Piecewise Ruled Approximation for Freeform Mesh Surfaces

TL;DR

This work addresses approximating arbitrary freeform triangle-mesh surfaces with piecewise ruled surfaces by introducing a joint optimization of mesh geometry and a ruling-direction tangent field. It proposes a novel local ruling model with a per-face ruling-variation parameter $\gamma_f$, enabling nonuniform ruling directions and better adherence to the target shape, followed by seam extraction to initialize rulings and a final seam-boundary refinement. The approach combines geodesic and curvature constraints with a robust, group-sparsity energy to encourage interior patches while preserving essential seam topology, yielding accurate approximations with limited seams. The results demonstrate strong accuracy across diverse topologies and curvature distributions, with improved efficiency and fabrication-friendly representations, showing potential for practical design and manufacturing pipelines.

Abstract

A ruled surface is a shape swept out by moving a line in 3D space. Due to their simple geometric forms, ruled surfaces have applications in various domains such as architecture and engineering. In the past, various approaches have been proposed to approximate a target shape using developable surfaces, which are special ruled surfaces with zero Gaussian curvature. However, methods for shape approximation using general ruled surfaces remain limited and often require the target shape to be either represented as parametric surfaces or have non-positive Gaussian curvature. In this paper, we propose a method to compute a piecewise ruled surface that approximates an arbitrary freeform mesh surface. We first use a group-sparsity formulation to optimize the given mesh shape into an approximately piecewise ruled form, in conjunction with a tangent vector field that indicates the ruling directions. Afterward, we utilize the optimization result to extract seams that separate smooth families of rulings, and use the seams to construct the initial rulings. Finally, we further optimize the positions and orientations of the rulings to improve the alignment with the input target shape. We apply our method to a variety of freeform shapes with different topologies and complexity, demonstrating its effectiveness in approximating arbitrary shapes.

Figures (21)

• Figure 1: Examples of ruled surfaces. Left: an art installation using tensioned strings, in the Science Museum in London. Right: the 'Ouverture au Monde' sculpture in Ouchy, Lausanne, by Ángel Duarte.
• Figure 2: The pipeline of our piecewise ruled surface approximation. Given an input reference surface mesh, we first optimize the mesh shape together with a vector field on it, to approximate a piecewise ruled surface and its rulings. The result is then used to extract an initial piecewise ruled surface. Lastly, we optimize the piecewise ruled surface to further align it with the reference surface and obtain the final result.
• Figure 3: Piecewise constant vector fields are insufficient for capturing the variation of ruling directions on a ruled surface. Here we show a ruled surface discretized as a triangle mesh, where the original rulings are a smooth family of straight lines connecting the top and bottom curves (a subset is displayed here in brown). We first represent the ruling directions using a piecewise constant vector field according to the original ruling directions at the face centroids, and use it to trace a polyline starting from a point on the bottom boundary. There is a notable deviation between the resulting polyline (displayed in green) and the original ruling starting at the same point (displayed in red) when they reach the top boundary. On the other hand, if we represent the ruling directions with our vector field model in Eq. \ref{['eq:PointRuilngVector']} and start tracing from the same point, the resulting polyline (displayed in blue) is much closer to the original ruling.
• Figure 4: With our local ruled surface model in Eq. \ref{['eq:LocalModel']}, the iso-$u$ lines are straight line segments that approximate the rulings around the point $\mathbf{p}$, whereas the parameter $\gamma$ captures the local variation of the rulings.
• Figure 5: A schematic illustration of the local parameterization in Eq. \ref{['eq:LocalParameterization']}, which is constructed by projecting the adjacent rulings of a point $\mathbf{p}$ onto its tangent plane $T_{\mathbf{p}}$.

Theorems & Definitions (1)

• proof