Smooth Surfaces via Nets of Geodesics

TL;DR

This paper introduces a method to compute smooth surfaces from a network of curves in 3D such that the curves are geodesics on the surface, aiding CAD design and reflecting artistic and cognitive insights.

Contribution

It presents a novel computational approach for constructing smooth surfaces with prescribed geodesic curves, bridging artistic, cognitive, and engineering applications.

Findings

Method efficiently computes surfaces with geodesic curves

Supports CAD implementation and artistic visualization

Aligns with cognitive science insights on visual perception

Abstract

The goal of this study is to provide a method for computing the following: Given a network of curves in 3d (satisfying a condition at the intersection points), compute efficiently a smooth surface such that the curves are geodesics on it. This work can serve as a base for engineers who wish to implement computations of such surfaces in Computer Aided Design (CAD) software or other applications. The motivation for this study was the following hypothesis and observation together with the desire to improve CAD interfaces. The hypothesis and observation is that artists draw projections of geodesics to illustrate 3d objects: for example projections of nets of curves can be seen in drawings of Rembrandt. In addition, this observation is supported by research in cognitive sciences: in a seminal work by the late David Knill he suggested that the human visual system incorporates a geodesic constraint in the processing of reflected contours.