Length Learning for Planar Euclidean Curves

TL;DR

This paper introduces ArcLengthNet, a neural network designed to learn the length of planar curves from data, demonstrating robustness to noise and discretization errors, and advancing geometric property reconstruction via deep learning.

Contribution

It presents a novel neural network model for learning curve lengths that reconstructs fundamental geometric axioms from data, with robustness to noise and discretization errors.

Findings

ArcLengthNet accurately predicts curve lengths from sine wave datasets.

The model is robust to additive noise and discretization errors.

It successfully reconstructs fundamental length axioms using supervised learning.

Abstract

In this work, we used deep neural networks (DNNs) to solve a fundamental problem in differential geometry. One can find many closed-form expressions for calculating curvature, length, and other geometric properties in the literature. As we know these concepts, we are highly motivated to reconstruct them by using deep neural networks. In this framework, our goal is to learn geometric properties from examples. The simplest geometric object is a curve. Therefore, this work focuses on learning the length of planar sampled curves created by a sine waves dataset. For this reason, the fundamental length axioms were reconstructed using a supervised learning approach. Following these axioms a simplified DNN model, we call ArcLengthNet, was established. The robustness to additive noise and discretization errors were tested.