Cayley-Klein geometry

👋 Introduction

🔑 Key points

• The gravitational/electromagnetic force between two objects is inversely proportional to the square of their distance.

• Distances and angles may be powerful for oriented measures. But quadrance and spread are more energy saving for non-oriented measures.

• Euclidean Geometry is a degenerate case.

Cayley-Klein Geometry

• Projective geometry can be further categorized by polarities.

• Except for degenerate cases, $(A^{\perp }{)}^{\perp }=A$ and $(a^{\perp }{)}^{\perp }=a$

• A fundamental cone $\mathcal{F}=(\mathbf{A},\mathbf{B})$ is defined by a pole/polar pair such that $[A^{\perp }]=\mathbf{A}\cdot [A]$ and $[a^{\perp }]=\mathbf{B}\cdot [a]$.

• To visualize Cayley-Klein Geometry, we may project the objects onto the 2D plane.

• In hyperbolic geometry, the projection of the fundamental cone onto the 2D plane is a unit circle, the so-called null circle. The distance and angle measurement can be negative outside the null circle.

• We can think of Euclidean geometry as a hyperbolic geometry in which the null circle is expanded toward infinity.

• In this section, we use the vector notation $p=[A]$ and $l=[a]$.

Fundamental Cone with pole and polar

{#fig:F}

Examples

• Let $p=[x,y,z]$ and $l=[a,b,c]$

• Hyperbolic geometry:

  • $\mathbf{A}\cdot p\equiv [x,y,-z]$
  • $\mathbf{B}\cdot l\equiv [a,b,-c]$

• Elliptic geometry:

  • $\mathbf{A}\cdot p\equiv [x,y,z]$
  • $\mathbf{B}\cdot l\equiv [a,b,c]$

• Euclidean geometry (degenerate conic):

  • $\mathbf{A}\cdot p\equiv [0,0,z]$
  • $\mathbf{B}\cdot l\equiv [a,b,0]$

• psuedo-Euclidean geometry (degenerate conic):

  • $\mathbf{A}\cdot p\equiv [0,0,z]$
  • $\mathbf{B}\cdot l\equiv [a,-b,0]$

]

Examples (cont'd)

• Perspective view of Euclidean geometry (degenerate conic):
  • Let $l$ be the line of infinity.
  • Let $p$ and $q$ are two complex conjugate points on $l$. Then
  • $\mathbf{A}\equiv l\cdot l^T$ (outer product)
  • $\mathbf{B}\equiv p\cdot q^T+q\cdot p^T$

Orthogonality

• A line $l$ is said to be perpendicular to a line $m$ if $l^{\perp }$ lies on $m$, i.e., $m^{\mathsf{T}}\mathbf{B}l=0$.

• To find a perpendicular line of $l$ that passes through $p$, join $p$ to the pole of $l$, i.e., join($p,l^{\perp }$). We call this the altitude line of $l$.

• For duality, a point $p$ is said to be perpendicular to point $q$ if $q^{\mathsf{T}}\mathbf{A}p=0$.

• A similar definition can be given for altitude points.

• Note that Euclidean geometry does not have the concept of the perpendicular point because every $p^{\perp }$ is the line of infinity.

Orthocenter of triangle

• Theorem 1 (Orthocenter and ortholine). The altitude lines of a non-dual triangle meet at a unique point $O$ called the orthocenter of the triangle.

• Despite the name "center", the orthocenter may be outside a triangle.

• Theorem 2. If the orthocenter of triangle $\{ABC\}$ is $O$, then the orthocenter of triangle $\{OBC\}$ is $A$.

An instance of orthocenter theorem

{#fig:orthocenter}

An instance of Theorem 2

{#fig:orthocenter2}

Involution

• Involution is closely related to geometric reflection.

• The defining property of an involution $\tau$ is that for every point $p$, $\tau (\tau (p))=p$.

