# Reconstruction of rational ruled surfaces from their silhouettes

## TL;DR

This paper presents algorithms for reconstructing rational ruled surfaces in 3D from a single silhouette, focusing on tangent developables and rational normal scrolls, by analyzing apparent contours and their singularities.

## Contribution

It introduces novel algorithms for reconstructing rational ruled surfaces from silhouettes, including tangent developables and rational normal scrolls, using singularity analysis.

## Key findings

- Successful reconstruction of tangent developables from silhouettes.
- Effective reconstruction of rational normal scrolls via apparent contours.
- Algorithms leverage singularity information for accurate 3D surface recovery.

## Abstract

We provide algorithms to reconstruct rational ruled surfaces in three-dimensional projective space from the `apparent contour' of a single projection to the projective plane. We deal with the case of tangent developables and of general projections to $\mathbb{p}^3$ of rational normal scrolls. In the first case, we use the fact that every such surface is the projection of the tangent developable of a rational normal curve, while in the second we start by reconstructing the rational normal scroll. In both instances we then reconstruct the correct projection to $\mathbb{p}^3$ of these surfaces by exploiting the information contained in the singularities of the apparent contour.