3 Orthogonality

When two ordinary circles $\pi_{1},\,\pi_{2}$ intersect at right angle, a direct application of Pythagoras theorem leads to $\left\Vert \omega\omega_{2}\right\Vert ^{2}-r_{1}^{2}-r_{2}^{2}=0$. When trying to restate this relation into a $\mathbb{P}_{\mathbb{R}}\mathbb{R}^{4}$ relation by using (2.1), we obtain the following :

Proposition 3.1   Define the Minkowski metric over $\mathbb{R}^{4}$ by :

$$mink\left(X_{1},\,X_{2}\right)=a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}-d_{1}d_{2}=^{\mathbf{t}}\!X_{2}\,.\,Mink\,.\,X_{1}$$ (3.1)

Then $mink\left(X_{1},\,X_{2}\right)=0$ is a characteristic property of orthogonal cycles.

Proof. Applied to the various types of cycles, we have :

$$\left(circle,\, circle\right) : -2\,mink\left(X_{1},\,X_{2}\right)=k_{1}k_{2}\left(\left\Vert \omega\omega_{2}\right\Vert ^{2}-r_{1}^{2}-r_{2}^{2}\right)$$

$$\left(line,\, circle\right) : mink\left(X_{1},\,X_{2}\right)=k_{1}k_{2}\left(x_{2}\cos\tau_{1}+y_{2}\sin\tau_{1}+p_{1}\right)$$

$$\left(line,\, line\right) : mink\left(X_{1},\,X_{2}\right)=k_{1}\, k_{2}\cos\left(\tau_{1}-\tau_{2}\right)$$ (3.2)

where the normalizing factor $k_{i}$ is $c_{i}+d_{i}$ for a circle and $\sqrt{a_{i}^{2}+b_{i}^{2}}$ for a line. $\qedsymbol$

Notation 3.2   The light cone $\mathbb{S}_{Mink}$ in Minkowski space $\mathbb{P}_{\mathbb{R}}\mathbb{R}^{4}$ has equation $a^{2}+b^{2}+c^{2}-d^{2}=0$. It doesnt contains any point at infinity. Therefore, it can be identified with the unit sphere $\mathbb{S}_{Eucl}$ in $\mathbb{R}^{3}\times\left\{ 1\right\}$ and $\mathcal{S}_{j}\in\mathbb{S}_{Mink}$, $S_{j}\in\mathbb{S}_{Eucl}$ will often be shortened into $\mathcal{S}_{j}\in\mathbb{S}$, $S_{j}\in\mathbb{S}$.

Theorem 3.3 (Orthogonality)   We have the following results :

1. Point $\omega_{1}$ belongs to cycle $\pi_{0}$ iff the point-circle $\left\{ \omega_{1}\right\}$ is orthogonal to $\pi_{0}$.
2. The bundle of all the cycles $\pi_{j}$ orthogonal to a given cycle $\pi_{0}$ is characterized by : $\mathcal{A}_{j}$ belongs to the polar plane of $\mathcal{A}_{0}$ relative to the light cone $\mathbb{S}_{Mink}$. When nothing is at infinity, point $A_{j}$ belongs to the polar plane of $A_{0}$ relative to the unit sphere $\mathbb{S}_{Eucl}$.
3. The set $shadow\left(\pi_{0}\right)=\left\{ shadow\left(\omega\right)\mid\omega\in\pi_{0}\right\}$ is the circle where the polar plane of $\mathcal{A}_{0}$ intersects $\mathbb{S}$. Conversely, as in Fig. 1, $\mathcal{A}_{0}$ is the vertex of the circular cone that is tangent to $\mathbb{S}$ along the circle $shadow\left(\pi_{0}\right)$.
4. Cycles $\pi$ orthogonal to two different cycles $\pi_{1}$ and $\pi_{2}$ form a pencil whose apex line is the polar respective to $\mathbb{S}$ of the apex line of pencil $P\left(\pi_{1},\pi_{2}\right)$.

Proof. Straightforward from(3.1) and (3.2). $\qedsymbol$

Remark 3.4   Real cycle $\pi_{0}$ is orthogonal to the imaginary circle $\pi_{1}$ ( $r_{1}^{2}<0$) when $\pi_{0}$ cuts the real circle $\pi_{2}=\pi\left(\omega_{1},\,\left\vert r_{1}\right\vert\right)$ along a diameter of $\pi_{2}$.

