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含假设输入数据为流形或从流形采样而来，且每点具有唯一切平面。然而几何数据通常包含尖锐特征（如棱边、角点、自交、分支节点及其他奇异性），导致标准方法在这些点处失效。为将几何处理推广至此类奇异空间，我们提出“切向爆破”——一种受代数几何启发的表示方法，通过将空间提升至环境空间与切平面格拉斯曼流形的乘积，恢复奇点处的结构。迭代该构造后，位置重合但切方向、曲率或更高阶接触不同的点得以分离。我们为切向爆破赋予乘积度量，并直接在提升域中定义离散微分算子（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应用验证该框架的有效性。