We introduce a new extrinsic discretization of tangent vector fields on triangle meshes that is continuous, with bounded derivatives that are continuous almost everywhere, supporting pointwise evaluation and integration of differential operators. We achieve this by building a continuous normal field over the mesh via Phong interpolation and using minimal Rodrigues rotations to transport vertex-based tangent vectors into triangle interiors. Unlike most existing discretizations, which typically sacrifice either continuity or the ability to evaluate derivatives pointwise, our approach supports both. Because it is pointwise evaluatable, and using the fact that the covariant derivative can be decomposed into its symmetric, antisymmetric, and scalar components, our discretization supports the construction of standard vector-field processing operators including the connection and Hodge Laplacians, Killing energy, divergence, curl, and the Lie bracket. This framework provides a simple and practical finite-element formulation for vector-field processing on meshes, supporting both integration-based operators and pointwise queries. To our knowledge, ours is the first discretization that jointly enables extrinsic continuous vector fields, bounded derivatives, and pointwise evaluation of this collection of operators.

翻译：我们在三角形网格上引入了一种新的切向矢量场外蕴离散化方法，该方法是连续的，且其导数在几乎处处有界且连续，支持逐点评估和微分算子积分。我们通过裴荣插值在网格上构建连续法向量场，并利用最小罗德里格斯旋转将基于顶点的矢量场传输至三角形内部。与多数现有离散化方法（通常牺牲连续性或逐点导数求值能力）不同，我们的方法同时支持两者。由于可逐点评估，且协变导数可分解为对称、反对称和标量分量，本离散化支持构建标准矢量场处理算子，包括联络与霍奇拉普拉斯算子、基林能量、散度、旋度及李括号。该框架为网格上的矢量场处理提供了一种简单实用的有限元表述，同时支持基于积分的算子和逐点查询。据我们所知，这是首个同时实现外蕴连续矢量场、有界导数及上述算系列子的逐点评估的离散化方法。