#x20D7;<sub>5</sub>σ = 0</td></tr><tr><td>Sphere</td><td>S</td><td>σ ∈ Λ<sup>4</sup></td><td>P ∈ L<sub>4,1</sub> ⇐⇒ p&#x20D7> • O&#x20D7;<sub>5</sub>π = 0</td></tr><tr><td>Plane</td><td>P</td><td>π ∈ Λ<sup>4</sup></td><td>e&#x20D7;<sub>∞</sub> • O&#x20D7;<sub>5</sub>π = 0</td></tr><tr><td rowspan="2">Sphere</td><td rowspan="2">S<sub>1</sub></td><td rowspan="2">σ<sub>1</sub></td><td>S<sub>1</sub> ⊥ S<sub>2</sub> ⇐⇒ O&#x20D7;<sub>5</sub>σ<sub>1</sub> • O&#x20D7;<sub>5</sub>σ<sub>2</sub> = 0</td></tr><tr><td>#(S<sub>1</sub> ∩ S<sub>2</sub>) > {1} ⇐⇒ |O&#x20D7;<sub>5</sub>σ<sub>1</sub> • O&#x20D7;<sub>5</sub>σ<sub>2</sub>| < 1</td></tr><tr><td rowspan="2">Sphere</td><td rowspan="2">S<sub>2</sub></td><td rowspan="2">σ<sub>2</sub></td><td>S<sub>1</sub> and S<sub>2</sub> are tangent ⇐⇒ |O&#x20D7;<sub>5</sub>σ<sub>1</sub> • O&#x20D7;<sub>5</sub>σ<sub>2</sub>| = 1</td></tr><tr><td>S<sub>1</sub> ∩ S<sub>2</sub> = ∅ ⇐⇒ |O&#x20D7;<sub>5</sub>σ<sub>1</sub> • O&#x20D7;<sub>5</sub>σ<sub>2</sub>| > 1</td></tr></tbody></table>

$$ \bullet (M; \omega) \oplus (N; -\omega) = (\omega \overrightarrow{NM}; 0); $$

$$ \bullet (\vec{u}; 0) \oplus (\vec{v}; 0) = (\vec{u} + \vec{v}; 0), $$

$$ \bullet \text{ if } \omega \neq 0 \text{ and } \omega + \mu \neq 0, \text{ then } (M; \omega) \oplus (N; \mu) = \text{ (Bar } \{ (M; \omega); (N; \mu) \}; \omega + \mu); $$

$$ \bullet \text{ if } \omega \neq 0 \text{ then } (M; \omega) \oplus (\vec{u}; 0) = (\mathcal{T}_{\frac{1}{\omega}\vec{u}}(M); \omega) \text{ where } \mathcal{T}_{\vec{w}} \text{ is the translation of } L_{4,1} \text{ of vector } \vec{w}. $$

In order to define $(L_{4,1}, \oplus, \odot)$ as a vector space, we define the multiplication by a scalar $\odot$ as follows:

$$ \bullet \text{ if } \omega \neq 0, \quad 0 \odot (M; \omega) = (\vec{0}; 0) $$

$$ \bullet \alpha \neq 0 \implies \alpha \odot (M; \omega) = (M; \alpha \omega) $$

$$ \bullet \alpha \odot (\vec{u}; 0) = (\alpha \vec{u}; 0) $$

For more details on the space of mass points, the reader can refer to books of Fiorot and Jeannin [21, 35] or the paper of Garnier and al. [30, 23].
## B.2 Rational quadratic Bézier curves in $\tilde{\mathbb{L}}_{4,1}$

### B.2.1 Definition

Let us recall the definition of quadratic Bernstein polynomials

$$ \left\{ \begin{array}{l} B_0(t) = (1-t)^2, \\ B_1(t) = 2t(1-t), \\ B_2(t) = t^2 \end{array} \right. \qquad (31) $$

with $t \in [0, 1]$.

Now, we can define a rational quadratic Bézier curves with three control mass points $(P_0; \omega_0)$, $(P_1; \omega_1)$ and $(P_2; \omega_2)$:

**Definition 3 :** Rational quadratic Bézier curve (BR curve) in $\tilde{\mathbb{L}}_{4,1}$.

Let $\omega_0$, $\omega_1$ and $\omega_2$ be three non-zero values.

Let $(P_0; \omega_0)$, $(P_1; \omega_1)$ and $(P_2; \omega_2)$ be three mass points in $\tilde{\mathbb{L}}_{4,1}$, these points are not collinear.

