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.

翻译：许多几何处理流程隐式假设输入数据是流形或从流形采样所得，且每个点具有唯一切平面。然而几何数据中常包含尖锐特征（如边缘、角点、自交、分支节点及其他奇点），导致标准方法在这些位置失效。为将几何处理拓展至此类奇异空间，我们提出"切向爆破"表示方法——该表示受代数几何启发，通过将空间提升至环境空间与切平面Grassmannian的乘积空间，在奇点处重建结构。重复该构造后，位置重合但切方向、曲率或高阶接触不同的点将获得明确分离。我们为切向爆破赋予乘积度量，并在提升域中直接定义离散微分算子（如梯度、散度和拉普拉斯算子）。我们通过测地线计算、分割、曲面参数化和曲率估计等任务验证了该框架的有效性。