1 Introduction

This paper is about pencil of circles. More precisely, the following two questions will be asked, and tentatively answered. How can we teach them to a computer, in order to get proofs, and how can we draw them, in order to obtain an intuitive perception of what happens. It is well known that $\mathbb{P}_{\mathbb{R}}\mathbb{R}^{4}$ gives an unified frame when dealing with families of circles. But our aim will be to emphasize how this frame can be split into computer proofs in $\mathbb{R}^{4}$ and human views in $\mathbb{R}^{3}\times\left\{ 1\right\}$, the ultimate goal being to provide an effective support for an intuitive perception of the question.

1.1 Proving is to convince a computer

Using a computer algebra tool is particularly useful when doing geometry. Obviously because computations are getting accurate and drawings are getting fast. But there is another benefit, maybe more important. When reformulating our topic in order to be understood by a computer, we are forced to explicit many underlying assumptions such as : what are the elementary objects, what are the allowed actions and how do they combine together. All these questions are not new, and were explicitly asked by Felix Klein in his Erlangen program (see, for example, , ).

But trying to convincing a computer moves these questions from the remote heaven to the immediate ground floor. When teaching geometry to a computer, one of the most difficult thing is to describes what could be "a point at infinity". This is not because the topic is really difficult. It is because we have so many different kinds of "infinities". We need an infinity ( $\left[1,0\right]$ in $\mathbb{P}_{\mathbb{C}}\mathbb{C}^{2}$) to describes that a given circle is is in reality a straight line, we need infinities ( $\left[a,b,0\right]$ in $\mathbb{P}_{\mathbb{R}}\mathbb{R}^{3}$) to describe the direction of such a straight line, and many other infinities to describe many other things (for example, a bundle of circles).

If we want to use simultaneously all these infinities, saying for example that $\pi=cycle\left(\omega,\,\infty,\,\alpha\right)$ is the straight line $\left(\infty\in\pi\right)$ through $\omega$ with direction $\alpha$, a clear unified frame is required that even a computer can understand. In what follows, we have chosen to describe every computation in the flat vector space $\mathbb{R}^{4}$ (not the projective one). It will be seen that this is not only the fastest way to teach a computer, but also that a formula with explicit homogeneous factors can be generalized quicker and safer.

1.2 Reusing our former knowledge ****

Another motivation is the following. It is well known that all the Cartesian equations of circles and straight lines can be summarized in an homogeneous equation namely :

$$\left(d_{0}+c_{0}\right)\left(x_{1}^{2}+y_{1}^{2}\right)-2a_{0}x_{1}-2b_{0}y_{1}+\left(d_{0}-c_{0}\right)=0$$ (1.1)

and even, using $\alpha=a+i\, b$, transformed in an homogeneous equation in $\mathbb{C}^{2}$ :

$$\omega\in\pi\,\Leftrightarrow\:^{\mathbf{t}}\!Z\,.\,\Pi\,.\,\overline{Z}=0$$ (1.2)

$$where\; Z=\left(\negmedspace\begin{array}{c} z_{1}\\ z_{2}\end{ar... ...overline{\alpha}\\ -\alpha & d-c\end{array}\negmedspace\right)\; describes\;\pi$$

From a computer centered point of view, translating properties of matrices $\Pi$ into properties of columns $X=\left\langle a,b,c,d\right\rangle$ is only introducing :

$$R=\left(\begin{smallmatrix}0&0&+1&1\\ -1&-i&0&0\\ -1&+i&0&0 \\ 0&... ...in{smallmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0 \\ 0&0&0&-1 \end{smallmatrix}\right)$$ (1.3)

and using :

$$X=R^{-1}\,.\,\operatorname{stack}\left(\Pi\right)\quad;\quad\det\Pi=-^{\mathbf{t}}\!X\,.\,Mink\,.\,X=-a^{2}-b^{2}-c^{2}+d^{2}$$ (1.4)

But, from an human centered point of view, using columns provides a better opportunity to reuse our informal knowledge of the usual geometric space. Indeed, quite all the elements of $\mathbb{P}_{\mathbb{R}}\mathbb{R}^{4}$ can be represented by a point in $\mathbb{R}^{3}\times\left\{ 1\right\}$ i.e. in the usual geometric space where we are living... and have developed nollens vollens an intuitive perception of geometrical objects.

On the third hand, any abstract mathematician is embodied in a concrete human being, with millennial experiment of the ordinary tridimensional space : drawing and measuring is unavoidable when dealing with abstract spaces.

Our purpose can be summarized as follows : use $\mathbb{R}^{3}$ to see and understand what happens, $\mathbb{R}^{4}$ to compute and prove, and $\mathbb{P}_{\mathbb{R}}\mathbb{R}^{4}$ to discover how all the special cases have been embodied, due to continuity of well stated (homogeneous) properties.

**** In other words, it is better to use a language that awakens the good reflexes in everyone's minds.

To fulfill these requirements, we will use the following notations.

