Reconstructing Curves from Points and Tangents

TL;DR

This paper investigates how incorporating tangent information improves the reconstruction of curves from samples, showing that fewer samples are needed when tangents are available compared to points alone.

Contribution

It introduces a theoretical analysis demonstrating that tangent data reduces the sampling rate needed for error-free curve reconstruction from O(D) to O(sqrt(D)).

Findings

Tangent information significantly reduces sampling requirements.

Error-free reconstruction is achievable with fewer samples when tangents are used.

Sampling rate drops from linear to square root order with tangent data.

Abstract

Reconstructing a finite set of curves from an unordered set of sample points is a well studied topic. There has been less effort that considers how much better the reconstruction can be if tangential information is given as well. We show that if curves are separated from each other by a distance D, then the sampling rate need only be O(sqrt(D)) for error-free reconstruction. For the case of point data alone, O(D) sampling is required.