# Spherical and hyperbolic conics

## TL;DR

This survey explores the metric properties of conics in spherical and hyperbolic geometries, highlighting their similarities to Euclidean conics and emphasizing the role of polarity in non-Euclidean contexts.

## Contribution

It provides a comprehensive overview of non-Euclidean conics, based on classical works, and clarifies their geometric properties and relationships with Euclidean conics.

## Key findings

- Non-Euclidean conics share metric properties with Euclidean conics.
- Polarity plays a significant role in non-Euclidean conic properties.
- Spherical and hyperbolic conics can be characterized as intersections with quadratic cones and affine conics.

## Abstract

This is a survey of metric properties of non-Euclidean conics, mainly based on works of Chasles and Story. A spherical conic is the intersection of the sphere with a quadratic cone; similarly, a hyperbolic conic is the intersection of the Beltrami-Cayley-Klein disk with an affine conic. Non-Euclidean conics have metric properties similar to those of Euclidean conics, and even more due to the polarity that works here better than in the Euclidean plane.