Many geometry processing techniques require the solution of partial differential equations (PDEs) on manifolds embedded in \(\mathbb {R}^2 \) or \(\mathbb {R}^3 \), such as curves or surfaces. Such manifold PDEs often involve boundary conditions (e.g., Dirichlet or Neumann) prescribed at points or curves on the manifold’s interior or along the geometric (exterior) boundary of an open manifold. However, input manifolds can take many forms (e.g., triangle meshes, parametrizations, point clouds, implicit functions, etc.). Typically, one must generate a mesh to apply finite element-type techniques or derive specialized discretization procedures for each distinct manifold representation. We propose instead to address such problems in a unified manner through a novel extension of the closest point method (CPM) to handle interior boundary conditions. CPM solves the manifold PDE by solving a volumetric PDE defined over the Cartesian embedding space containing the manifold, and requires only a closest point representation of the manifold. Hence, CPM supports objects that are open or closed, orientable or not, and of any codimension. To enable support for interior boundary conditions we derive a method that implicitly partitions the embedding space across interior boundaries. CPM’s finite difference and interpolation stencils are adapted to respect this partition while preserving second-order accuracy. Additionally, we develop an efficient sparse-grid implementation and numerical solver that can scale to tens of millions of degrees of freedom, allowing PDEs to be solved on more complex manifolds. We demonstrate our method’s convergence behaviour on selected model PDEs and explore several geometry processing problems: diffusion curves on surfaces, geodesic distance, tangent vector field design, harmonic map construction, and reaction-diffusion textures. Our proposed approach thus offers a powerful and flexible new tool for a range of geometry processing tasks on general manifold representations.

许多几何处理技术需要求解嵌入在 \(\mathbb {R}^2 \) 或 \(\mathbb {R}^3 \) 中的流形（例如曲线或曲面）上的偏微分方程 (PDE)。此类流形偏微分方程通常涉及在流形内部或沿着开流形的几何（外部）边界的点或曲线处规定的边界条件（例如，狄利克雷或诺伊曼）。然而，输入流形可以采用多种形式（例如，三角形网格、参数化、点云、隐函数等）。通常，必须生成网格以应用有限元类型技术或为每个不同的流形表示导出专门的离散化过程。相反，我们建议通过最近点法（CPM）的新颖扩展来处理内部边界条件，以统一的方式解决这些问题。 CPM 通过求解在包含流形的笛卡尔嵌入空间上定义的体积 PDE 来求解流形 PDE，并且仅需要流形的最近点表示。因此，CPM 支持开放或封闭、可定向或不可定向以及任何余维的对象。为了支持内部边界条件，我们推导了一种跨内部边界隐式划分嵌入空间的方法。 CPM 的有限差分和插值模板经过调整以尊重此划分，同时保留二阶精度。此外，我们开发了一种高效的稀疏网格实现和数值求解器，可以扩展到数千万个自由度，从而可以在更复杂的流形上求解偏微分方程。 我们展示了我们的方法在选定模型偏微分方程上的收敛行为，并探索了几个几何处理问题：曲面上的扩散曲线、测地距离、切向量场设计、调和图构建和反应扩散纹理。因此，我们提出的方法为一般流形表示上的一系列几何处理任务提供了强大且灵活的新工具。