Proposition 3.5   In $\mathbb{R}^{4}$, the orthogonal projector onto the space whose projective is the polar plane of $apex\left(\pi_{0}\right)$ is $id-p$ while projector $p$ onto $Vect\left(X_{0}\right)$ is :

$$p\left(X\right)=\left(mink\left(X,\,X_{0}\right)/mink\left(X_{0},\,X_{0}\right)\right)\, X_{0}$$

Proof. Both $p\circ p=p$ and $mink\left(p\left(X\right),\,X-p\left(X\right)\right)=0$ are straightforward. $\qedsymbol$

Remark 3.6   A matrix formulation of the preceding is

$$\boxed{p}=\frac{1}{mink\left(X_{0},\,X_{0}\right)}\, X_{0}\,.\,^{\mathbf{t}}\!X_{0}\,.\,Mink$$

where matrix $Mink$ is defined in (1.3).

Definition 3.7   The Gram matrix $\operatorname{G}_{p,q,\cdots,r}$ of $X_{p},\, X_{q},\cdots,\, X_{r}\in\mathbb{R}^{4}$ is the matrix of all the Minkowski products. In this context, notation $W_{pq}=mink\left(X_{p},\,X_{q}\right)$ and $w_{p}^{2}=mink\left(X_{p},\,X_{q}\right)$ will be used, leading to 

$$\operatorname{G}_{pq}=\left(\begin{array}{cc} w_{p}^{2} & W_{pq}\\ W_{pq} & w_{q}^{2}\end{array}\right)$$

Proposition 3.8   Two cycles $\pi_{1},\pi_{2}$ are (respectively) secant, tangent or external when $\operatorname{signum}\det\operatorname{G}_{12}$ is (respectively) $+1$, 0 or $-1$.

Proof. Necessary to understand Theorem 3.9. Will be proved as a Corollary. $\qedsymbol$

Theorem 3.9 (Classification)   Quantity $\sigma\left(P\right)\doteq\operatorname{signum}\det\operatorname{G}_{12}$ is the same for any choice of two cycles $\pi_{1}\neq\pi_{2}$ in a given pencil P. According to this value, we have the following classification :

• [ $\sigma\left(P\right)=0$] tangent pencil of all the cycles containing a given point $\omega_{0}$ and tangent at $\omega_{0}$ to a line $\Delta_{1}$ containing $\omega_{0}$. Archetype : $\omega_{0}=\infty$ and $P$ is "all the lines parallel to a given line $\Delta_{1}$".
• [ $\sigma\left(P\right)=-1$] $isotomic$ pencil, generated by two different point-circles $\left\{ \omega_{1}\right\}$ and $\left\{ \omega_{2}\right\}$ ( $\omega_{i}$ are the limit points of $P$). Archetype : $\omega_{2}=\infty$ and $P$ is, apart from $\left\{ \infty\right\}$, "all the circles centered at a finite point $\omega_{1}$".
• [ $\sigma\left(P\right)=+1$] $isoptic$ pencil of all the cycles going through two different points $\omega_{1}$ and $\omega_{2}$ (the base points). Archetype : $\omega_{2}=\infty$ and $P$ is "all the lines through a finite point $\omega_{1}$".

When $P$ is a tangent pencil, so is $P^{\perp}$ (using $\omega_{0}$ and $\Delta_{1}^{\perp}$ orthogonal to $\Delta_{1}$ at $\omega_{0}$). When $P$ is $isoptic\left(\omega_{1},\omega_{2}\right)$ then $P^{\perp}$ is $isotomic\left(\omega_{1},\omega_{2}\right)$ and conversely.

Proof. To see that $\sigma\left(P\right)$ is well defined, consider $X_{3}=k_{13}X_{1}+k_{23}X_{2},\, X_{4}=k_{14}X_{1}+k_{24}X_{2}$. Then $-G_{12}$ is the discriminant of $mink\left(X_{3},\,X_{3}\right)$, i.e. the condition for the apex line of $P$ cuts $\mathbb{S}$ : this will not change when choosing two other points on this line. Another proof is $\operatorname{G}_{34}=^{\mathbf{t}}\!\left(k\right)\,.\,\operatorname{G}_{12}\,.\,\left(k\right)$ while $\pi_{3}\neq\pi_{4}$ implies $\det\left(k\right)\neq0$.

