Extensions of the D'Ocagne-Koenderink Theorem to Singular Surfaces

• Grant number: 24/20213-3
• Support Opportunities: Scholarships abroad - Research
• Start date: July 09, 2025
• End date: July 28, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Luciana de Fátima Martins
• Grantee: Luciana de Fátima Martins
• Host Investigator: Kentaro Saji
• Host Institution: Instituto de Biociências, Letras e Ciências Exatas (IBILCE). Universidade Estadual Paulista (UNESP). Campus de São José do Rio Preto. São José do Rio Preto , SP, Brazil
• Institution abroad: Kobe University, Japan
• Associated research grant: 19/07316-0 - Singularity theory and its applications to differential geometry, differential equations and computer vision, AP.TEM

Abstract

The project aims to conduct research in the field of Geometry/Topology, using tools from Singularity Theory and Differential Geometry, developed in collaboration with Prof. Dr. Kentaro Saji from the Department of Mathematics at Kobe University, Japan. We aim to contribute to the investigation of the geometry of singular surfaces in the Euclidean space R3 through their orthogonal projection onto a plane. Let M be a surface in R3, and consider the orthogonal projection of its points onto a plane along a direction v. This mapping becomes singular when v is a tangent direction to M. It serves as an essential tool for classifying the type of contact between M and lines parallel to the direction v. The singular set of the orthogonal projection restricted to M is called the contour generator, and its projection is known as the apparent contour. When M is regular, D'Ocagne in 1895 and Koenderink in 1984 demonstrated a result presenting a formula that relates the Gaussian curvature of M with the curvatures of the normal section of M in the direction v and the apparent contour, assuming the contour is regular. This project aims to obtain results analogous to those of D'Ocagne- Koenderink, considering generic rank-1 surfaces in R3 that are wave fronts, such as cuspidal edges and swallowtails.