Let us define two sets

$$ \left\{ \begin{array}{l} I = \{i \in [0, 2] \mid \omega_i \neq 0\} \\ J = \{i \in [0, 2] \mid \omega_i = 0\} \end{array} \right. \qquad (32) $$

Let us define the function $\omega_f$ as follows

$$ \omega_f: \begin{array}{r} [0, 1] \\ t \end{array} \longrightarrow \mathbb{R} \\ \qquad \qquad \qquad \qquad \sum_{i \in I} \omega_i \times B_i(t) \qquad (33) $$

A mass point $(M; \omega)$ or $(\vec{u}; 0)$ belongs to the quadratic Bézier curve defined by the three control mass points $(P_0; \omega_0)$, $(P_1; \omega_1)$ and $(P_2; \omega_2)$, if there is a parameter $t_0$ in $[0; 1]$ such that:

* if $\omega_f(t_0) \neq 0$ then we have

$$ \left\{ \begin{array}{l} \overrightarrow{OM} = \frac{1}{\omega_f(t_0)} \left( \sum_{i \in I} \omega_i B_i(t_0) \overrightarrow{OP_i} \right) \\ + \frac{1}{\omega_f(t_0)} \left( \sum_{i \in J} B_i(t_0) \overrightarrow{P_i} \right) \\ \omega = \omega_f(t_0) \end{array} \right. \qquad (34) $$

* if $\omega_f(t_0) = 0$ then we have

$$ \begin{aligned} \vec{u} = & \sum_{i \in I} \omega_i B_i(t_0) \overrightarrow{OP_i} \\ & + \sum_{i \in J} B_i(t_0) \overrightarrow{P_i} \end{aligned} \qquad (35) $$

Note that a Bézier curve with mass points of control mixes affine properties

$$ \frac{1}{\sum_{i \in I} \omega_i B_i(t_0)} \left( \sum_{i \in I} B_i(t_0) \omega_i \overrightarrow{OP_i} \right) \qquad (36) $$

and vector properties

$$ \frac{1}{\omega_f(t_0)} \left( \sum_{i \in J} B_i(t_0) \vec{P}_i \right) \qquad (37) $$

### B.2.2 Some properties

If $J = \emptyset$, we do not modify the Bézier curve if we multiply all the weights by a non-zero constant value. More generally, the following lemma holds:

**Lemma 1** Let $(P_0; \omega_0)$, $(P_1; \omega_1)$ and $(P_2; \omega_2)$ be three mass points $\tilde{\mathbb{L}}_{4,1}$.

Let $\lambda$ be a non-zero value.

If $\sum_{i \in I} \omega_i B_i(t_0) \neq 0$, we have

$$ \begin{aligned} & BR \left\{ (P_i; \omega_i)_{i \in I} ; (\vec{P}_j; 0)_{j \in J} \right\} = \\ & BR \left\{ (P_i; \lambda \omega_i)_{i \in I} ; (\lambda \vec{P}_j; 0)_{j \in J} \right\} \end{aligned} \qquad (38) $$

**Proof:**
$$
\begin{align*}
& \frac{1}{\sum_{i \in I} \omega_i \times B_i(t_0)} \left( \sum_{i \in I} \omega_i B_i(t_0) \overrightarrow{OP_i} \right) + \\
& \frac{1}{\sum_{i \in I} \omega_i \times B_i(t_0)} \left( \sum_{j \in J} B_j(t_0) \overrightarrow{P_i} \right) = \\
& \frac{1}{\sum_{i \in I} \lambda \omega_i \times B_i(t_0)} \left( \sum_{i \in I} \lambda \omega_i B_i(t_0) \overrightarrow{OP_i} \right) + \\
& \frac{1}{\sum_{i \in I} \lambda \omega_i \times B_i(t_0)} \left( \sum_{j \in J} \lambda B_j(t_0) \overrightarrow{P_i} \right)
\end{align*}
$$

<figure><img src="image_12.png" /></figure>

Without loss of generality, using Theo-
rem 5, if a weight is equal to 0, the others
weights belong to {0, 1}. The table 4 gives
the type of the conic defined by a quadratic
Bézier curve with mass points.

B.2.3 Homographic Parameter Change

The Bézier curve and the Bézier curve ob-
tained by this homographic parameter change
model two different arcs of the same given
conic.

