Tangent Blow-Ups for Processing Non-Manifold Geometry

TL;DR

The paper introduces the tangent blow-up, a novel geometric representation inspired by algebraic geometry, to handle singularities in non-manifold geometry processing by lifting data to a higher-dimensional space.

Contribution

It proposes the tangent blow-up method to explicitly represent and process singularities in geometry data, enabling standard differential operators to be applied.

Findings

Enables well-defined differential operators at singularities.

Improves geodesic computation and surface segmentation.

Facilitates curvature estimation on non-manifold geometries.

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.