# Intersections of hyperplanes and conic sections in $\mathbf{R}^n$

## TL;DR

This paper derives explicit formulas for the intersection of hyperplanes with various n-dimensional conic sections, enabling efficient computation of intersection parameters and vectors.

## Contribution

It provides closed-form expressions for intersections of hyperplanes with n-dimensional conic sections, including a class of hyperboloids with efficient parameter computation.

## Key findings

- Closed-form formulas for intersection parameters and vectors.
- Identification of hyperboloid class with efficient intersection computation.
- Applicable to symmetric conic sections with arbitrary orientation and center.

## Abstract

Closed form expressions are given for computing the parameters and vectors that identify and define the $n-1$ dimensional conic section that results from the intersection of a hyperplane with an $n$-dimensional conic section: cone, hyperboloid of two sheets, ellipsoid or paraboloid. The conic sections are assumed to be symmetric about their major axis, but may have any orientation and center. A class of hyperboloids are identified with the property that the parameters and vectors of the intersection of all hyperboloids in a subset of the class can be computed efficiently.