M\"obius Photogrammetry

TL;DR

This paper introduces M"obius Photogrammetry, a novel approach to reconstructing 3D point sets from orthogonal projections known up to M"obius transformations, with applications to pentapod mobility analysis.

Contribution

It formulates a new reconstruction problem involving M"obius transformations and provides theoretical results linking pentapod mobility to geometric configurations.

Findings

Reconstruction possible with partial M"obius projection data.

Pentapod mobility implies collinearity or similarity of platform and base.

Established conditions for geometric configurations based on mobility.

Abstract

Motivated by results on the mobility of mechanical devices called pentapods, this paper deals with a mathematically freestanding problem, which we call M\"obius Photogrammetry. Unlike traditional photogrammetry, which tries to recover a set of points in three-dimensional space from a finite set of central projection, we consider the problem of reconstructing a vector of points in $\mathbb{R}^3$ starting from its orthogonal parallel projections. Moreover, we assume that we have partial information about these projections, namely that we know them only up to M\"obius transformations. The goal in this case is to understand to what extent we can reconstruct the starting set of points, and to prove that the result can be achieved if we allow some uncertainties in the answer. Eventually, the techniques developed in the paper allow us to show that for a pentapod with mobility at least two either some anchor points are collinear, or platform and base are similar, or they are planar and affine equivalent.