• Theorem: Let $\tau$ be an involution. Then

  1. there is a line $m$ for which $\tau (p)=p$ for every poiny $p$ incident with $m$.
  2. there is a point $o$ for which $\tau (l)=l$ for every line $l$ incident with $o$.

• We call the line $m$ the mirror and the point $o$ the center of the involution.

• If $o$ is at the line of infinity (Euclidean Geometry), then we get an undistorted Euclidean line reflection on $m$.

• If we choose $o=m^{\perp }$, then the fundamental cone is invariant.

Involution (cont'd)

• Theorem: The point transformation matrix $T$ of the projective involution $\tau$ with center $o$ and mirror $m$ is given by $$(o^{\mathsf{T}}m)\mathrm{I}-2o{m}^{\mathsf{T}}$$

• In other words, $T\cdot p$ = $(o^{\mathsf{T}}m)p-2(m^{\mathsf{T}}p)o$.

🐍 Python Code

from proj_geom import *

def is_perpendicular(l, m):
    return m.incident(dual(l))

def altitude(p, l):
    return p * dual(l)

def orthocenter(a1, a2, a3):
    t1 = altitude(a1, a2*a3)
    t2 = altitude(a2, a1*a3)
    return t1*t2

class reflect:
    def __init__(self, m, O):
        self.m = m
        self.O = O
        self.c = dot(m, O)

    def __call__(self, p):
        return pk_point(self.c, p, -2 * dot(self.m, p), self.O)

Mirror Image

{#looking-at-self}

Basic measurement

Quadrance and Spread for the general cases

• Let $\Omega (x)=x\cdot x^{\perp }$.

• $\Omega (A)=A\cdot A^{\perp }=[A{]}^{\mathsf{T}}\mathbf{A}[A]$.

• $\Omega (a)=a\cdot a^{\perp }=[a{]}^{\mathsf{T}}\mathbf{B}[a]$.

• The quadrance $q(A,B)$ between points $A$ and $B$ is: $$q(A,B)\equiv \Omega (AB)/\Omega (A)\Omega (B)$$

• The spread $s(l,m)$ between lines $l$ and $m$ is $$s(l,m)\equiv \Omega (lm)/\Omega (l)\Omega (m)$$

• Note: they are invariant to any projective transformations.

🐍 Python Code

import numpy as np
from fractions import *

def omega(l):
    return dot(l, dual(l))

def measure(a1, a2):
    omg = omega(a1*a2)
    if isinstance(omg, int):
        return Fraction(omg, omega(a1) * omega(a2))
    else:
        return omg / (omega(a1) * omega(a2))

def quadrance(a1, a2):
    return measure(a1, a2)

def spread(l1, l2):
    return measure(l1, l2)

versus raditional Distance and Angle

• Hyperbolic:
  • $q(A,B)={\sinh }^2(d(A,B))$
  • $s(l,m)={\sin }^2(\theta (l,m))$
• Elliptic:
  • $q(A,B)={\sin }^2(d(A,B))$
  • $s(l,m)={\sin }^2(\theta (l,m))$
• Euclidean:
  • $q(A,B)=d^2(A,B)$
  • $s(l,m)={\sin }^2(\theta (l,m))$

Measure the dispersion among points on the unit sphere

The usual way:

nsimplex, n = K.shape
maxd = 0
mind = 1000
for k in range(nsimplex):
  p = X[K[k,:],:]
  for i in range(n-1):
    for j in range(i+1, n):
      dot = dot(p[i,:], p[j,:])
*     q = 1.0 - dot*dot
*     d = arcsin(sqrt(q))
      if maxd < d:
        maxd = d
      if mind > d:
        mind = d
*dis = maxd - mind

A better way:

nsimplex, n = K.shape
maxd = 0
mind = 1000
for k in range(nsimplex):
  p = X[K[k,:],:]
  for i in range(n-1):
    for j in range(i+1, n):
      dot = dot(p[i,:], p[j,:])

*     q = 1.0 - dot*dot
      if maxq < q:
        maxq = q
      if minq > q:
        minq = q
*dis = arcsin(sqrt(maxq)) \
*      - arcsin(sqrt(minq))