1.3 Notations

Notation 1.1   Everything with a over-arrow lives in $\mathbb{R}^{3}\times\left\{ 0\right\}$ (the usual vector space), every figure is drawn in $\mathbb{R}^{3}\times\left\{ 1\right\}$ (the usual barycentric space), every explicit 4-uple lives in $\mathbb{R}^{4}$ (the flat -non projective- vector space) and everything else lives in or acts over $\mathbb{P}_{\mathbb{R}}\mathbb{R}^{4}$. Objects and properties are synchronized by subscripts, without further mention and $A_{j}\in\mathbb{R}^{3}\times\left\{ 1\right\}$, $X_{j}\in\mathbb{R}^{4}$, $\mathcal{A}_{j}\in\mathbb{P}_{\mathbb{R}}\mathbb{R}^{4}$ is ever assumed..

Remark 1.2   Sets $\mathbb{R}^{3}\times\left\{ 0\right\}$ and $\mathbb{R}^{3}\times\left\{ 1\right\}$ are obviously two copies of the usual space $\mathbb{R}^{3}$ where we are living. Nevertheless, we want to focus onto the fact that $\mathbb{R}^{3}\times\left\{ 1\right\}$ is (in many aspects) a subset of $\mathbb{P}_{\mathbb{R}}\mathbb{R}^{4}$, while is $\mathbb{R}^{3}\times\left\{ 0\right\}$ is certainly not "quite a subset" of the same set (only $\mathbb{P}_{\mathbb{R}}\mathbb{R}^{3}\times\left\{ 0\right\}$ would be).

Definition 1.3   Letter $\omega$ denotes a point in $\mathbb{P}_{\mathbb{C}}\mathbb{C}^{2}\simeq\overline{\mathbb{C}}$. Such a point $\omega_{j}$ is either $\infty$, the class of $\left[1,\,0\right]$ or the class of $\left[z_{j},\,1\right]$ where $z_{j}\doteq x_{j}+i\, y_{j}=\rho_{j}\exp\left(i\,\tau_{j}\right)\in\mathbb{C}$ (ordinary point). Its shadow, i.e. its associate on the Riemann sphere $\mathbb{S}$ (cf infra) will be noted either $S_{j}\in\mathbb{S}\subset\mathbb{R}^{3}\times\left\{ 1\right\}$ or $\mathcal{S}_{j}\in\mathbb{P}_{\mathbb{R}}\mathbb{R}^{4}$.

Remark 1.4   By exception, point $\left(0,0\right)\in\mathbb{R}^{2}$ is noted $O$ when it plays a special role. To avoid confusions between 0 and $O$, the shadow of $O$ is noted $N$ (north pole) and their relatives indexed by $n$ (north).

Definition 1.5   Letter $\pi$ denotes a cycle i.e. either a circle or a straight line.

1. Apart from special circle $\left\{ \infty\right\}$ (the point circle at infinity), an ordinary circle $\pi_{k}$ lives in $\mathbb{R}^{2}$, has an ordinary point $\omega_{k}$ for center, a real radius $r_{k}$ and a standardized equation $\omega_{j}\in\pi_{k}$ iff $\pi_{k}\left(\omega_{j}\right)=0$ where :

   $$\pi_{k}\left(\omega_{j}\right)=\left(x_{j}-x_{k}\right)^{2}+\left... ...t)^{2}-r_{k}^{2}=\left(x_{j}^{2}+y_{j}^{2}\right)-2x_{j}x_{k}-2y_{j}y_{k}+p_{k}$$ (1.5)

   
   

2. A line is $\pi_{k}\doteq\Delta_{k}\cup\left\{ \infty\right\}$ where $\Delta_{k}$ is a straight line living in $\mathbb{R}^{2}$. Its standardized equation (for an ordinary point $\omega_{j}$) is :

   $$\pi_{k}\left(\omega_{j}\right)=x_{j}\cos\tau_{k}+y_{j}\sin\tau_{k}-p_{k}$$ (1.6)

   
   

Remark 1.6   As indicated before, the concept of cycle has been introduced to take into account the fact that all the Cartesian equations of circles and straight lines can be summarized in the following homogeneous equation :

$$\left(d_{0}+c_{0}\right)\left(x_{1}^{2}+y_{1}^{2}\right)-2a_{0}x_{1}-2b_{0}y_{1}+\left(d_{0}-c_{0}\right)=0$$ (1.7)

Proposition 1.7   Four points $\omega_{i}\in\mathbb{P}_{\mathbb{C}}\mathbb{C}^{2}$, at least three of them being different, are on the same cycle if and only if their cross-ratio belongs to $\mathbb{R}\cup\left\{ \infty\right\}$.

Proof. [New proof of a well known fact] Use a computer and see that :

$$numer(\overline{\gamma}-\gamma)=\left\vert\begin{array}{cccc} z_{... ...t(z_{3}-z_{1}\right)}{\left(z_{4}-z_{1}\right)\left(z_{3}-z_{2}\right)}\qedhere$$

$\qedsymbol$

Proposition 1.8   Cross-ratio is invariant under the "circular group" $\mathbb{G}_{circ}$ generated by $z\mapsto\overline{z}$, $z\mapsto1/z$, and the $z\mapsto a\, z+b$ where $a\neq0$. By Proposition 1.7, any $\psi\in\mathbb{G}_{circ}$ transforms a $cycle$ into a cycle, eponymous property.