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Using Algebraic Geometry to Reconstruct a Darboux Cyclide from a Calibrated Camera Picture

Eriola Hoxhaj, Jean Michel Menjanahary, Josef Schicho

TL;DR

The paper addresses single-view recognition of a Darboux cyclide by converting the visual apparent contour into the discriminant $\Delta_w(F)$ of a quartic surface equation $F(x,y,z,w)$ and exploiting the cyclide's singularity along the Euclidean absolute conic. The authors develop a pipeline that (i) analyzes the contour $\mathrm{R}$ and apparent contour $\mathrm{B}$ under a calibrated camera, (ii) computes the local and global conductor to recover the map $\mathrm{B}\dashrightarrow\mathrm{R}$ via generators $G_0$ and $G_1$, and (iii) reconstructs the surface equation $F$ from the contour data using a linear system anchored by a cubic $H_0$ and a quartic $H_1$, followed by a projective normalization to place the singular conic at infinity. Key contributions include detailed local conductor formulas for special points, an explicit practical algorithm that yields $F$ up to scaling, and demonstration via computer algebra that the surface can be recovered efficiently from the apparent contour with a calibrated camera. This advances single-view algebraic surface recognition by leveraging the Euclidean absolute conic and conductor theory, with potential extensions to multi-view or non-generic camera configurations. The approach provides a concrete, scalable route from image contours to exact quartic surface equations in a geometrically constrained setting.

Abstract

The task of recognizing an algebraic surface from a single apparent contour can be reduced to the recovering of a homogeneous equation in four variables from its discriminant. In this paper, we use the fact that Darboux cyclides have a singularity along the absolute conic in order to recognize them up to Euclidean similarity transformations.

Using Algebraic Geometry to Reconstruct a Darboux Cyclide from a Calibrated Camera Picture

TL;DR

The paper addresses single-view recognition of a Darboux cyclide by converting the visual apparent contour into the discriminant of a quartic surface equation and exploiting the cyclide's singularity along the Euclidean absolute conic. The authors develop a pipeline that (i) analyzes the contour and apparent contour under a calibrated camera, (ii) computes the local and global conductor to recover the map via generators and , and (iii) reconstructs the surface equation from the contour data using a linear system anchored by a cubic and a quartic , followed by a projective normalization to place the singular conic at infinity. Key contributions include detailed local conductor formulas for special points, an explicit practical algorithm that yields up to scaling, and demonstration via computer algebra that the surface can be recovered efficiently from the apparent contour with a calibrated camera. This advances single-view algebraic surface recognition by leveraging the Euclidean absolute conic and conductor theory, with potential extensions to multi-view or non-generic camera configurations. The approach provides a concrete, scalable route from image contours to exact quartic surface equations in a geometrically constrained setting.

Abstract

The task of recognizing an algebraic surface from a single apparent contour can be reduced to the recovering of a homogeneous equation in four variables from its discriminant. In this paper, we use the fact that Darboux cyclides have a singularity along the absolute conic in order to recognize them up to Euclidean similarity transformations.
Paper Structure (6 sections, 11 theorems, 7 equations, 3 figures, 1 algorithm)

This paper contains 6 sections, 11 theorems, 7 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Calibrated Camera
  3. Darboux Cyclides
  4. The contour and apparent contour
  5. The conductor of the apparent contour in the contour
  6. From the contour to the Surface

Key Result

Proposition 3.1

Let $\mathrm{D}\subset\mathbb{P}^3$ be an irreducible Darboux cyclide. Then the absolute conic $\mathrm{C}_A$ is either a nodal or a cuspidal curve of $\mathrm{D}$. Also, the absolute conic is the only singular curve of $\mathrm{D}$.

Figures (3)

  • Figure 1: Reconstruction of a torus from a single picture. From the photo with a calibrated camera, the three angles $a,b,c$ are determined and drawn in a plane. Next, we draw a circle tangential to the rays forming the angle $a$. Next, we draw a second circle with equal radius tangential to the rays forming the angle $c$. Next, we rotate both circles around the symmetry line of the two midpoints $p,q$ and obtain the torus.
  • Figure 2: Some Darboux cyclides with $\mathrm{C}_A$ being a nodal curve, together with the real part of their apparent contour, a curve of degree 8. The complex part of the apparent contour is not visible: it is the elliptic absolute conic $\mathrm{C}_E$. The equation of $\mathrm{C}_E$ can be assumed to be known because of the assumption that we have a calibrated camera.
  • Figure 3: Some Darboux cyclides with $\mathrm{C}_A$ being a cuspidal curve, together with the real part of their apparent contour, a curve of degree 6.

Theorems & Definitions (27)

  • Remark 2.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 4.1
  • proof
  • Example 4.2
  • Proposition 4.3
  • proof
  • ...and 17 more