# Conformal Coordinates for Molecular Geometry: From 3D to 5D

## TL;DR

This paper introduces a new 5D coordinate system for molecules that simplifies and speeds up distance calculations.

## Contribution

The novel use of conformal coordinates in molecular geometry introduces a more efficient computational framework.

## Key findings

- Conformal coordinates in ℝ5 allow for efficient interatomic distance calculations.
- The C-matrix reduces computational operations compared to traditional methods.
- The model outperforms standard Euclidean and homogeneous models in efficiency.

## Abstract

This paper introduces the conformal model (an extension of the homogeneous coordinate system) for molecular geometry, where 3D space is represented within ℝ5 with an inner product different from the usual one. This model enables efficient computation of interatomic distances using what we call the Conformal Coordinate Matrix (C‐matrix). The C‐matrix not only simplifies the mathematical framework but also reduces the number of operations required for distance calculations compared to traditional methods.

The figure illustrates the conformal representation of ℝ3 in a non‐Euclidean five‐dimensional space ℍ. Each point x∈ℝ3 is described with two additional vectors, e0 and e∞, so that Euclidean distances in ℝ3 can be recovered via inner products in ℍ. Using the C‐matrix, this encoding enables interatomic distances to be computed more directly and efficiently than in the standard Euclidean model (ℝ3) or in the homogeneous model (ℝ4).