Algorithms, unaffected by the Schwarz paradox, approximating tangent planes and area of smooth surfaces via inscribed triangular polyhedra

TL;DR

This paper introduces an algorithm for approximating tangent planes and surface areas of smooth surfaces using inscribed triangles, overcoming limitations posed by the Schwarz paradox and ensuring convergence regardless of triangle shape or position.

Contribution

The paper presents a novel algorithm that reliably approximates tangent bivectors and surface areas through inscribed triangles, unaffected by the Schwarz paradox.

Findings

Algorithm converges to true tangent planes and areas.

Approximations are valid regardless of triangle shape or position.

Method overcomes limitations of classical approaches affected by the Schwarz paradox.

Abstract

In this work we provide an algorithm approximating the tangent bivector at a point of a smooth surface through inscribed triangles converging to the point, regardless their form or position with respect to the tangent plane. This result is obtained approximating Jacobian determinants of smooth plane transformations at a point x through nondegenerate triangles converging to x. We can also approximate the area of a portion of a smooth surface, through a slightly modified notion of area of inscribed triangular polyhedra approaching the surface (without any kind of constraint due to the Schwarz paradox).