Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate multivalued smooth function \(f: {\mathbb {R}}^d\rightarrow {\mathbb {R}}^{d-n}\). A natural (and efficient) way to approximate an isomanifold is to consider its piecewise-linear (PL) approximation based on a triangulation \(\mathcal {T}\) of the ambient space \({\mathbb {R}}^d\). In this paper, we give conditions under which the PL approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently fine and thick triangulation \(\mathcal {T}\). This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the Fréchet distance between the original isomanifold and its PL approximation. Finally, we show analogous results for the PL approximation of an isomanifold with boundary.

等值线是等值面到任意维度和余维的泛化，即定义为一些多元多值平滑函数的零集的流形\(f: {\mathbb {R}}^d\rightarrow {\mathbb {R}}^{dn }\)。一种自然（且有效）逼近同形流形的方法是基于环境空间\({\mathbb {R}}^d )的三角剖分\(\mathcal {T}\)来考虑其分段线性 (PL) 逼近\)。在本文中，我们给出了一个等分形的 PL 近似在拓扑上等价于该等分形的条件。这些条件很容易满足，因为它们总是可以通过采用足够细和粗的三角剖分来满足\(\mathcal {T}\). 这与之前关于流形三角剖分的结果形成对比，在流形三角剖分中，在任意维度上，需要微妙的扰动来保证拓扑正确性，这导致实践中存在很大的局限性。我们进一步给出了原始等流形与其 PL 近似值之间的 Fréchet 距离的界限。最后，我们展示了具有边界的等流形的 PL 近似的类似结果。