Tangent Blow-Ups for Processing Non-Manifold Geometry

2026-05-18

AI Summary

This work addresses the limitations of traditional geometric processing methods, which rely on manifold assumptions and struggle with non-manifold geometries featuring singular structures such as sharp features, self-intersections, or branches. The authors propose a novel “tangent blow-up” representation, introducing for the first time the algebraic geometry concept of blow-up into geometric processing. By jointly embedding each spatial point together with its tangent plane into the product space of Euclidean space and a Grassmannian manifold, the method iteratively disambiguates coincident points that differ in tangential or higher-order contact. Within this lifted domain, discrete gradient, divergence, and Laplace operators are rigorously defined. This structured representation enables a natural extension of classical differential operators to singular points, demonstrating effectiveness in tasks including geodesic computation, segmentation, parameterization, and curvature estimation.

Abstract

Many geometry processing pipelines implicitly assume their input data is a manifold, or is sampled from one, with a unique tangent plane at every point. Geometric data, however, routinely contains sharp features like edges, corners, self-intersections, branching junctions, and other singularities, rendering standard methods ill-defined at these points. To bring geometry processing to these and other singular spaces, we introduce the ``tangent blow-up,'' a representation inspired by algebraic geometry that restores structure at singularities by lifting to the product of the ambient space and the Grassmannian of tangent planes. After iterating this construction, points that coincide in position but differ in tangent direction, curvature, or higher-order contact become well-separated. We equip the tangent blow-up with a product metric and define discretized differential operators, such as the gradient, divergence, and Laplacian, directly in the lifted domain. We demonstrate our framework across geodesic computation, segmentation, surface parameterization, and curvature estimation.

Problem

• non-manifold geometry
• singularities
• tangent blow-up
• geometry processing
• Grassmannian

Innovation

• tangent blow-up
• non-manifold geometry
• Grassmannian
• discrete differential operators
• singularities