When apex line of $P$ cuts $\mathbb{S}$ in two points $S_{1},\, S_{2}$, then $P$ is $isotomic\left(\omega_{1},\omega_{2}\right)$ by the very definition of a pencil. When $apex\left(P\right)$ is tangent to $\mathbb{S}$ at $S_{0}\neq S_{\infty}$, this line cuts the south plane at some $\mathcal{A}_{1}=apex\left(\Delta_{1}\right)$, $\omega_{0}\in\Delta_{1}$ and $P$ is a tangent pencil. When $apex\left(P\right)\cap\mathbb{S}=\emptyset$, the polar of $apex\left(P\right)$ cuts $\mathbb{S}$ in two points $S_{1},\, S_{2}$ and $P^{\perp}=isotomic\left(\omega_{1},\omega_{2}\right)$. For each $\pi\in P$ we have $\left\{ \omega_{i}\right\} \perp\pi$ for $i=1,2$ so that $\omega_{i}\in\pi$ while the converse is clear. $\qedsymbol$

Remark 3.10   Pencils are sometimes described as "elliptic" or "hyperbolic". Topologically, any quadric is connected in a projective space and therefore elliptic. Moreover, pencils are linear, i.e. are straight lines and not quadrics. Using Apollonius as eponym for the $isotomic$ circles (English use) is questionnable since quite all circles are Apollonian, while using Poncelet (French use) is questionnable since Poncelet's article is .

Proposition 3.11   Any $\psi\in\mathbb{G}_{circ}$ transforms not only cycles into cycles (Proposition 1.8) but also orthogonal pencils into orthogonal pencils.

Proof. A pencil $P$ is characterized by two cycles that belong to $P^{\perp}$. But homographies are conformal by differentiability while the property is obvious for $z\mapsto\overline{z}$. $\qedsymbol$

Proposition 3.12   When $P$ and $P^{\perp}$ are tangent pencils, they are the images of the Cartesian coordinate lines under the transformation $z\mapsto\omega_{0}+\exp i\left(\tau_{1}+\pi/2\right)/\overline{z}$ (notations of Theorem 3.9).

Theorem 3.13 (Bifocal coords)   Pencils $isotomic\left(\omega_{1},\,\omega_{2}\right)$ and $isoptic\left(\omega_{1},\,\omega_{2}\right)$ are the images, under a suitable homography $\psi$, of the polar coordinates grid. More precisely, coordinate circle $\left\vert\zeta\right\vert=\rho$ is transformed into the $isotomic\left(\rho\right)$ cycle characterized by $\left\Vert \omega\,\omega_{2}\right\Vert /\left\Vert \omega\,\omega_{1}\right\Vert =\rho$ while a coordinate ray, i.e. half a line from 0, characterized by $\zeta/\left\vert\zeta\right\vert=\exp i\vartheta$, goes onto "half" an $isoptic$ cycle. This $isoptic\left(\vartheta\right)$ arc, bounded by $\omega_{1}$ and $\omega_{2}$, is characterized by $\left(\overrightarrow{\omega\,\omega_{1}},\,\overrightarrow{\omega\,\omega_{2}}\right)=\vartheta$ .

Proof. Use $\psi$ defined by $0\mapsto\omega_{2},\,\infty\mapsto\omega_{1},\,1\mapsto\infty$ and consider $\omega_{0}\doteq\psi\left(\zeta\right)$ where $\zeta_{0}$ is the ordinary point $\zeta_{0}=\rho_{0}\,\exp i\vartheta_{0}$. By Proposition 1.8, we have :

$$\zeta_{0}=\rho_{0}\exp i\vartheta_{0}=\gamma\left(0,\,\infty,\,\r... ...,\omega_{0},\,\infty\right)=\frac{\omega_{2}-\omega_{0}}{\omega_{1}-\omega_{0}}$$ (3.3)

Let $\omega$ be any point on the whole $isoptic$ cycle through $\omega_{1},\omega_{2}$ and $\omega_{0}$ except from the basis points. By Proposition 1.7, $\omega$ comes from a $\zeta$ such that $\rho\exp i\vartheta=\mu\,\rho_{0}\exp i\vartheta_{0}$ where $\mu$ is real. Therefore, we have either $\zeta=\rho\exp i\vartheta_{0}$ or $\zeta=-\rho\exp i\vartheta_{0}=\rho\exp i\left(\vartheta_{0}+\pi\right)$. In the first case, $\zeta$ and $\zeta_{o}$ are on the same half line bounded by 0 and $\infty$ and, by continuity $\omega_{0}$ and $\omega$ are on the same arc bounded by $\omega_{1}$ and $\omega_{2}$. From now on this arc will be referred as the $isoptic\left(\vartheta\right)$ arc while the union of both arcs will be referred as the $isoptic\left(\vartheta+k\pi\right)$ cycle.

