Hyperbolic/Elliptic Geometry

👋 Introduction

In Hyperbolic Geometry,
- two parallel lines meet outside the null circle.
- Given a line $l$ , there are more one parallel lines that pass through a point $p$ .
Notations: To distinguish with Euclidean geometry, lines are written as capital letters.

For efficiency, quadrance and spread can also be written as follows.
The quadrance $q (a, b)$ between points $a$ and $b$ is: $q (a, b) \equiv 1 - (a \cdot b^{⊥})^{2} / (a \cdot a^{⊥}) (b \cdot b^{⊥})$
The spread $S (L, M)$ between lines $L$ and $M$ is $S (L, M) \equiv 1 - (L \cdot M^{⊥})^{2} / (L \cdot L^{⊥}) (M \cdot M^{⊥})$
Note: In Hyperbolic Geometry, the quadrance of two points inside the null circle is negative.

Hyperbolic:
- Distance: $q (a, b) = sinh^{2} (d (a, b))$
- Angle: $S (L, M) = sin^{2} (θ (L, M))$
Elliptic:
- Distance: $q (a, b) = sin^{2} (d (a, b))$
- Angle: $S (L, M) = sin^{2} (θ (L, M))$

Spread Law $q_{1} / S_{1} = q_{2} / S_{2} = q_{3} / S_{3} .$
(Compare with the sine law in traditional Hyperbolic Geometry): $s i n h d_{1} / s i n θ_{1} = s i n h d_{2} / s i n θ_{2} = s i n h d_{3} / s i n θ_{3} .$
(Compare with the sine law in traditional Elliptic Geometry): $s i n d_{1} / s i n θ_{1} = s i n d_{2} / s i n θ_{2} = s i n d_{3} / s i n θ_{3} .$

Let $a_{1}$ , $a_{2}$ and $a_{3}$ are points with $q_{1} \equiv q (a_{2}, a_{3})$ , $q_{2} \equiv q (a_{1}, a_{3})$ and $q_{3} \equiv q (a_{1}, a_{2})$ . Let $L_{1}$ , $L_{2}$ and $L_{3}$ are lines with $S_{1} \equiv S (L_{2}, L_{3})$ , $S_{2} \equiv S (L_{1}, L_{3})$ and $S_{3} \equiv S (L_{1}, L_{2})$ .
Theorem (Triple quad formula): If $a_{1}$ , $a_{2}$ and $a_{3}$ are collinear points then $(q_{1} + q_{2} + q_{3})^{2} = 2 (q_{1}^{2} + q_{2}^{2} + q_{3}^{2}) + 4 q_{1} q_{2} q_{3}$
Theorem (Triple spread formula): If $L_{1}$ , $L_{2}$ and $L_{3}$ are concurrent lines then $(S_{1} + S_{2} + S_{3})^{2} = 2 (S_{1}^{2} + S_{2}^{2} + S_{3}^{2}) + 4 S_{1} S_{2} S_{3} .$

Theorem (Cross law) $(S_{1} S_{2} q_{3} - (S_{1} + S_{2} + S_{3}) + 2)^{2} = 4 (1 - S_{1}) (1 - S_{2}) (1 - S_{3}) .$
Theorem (Cross dual law) $(q_{1} q_{2} S_{3} - (q_{1} + q_{2} + q_{3}) + 2)^{2} = 4 (1 - q_{1}) (1 - q_{2}) (1 - q_{3}) .$
Note:
- Given three quadrances, three spreads can be uniquely determined. Same as Euclidean Geometry.
- Given three spreads, three quadrances can be uniquely determined. Not true in Euclidean Geometry.

Theorem (Pythagoras): If $L_{1}$ and $L_{2}$ are perpendicular lines ( $S (L_{1}, L_{2}) = 1$ ) then $q_{3} = q_{1} + q_{2} - q_{1} q_{2} .$
Theorem (Thales): Suppose that ${a_{1} a_{2} a_{3}}$ is a right triangle with $S_{3} = 1$ . Then $S_{1} = q_{1} / q_{3}$ and $S_{2} = q_{2} / q_{3}$ .

Theorem (Right parallax): If a right triangle ${a_{1} a_{2} a_{3}}$ has spreads $S_{1} = 0$ , $S_{2} = S$ and $S_{3} = 1$ , then it will have only one defined quadrance $q_{1} = q$ given by $q = (S - 1) / S .$
We may restate this result in the form: $S = 1/ (1 - q) .$

Theorem (Triangle proportions): Suppose that $d$ 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 $R_{1} / R_{2} = (S_{1} / S_{2}) (r_{1} / r_{2}) = (q_{1} / q_{2}) (r_{1} / r_{2}) .$