] ]

Spread law and Thales Theorem

• Spread Law $$q_1/s_1=q_2/s_2=q_3/s_3.$$

• (Compare with the sine law in Euclidean Geometry): $$d_1/\sin {\theta }_1=d_2/\sin {\theta }_2=d_3/\sin {\theta }_3.$$

• Theorem (Thales): Suppose that $\{a_1{a}_2{a}_3\}$ is a right triangle with $s_3=1$. Then $$s_1=q_1/q_3\text{and}{s}_2=q_2/q_3$$

• Note: in some geometries, two lines being perpendicular does not imply they have a right angle ($s=1$).

Triangle proportions

• Theorem (Triangle proportions): Suppose $d$ is a point lying on the line $a_1{a}_2$. Define the quadrances $r_1\equiv q(a_1,d)$ and $r_1\equiv q(a_2,d)$, and the spreads $R_1\equiv s(a_3{a}_1,a_{3}d)$ and $R_2\equiv s(a_3{a}_2,a_{3}d)$. Then $$R_1/R_2=(s_1/s_2)(r_1/r_2)=(q_1/q_2)(r_1/r_2).$$

Midpoint and Angle Bisector

• There are two angle bisectors for two lines.
• In general geometries, There are also two midpoints for two points.
• Let $r$ be the midpoint of $p$ and $q$.
• Then $r$ = $\sqrt{\Phi (p)}q\pm \sqrt{\Phi (q)}p$.
• Let $b$ be the angle bisector of $l$ and $m$.
• Then $b$ = $\sqrt{\Phi (m)}l\pm \sqrt{\Phi (l)}m$.
• Note:
  • The midpoint could be irrational in general.
  • The midpoint could even be complex, even both points are real.
  • The bisectors of two lines are perpendicular.
  • In Euclidean geometry, the other midpoint is on the line of infinity.

Midpoint in Euclidean geometry

• Let $l$ be the line of infinity.
• $\mathbf{A}\equiv l\cdot l^T$
• $\Phi (p)$ = $p^{\mathsf{T}}\mathbf{A}p$ = $(p^{\mathsf{T}}l{)}^2$.
• Then, the midpoint $r$ = $(q^{\mathsf{T}}l)p\pm (p^{\mathsf{T}}l)q$.
• One midpoint $(q^{\mathsf{T}}l)p-(p^{\mathsf{T}}l)q$ in fact lies on $l$.

Constructing angle bisectors using a conic

1. For each line, construct the two tangents $(t_f^1,t_f^2)$ and $(t_g^1,t_g^2)$ of its intersection points with the fundamental conic to that conic.
2. The following lines are the two angle bisectors:
   • join(meet($t_f^1,t_g^1$), meet($t_f^2,t_g^2$))
   • join(meet($t_f^1,t_g^2$), meet($t_f^2,t_g^1$))

Remark: the tangents in elliptic geometry have complex coordinates. However, the angle bisectors are real objects again.

Constructing a pair of angle bisectors

{#fig:bisector}

Angle Bisector Theorem

• Let $a$, $b$, $c$ be three lines, none of which is tangent to the fundamental cone.

• Then one set of angle bisector $m_{ab}^1,m_{bc}^1,m_{ac}^1$ are concurrent.

• Furthermore, the points meet($m_{ab}^2,c$), meet($m_{bc}^2,a$), meet($m^2\ast ac,b$*) are collinear.

An instance of complete angle bisector theorem

{#fig:bisectortheorem}

Midpoint theorem

• Let $p$, $q$, $r$ be three points, none of which lies on the fundamental cone.

• Then one set of midpoints $m_{pq}^1$, $m_{qr}^1$, $m_{pr}^1$_ are collinear.

• Furthermore, the lines join($m_{pq}^2,r$), join($m_{qr}^2,p$), join($m^2\ast pr,q$*) meet at a point.

backup

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