By Proposition 3.11, each cycle $\pi$ of the $isotomic\left(\omega_{1},\,\omega_{2}\right)$ pencil is the image by $\psi$ of a circle $\left\vert\zeta\right\vert=\rho$, and thus characterized by the value of $\rho$, while $\left\Vert \omega\,\omega_{2}\right\Vert /\left\Vert \omega\,\omega_{1}\right\... ...rt\left(\omega_{2}-\omega\right)/\left(\omega_{1}-\omega\right)\right\vert=\rho$ comes from (3.3). $\qedsymbol$

Fig. 2: Isoptic arc and isotomic cycle

Remark 3.14   Using $isoptic$ cycles, characterized only by $\tan\vartheta$, together with $isotomic$ cycles don't provide a system of coordinates since two cycles, one in each pencil, intersect in two points. The required uniqueness is only obtained when using $isoptic$ arcs (characterized by $\exp i\vartheta$).

Proposition 3.15   Transformation $\psi$ defined by $isoptic\left(\vartheta\right)\,\cap\,isotomic\left(\rho\right)\mapsto isoptic\left(\vartheta+\pi\right)\,\cap\,isotomic\left(\rho\right)$ belongs to $\mathbb{G}_{circ}^{+}$. Conversely, any involutary element of $\mathbb{G}_{circ}^{+}$ is either such a transformation or the identity.

Proof. An homography with exactly one fixed point cannot be involutary. Otherwise, both affirmations are equivalent to :

$$\left(\omega_{2}-\psi\left(\omega\right)\right)\div\left(\omega_{... ...ight)=-\left(\omega_{2}-\omega\right)\div\left(\omega_{1}-\omega\right)\qedhere$$

$\qedsymbol$

Proposition 3.16   For three cycles $\pi_{u},\,\pi_{v},\,\pi_{w}$, define $X_{\perp}=X_{u}\wedge X_{v}\wedge X_{w}$ by the coefficients of $a_{0},b_{0},c_{0},-d_{0}$ in $\det\left(X_{0},X_{u,}X_{v}X_{w}\right)$ -generalized cross product- and $Gram_{u,v,w}$ as in Definition 3.7. The three cycles are from the same pencil if and only if $X_{\perp}=\left[0,0,0,0\right]$. Otherwise $\pi_{\perp}$is the unique cycle orthogonal to the given three, and is real according to the sign of $mink\left(X_{\perp},\,X_{\perp}\right)=-Gram_{u,v,w}$.

Proof. When the three cycles are from the same pencil, everything vanishes. When $X_{\perp}\neq\left[0,0,0,0\right]$, the four points form a basis of $\mathbb{R}^{4}$ and $X_{u},\, X_{v},\, X_{w}$ define a plane whose polar relative to $\mathbb{S}$ is $X_{\perp}$. Cycle $\pi_{\perp}$ is real when its apex is outside $\mathbb{S}$ while $mink\left(X_{\perp},\,X_{\perp}\right)=-Gram_{u,v,w}$ generalizes the usual cross product formula... and is easy to verify. $\qedsymbol$

Remark 3.17   Let $\omega_{1},\omega_{2}$ be two ordinary points, $2\, r=\left\Vert \omega_{1}\,\omega_{2}\right\Vert \neq0$, $\omega_{0}$ the middle of $\left[\omega_{1,}\omega_{2}\right]$. Choose $X_{\infty}=\left[0,0,-1,1\right]$ and $X_{1},X_{2}$ such that $c_{i}+d_{i}=2$ (normalized form). Define :

$$X_{p},\, X_{m}=\frac{1}{2}\left(X_{1}\pm X_{2}\right)\:;\quad X_{... ...}\wedge X_{s}\:;\quad X_{4}=\frac{1}{4\, r^{2}}\, X_{p}\wedge X_{m}\wedge X_{3}$$

Then $\pi_{3}$ is line $\left(\omega_{1},\,\omega_{2}\right)$, $\pi_{m}$ its perpendicular through $\omega_{0}$, $\pi_{4}$ is circle $\left(\omega_{0},\, r\right)$ and $\pi_{p}$ is circle $\left(\omega_{0},\, i\, r\right)$. Pencil $isoptic\left(\omega_{1},\,\omega_{2}\right)$ is $P\left(\pi_{3},\,\pi_{4}\right)$ while $isotomic\left(\omega_{1},\,\omega_{2}\right)$ is $P\left(\pi_{m},\,\pi_{p}\right)$. This decomposition is useful since $\boxed{\operatorname{G}_{34mp}}=4\, r^{2}\, Mink$.