Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-Euclidean Cross-Ratios and Carnot's Theorem for Conics

Michael Perez Palapa, Kai Williams

Abstract

When considering geometry, one might think of working with lines and circles on a flat plane as in Euclidean geometry. However, doing geometry in other spaces is possible, as the existence of spherical and hyperbolic geometry demonstrates. Despite the differences between these three geometries, striking connections appear among the three. In this paper, we illuminate one such connection by generalizing the cross-ratio, a powerful invariant associating a number to four points on a line, into non-Euclidean geometry. Along the way, we see how projections between these geometries can allow us to directly export results from one geometry into the others. The paper culminates by generalizing Carnot's Theorem for Conics - a classical result relating when six points on a triangle lie on a conic - into spherical and hyperbolic geometry. These same techniques are then applied to Carnot's Theorem for higher degree curves.

Non-Euclidean Cross-Ratios and Carnot's Theorem for Conics

Abstract

When considering geometry, one might think of working with lines and circles on a flat plane as in Euclidean geometry. However, doing geometry in other spaces is possible, as the existence of spherical and hyperbolic geometry demonstrates. Despite the differences between these three geometries, striking connections appear among the three. In this paper, we illuminate one such connection by generalizing the cross-ratio, a powerful invariant associating a number to four points on a line, into non-Euclidean geometry. Along the way, we see how projections between these geometries can allow us to directly export results from one geometry into the others. The paper culminates by generalizing Carnot's Theorem for Conics - a classical result relating when six points on a triangle lie on a conic - into spherical and hyperbolic geometry. These same techniques are then applied to Carnot's Theorem for higher degree curves.
Paper Structure (4 sections, 15 theorems, 35 equations, 11 figures)

This paper contains 4 sections, 15 theorems, 35 equations, 11 figures.

Table of Contents

  1. Non-Euclidean Geometry and other preliminaries
  2. Non-Euclidean Cross-Ratios
  3. From Cross-ratios to Carnot's Theorem
  4. Higher Degree Curves

Key Result

Theorem 1

Let $\triangle ABC$ be an arbitrary triangle in $\mathbb{E}^2$. Let $A_1,A_2$, $B_1,B_2$, and $C_1,C_2$ be points on the sides $\overleftrightarrow{BC}$, $\overleftrightarrow{AC}$, and $\overleftrightarrow{AB}$ respectively. Then, $A_1$, $A_2$, $B_1$, $B_2$, $C_1$, and $C_2$ lie on a conic if and on where we measure the signed length of each segment.

Figures (11)

  • Figure 1: $A_1, A_2, B_1, B_2, C_1$, and $C_2$ lie on a conic if and only if the above equation holds.
  • Figure 2: A geodesic in $\mathbb{S}^2$
  • Figure 3: A geodesic in $\mathbb{H}^2$
  • Figure 4: An ellipse in $\mathbb{S}^2$
  • Figure 5: The Cross-Ratio on the geodesic $\ell$
  • ...and 6 more figures

Theorems & Definitions (29)

  • Theorem 1: Carnot's Theorem for Conic
  • Definition
  • Definition
  • Remark
  • Theorem 2
  • Definition
  • Lemma 3
  • proof
  • Definition
  • Lemma 4
  • ...and 19 more