Orthogonal Projections Based on Hyperbolic and Spherical n-Simplex

TL;DR

This paper introduces a method for orthogonal projection along geodesics onto k-planes in hyperbolic and spherical n-spaces, utilizing edge and Gram matrices of n-simplices to compute distances and foot points.

Contribution

It presents a novel approach for orthogonal projection in hyperbolic and spherical geometries using n-simplex matrices, extending classical Euclidean methods.

Findings

Derived formulas for orthogonal projection in hyperbolic and spherical spaces.

Provided methods to compute distances from points to k-planes.

Calculated perpendicular foot points in non-Euclidean geometries.

Abstract

In this paper, orthogonal projection along a geodesic to the chosen k-plane is introduced using edge and Gram matrix of an n-simplex in hyperbolic or spherical n-space. The distance from a point to k-plane is obtained by the orthogonal projection. It is also given the perpendicular foots from a point to k-plane of hyperbolic and spherical n-space.