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Smooth Surfaces via Nets of Geodesics

Tom Gilat

TL;DR

This work tackles the problem of reconstructing a smooth surface from a 3D network of curves under the geodesic constraint by formulating and solving an energy-minimization problem that minimizes the total Gaussian curvature squared, expressed in isothermal coordinates as $K=-e^{-2f}\Delta f$ with a heuristic biharmonic reduction $f_{xxxx}+2f_{xxyy}+f_{yyyy}=0$. A coordinate-free framework links projection and isothermal charts, yielding a PDE system for the surface’s height and conformal factor, along with explicit Dirichlet and Neumann boundary conditions for the boundary contour. The proposed patch-based algorithm solves a sequence of problems per cell: (i) solve a biharmonic equation for the conformal factor $f$, (ii) prescribe curvature via a curvature Monge–Ampère equation to obtain a height function $h$, and (iii) validate via the discrete Laplace–Beltrami operator, with refinement by subdividing cells if necessary. The approach aims to enable CAD-amenable reconstruction of geodesic nets into smooth surfaces, with potential applications in sketch-to-3D, architectural design, and geometric modelling, supported by numerical experiments and pragmatic boundary-condition formulations.

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.

Smooth Surfaces via Nets of Geodesics

TL;DR

This work tackles the problem of reconstructing a smooth surface from a 3D network of curves under the geodesic constraint by formulating and solving an energy-minimization problem that minimizes the total Gaussian curvature squared, expressed in isothermal coordinates as with a heuristic biharmonic reduction . A coordinate-free framework links projection and isothermal charts, yielding a PDE system for the surface’s height and conformal factor, along with explicit Dirichlet and Neumann boundary conditions for the boundary contour. The proposed patch-based algorithm solves a sequence of problems per cell: (i) solve a biharmonic equation for the conformal factor , (ii) prescribe curvature via a curvature Monge–Ampère equation to obtain a height function , and (iii) validate via the discrete Laplace–Beltrami operator, with refinement by subdividing cells if necessary. The approach aims to enable CAD-amenable reconstruction of geodesic nets into smooth surfaces, with potential applications in sketch-to-3D, architectural design, and geometric modelling, supported by numerical experiments and pragmatic boundary-condition formulations.

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.

Paper Structure

This paper contains 16 sections, 2 theorems, 14 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background: minimal Gaussian curvature surface
  3. Preliminary definitions and assumptions
  4. Main analysis
  5. Formula for Neumann boundary condition
  6. PDE system for minimal Gaussian curvature surface
  7. Background: experimental part
  8. The full PDE of a cell in the net
  9. Solving the biharmonic equation
  10. The curvature Monge-Ampère equation
  11. Applying the Laplace-Beltrami operator
  12. Test case: the Neumann boundary condition formula
  13. Method Summary
  14. Acknowledgments
  15. Data availability
  16. ...and 1 more sections

Key Result

Theorem 1

Given a smooth simple closed curve $\Gamma$ for which Properties (1) and (2) hold. Let $\widetilde{S}$ and $\widetilde{H}$ be as in the statement of Property (1) and (2), with respect to $\Gamma$. Let $g=e^{2f(x,y)}(dx^2+dy^2)$ be a Riemannian metric on $\widetilde{S}$ for a coordinate chart which m

Figures (5)

  • Figure 1: An Elephant, Rembrandt 1637 (Source: Wikimedia). Projections of nets of curves can be seen in the midsection area, on the head, on the body to the left of the ear and in other areas.
  • Figure 2: Logic behind Neumann boundary condition formula.
  • Figure 3: Computed minimal Gaussian curvature surfaces. Neumann boundary condition is naively set to zero allowing a simpler computing approach compromising accuracy.
  • Figure 4: Minimal Gaussian curvature surfaces on adjacent cells.
  • Figure 5: Computed curvature for a planar circle contour

Theorems & Definitions (9)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • proof : Proof (PDE for $f$ inside the domain)
  • proof : Proof (Dirichlet Boundary Condition)
  • Theorem 2
  • proof