Theorem 5 : Homographic Parameter Change

Let $\gamma$ be a Bézier curve with control mass
points $(P_0; \omega_0)$, $(P_1; \omega_1)$ and $(P_2; \omega_2)$ laying on
the conic C. Let $a$, $b$, $c$ and $d$ be four real
numbers satisfying

$$
\left| \begin{array}{cc} a & b \\ c & d \end{array} \right| \neq 0 \qquad (39)
$$

Let $h$ be defined by

$$
\begin{align*}
h: \quad \bar{\mathbb{R}} &\longrightarrow \bar{\mathbb{R}} \\
u &\longmapsto \frac{a(1-u)+bu}{c(1-u)+du} \tag{40}
\end{align*}
$$

then $\gamma \circ h$ is a Bézier curve of control mass
points $(Q_0; \varpi_0)$, $(Q_1; \varpi_1)$ and $(Q_2; \varpi_2)$ with

$$
\left\{
\begin{array}{l}
(Q_0; \varpi_0) = \\
(c-a)^2 \odot (P_0; \omega_0) \oplus \\
2a(c-a) \odot (P_1; \omega_1) \oplus \\
a^2 \odot (P_2; \omega_2) \\
\\
(Q_1; \varpi_1) = \\
(c-a)(d-b) \odot (P_0; \omega_0) \oplus \\
(bc-2ab+ad) \odot (P_1; \omega_1) \oplus \\
a b \odot (P_2; \omega_2) \\
\\
(Q_2; \varpi_2) = \\
(d-b)^2 \odot (P_0; \omega_0) \oplus \\
2b(d-b) \odot (P_1; \omega_1) \oplus \\
b^2 \odot (P_2; \omega_2)
\end{array}
\right.
\tag{41}
$$

Proof: see [20, 41].

The following corollary of Theorem 5 offers
to keep the endpoints.

Corollary 1 : Homographic Parameter Change with 0 and 1 unmodified.

Let $\gamma$ be a Bézier curve with mass control
points $(P_0; \omega_0)$, $(P_1; \omega_1)$ and $(P_2; \omega_2)$ laying on
the conic C. Let $b$ and $c$ be two non-zero num-
bers. Let $h$ be defined by :

$$
\begin{array}{rcl}
h: & \bar{\mathbb{R}} & \longrightarrow \bar{\mathbb{R}} \\
   & u & \longmapsto \frac{bu}{c(1-u)+bu}
\end{array}
\qquad (42)
$$

then $\gamma \circ h$ is a Bézier curve with mass control
points $(Q_0; \varpi_0)$, $(Q_1; \varpi_1)$ and $(Q_2; \varpi_2)$ on the
same conic C with

$$
\left\{
\begin{array}{l}
(Q_0; \varpi_0) = c^2 \odot (P_0; \omega_0) \\
(Q_1; \varpi_1) = b c \odot (P_1; \omega_1) \\
(Q_2; \varpi_2) = b^2 \odot (P_2; \omega_2)
\end{array}
\right.
\qquad (43)
$$

Proof: see [20].
<table><caption>Table 4: Type of the conic according the weights of the control mass points.</caption><thead><tr><th>Weights (ω<sub>0</sub>, ω<sub>1</sub>, ω<sub>2</sub>)</th><th>Δ = ω<sub>1</sub><sup>2</sup> - ω<sub>2</sub>ω<sub>0</sub></th><th>Euclidean type of the conic</th></tr></thead><tbody><tr><td rowspan="6">(1; ω<sub>1</sub>; 1)</td><td>+</td><td>ω<sub>1</sub> > 1: connected hyperbola arc</td></tr><tr><td>0</td><td>ω<sub>1</sub> = 1: connected parabola arc</td></tr><tr><td>-</td><td>ω<sub>1</sub> = 0: semi-ellipse</td></tr><tr><td>-</td><td>0 &lt; |ω<sub>1</sub>| &lt; 1: ellipse arc</td></tr><tr><td>0</td><td>ω<sub>1</sub> = -1: not connected parabola arc</td></tr><tr><td>+</td><td>ω<sub>1</sub> &lt; -1: not connected hyperbola arc</td></tr><tr><td>(0; 1; 1) or (1; 1; 0)</td><td>1</td><td>connected hyperbola arc</td></tr><tr><td>(0; 0; 1) or (1; 0; 0)</td><td>0</td><td>connected parabola arc</td></tr><tr><td>(0; 1; 0)</td><td>1</td><td>hyperbola branch</td></tr></tbody></table>

The denominator of a rational quadratic Bézier curve defined by the mass points $(P_0; \omega_0)$, $(P_1; \omega_1)$ and $(P_2; \omega_2)$ is

$$ (\omega_0 - 2\omega_1 + \omega_2) t^2 + 2(\omega_1 - \omega_0) t + \omega_0 \quad (44) $$

and the sign of the discriminant of this polynomial is

$$ \omega_1^2 - \omega_2 \omega_0 \quad (45) $$

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