Geometric orthogonality condition for circles and hyperbolas¶
Exercise I.6.3 [1] $ \newcommand{\scalar}[3][]{\left\langle #2,#3 \right\rangle_{#1}} \newcommand{\Space}[3][]{\mathbb{#2}^{#3}_{#1}} \newcommand{\Space}[3][]{\mathbb{#2}^{#3}_{#1}} \newcommand{\cycle}[3][]{{#1 C^{#2}_{#3}}} \newcommand{\realline}[3][]{#1 R^{#2}_{#3}} \newcommand{\bs}{\breve{\sigma}} \newcommand{\zcycle}[3][]{#1 Z^{#2}_{#3}} \newcommand{\SL}[1][2]{\mathrm{SL}_{#1}(\Space{R}{})} \newcommand{\rs}{\sigma_r} \newcommand{\lvec}[1]{\overrightarrow{#1}} \newcommand{\rmi}{\mathrm{i}} \newcommand{\alli}{\iota} \newcommand{\rme}{\mathrm{e}} \newcommand{\rmd}{\mathrm{d}} \newcommand{\rmh}{\mathrm{j}} \newcommand{\rmp}{\varepsilon} \newcommand{\modulus}[2][]{\left| #2 \right|_{#1}} \newcommand{\sperp}{⋋} $ Check the following geometrical meaning for $\sigma$-orthogonality of $\sigma$-cycles.
- (e,h) Let $\sigma=\pm 1$. Then, two cycles in $\Space[\sigma]{R}{}$ (that is, circles or hyperbolas) are $\sigma$-orthogonal if slopes $S_1$ and $S_2$ of their tangents at the intersection point satisfy the condition \begin{equation} \label{eq:ortho-slopes-rel} S_1 S_2=\sigma. \end{equation} The geometrical meaning of this condition can be given either in terms of angles (A) or centres (C):
(A) For the case $\sigma=-1$ (circles), equation implies orthogonality of the tangents. For $\sigma=1$, two hyperbolas are h-orthogonal if lines with the slopes $\pm 1$ bisect the angle of intersection of the hyperbolas.
Hiint. Define a cycle $\cycle{}{\sigma}$ by the condition that it passes a point $(u,v)\in\Space[\sigma]{R}{}$. Define a second cycle $\cycle[\tilde]{}{\sigma}$ by both conditions: it passes the same point $(u,v)$ and is orthogonal to $\cycle{}{\sigma}$. Then use the implicit derivative formula to find the slopes of tangents to $\cycle{}{\sigma}$ and $\cycle[\tilde]{}{\sigma}$ at $(u,v)$. A script calculating this in CAS is also provided.
(C) In the cases $\sigma=\pm 1$, the tangent to one cycle at the intersection point passes the centre of another cycle.
Hint. This fact is clear for circles from inspection of, say, Fig. For hyperbolas, it is enough to observe that the slope of the tangent to a hyperbola $y=1/x$ at a point $(x, 1/x)$ is $-1/x^2$ and the slope of the line from the centre $(0,0)$ to the point $(x,1/x)$ is $1/x^2$, so the angle between two lines is bisected by a vertical/horizontal line. All our hyperbolas are obtained from $y=1/x$ by rotation of $\pm 45 ^\circ$ and scaling.
- (p) Let $\sigma=0$ and a parabola $\cycle{}{p}$ have two real roots $u_1$ and $u_2$. If a parabola $\cycle[\tilde]{}{p}$ is $p$-orthogonal to $\cycle{}{p}$, then the tangent to $\cycle[\tilde]{}{p}$ at a point above one of the roots $u_{1,2}$ passes the p-centre $(\frac{u_1+u_2}{2},0)$ of $\cycle{}{p}$.
Solution. Define a cycle passing point (u,v)
from init_cycle import *
C3=C.subject_to(C.passing(P))
Define a cycle passing the same point and being orthogonal to C3
C4=C1.subject_to([C1.passing(P), C1.is_orthogonal(C3)])
Evaluate tangent slope of C3 at P1
Sl3=C3.val(P1).diff(u1)/C3.val(P1).diff(v1)
Evaluate tangent slope of C4 at P1
Sl4=C4.val(P1).diff(u1)/C4.val(P1).diff(v1)
Check the orthogonality relation for circles and hyperbolas this is the case of $\sigma=1$ or $-1$, then $\sigma^2=1$.
"Tangents of orthogonal circles at an intersection point are perpendicular" \
+ "and slopes of orthogonal hyperbolas at an intersection point are reciprocal: %s" % \
(Sl3*Sl4-sign).subs({u1 : u, v1 : v}).normal()\
.subs({pow(sign,2) : 1}, subs_options.algebraic).is_zero()
Check the geometric orthogonality condition for parabolas
Define a point above the root of the first parabola
P2=[C.roots()[0],v]
Define a cycle passing this point and being orthogonal to C
C5=C1.subject_to([C1.passing(P2), C1.is_orthogonal(C)])
Calculate the slop of the parabola
Sl5=C5.val(P1).diff(u1)/C5.val(P1).diff(v1)
"Tangent to first parabola above second's root passes second's centre: %s" %\
(Sl5-v/(C.center()[0]-P2[0])).subs({u1 : P2[0], v1 : P2[1], sign : 0})\
.numer().normal().is_zero()
This notebook is a part of the MoebInv notebooks project [2] .
References¶
Vladimir V. Kisil. Geometry of Möbius Transformations: Elliptic, Parabolic and Hyperbolic Actions of $SL_2(\mathbb{R})$. Imperial College Press, London, 2012. Includes a live DVD.
Vladimir V. Kisil, MoebInv notebooks, 2019.