Generalized altitudes and their bounds

2026-07-07

AI Summary

This work proposes the notion of “generalized height” in simplices, extending the classical concept—defined as the distance from a vertex to its opposite facet—to arbitrary pairs of opposite faces. The relative position of such face pairs is uniformly characterized via the angle between their affine subspaces. By leveraging generalized cross products and Gram determinants, the authors establish an algebraic relationship between generalized heights and classical heights, and prove that all generalized heights are bounded below by the minimal classical height. This result demonstrates that the standard thickness assumption in simplex quality analysis naturally extends to a broader class of geometric quantities, offering a new theoretical framework for assessing simplex quality and informing triangulations of Riemannian manifolds, thereby supporting the generation of high-quality meshes.

Abstract

We introduce generalized altitudes of a simplex, extending the usual vertex-to-opposite-face altitude to arbitrary pairs of opposite faces. These quantities encode the relative position of the affine spans of such faces and yield a uniform formula for the angle between them. We also derive an equivalent algebraic expression in terms of generalized cross products and Gram determinants, linking the construction to standard determinant-based tools. Finally, we prove that every generalized altitude is bounded below by a quantity controlled by the ordinary height of the simplex. Thus, classical height or thickness assumptions imply control over this broader family of geometric quantities. The results provide a compact framework for studying simplex quality and are motivated by applications to triangulation criteria for Riemannian manifolds.

Problem

• generalized altitudes
• simplex
• opposite faces
• height bounds
• triangulation criteria

Innovation

• generalized altitudes
• simplex geometry
• Gram determinants
• generalized cross products
• triangulation criteria