We propose a novel method to generate a small set of ruled surfaces that do not collide with the input shape for linear hot-wire rough machining. Central to our technique is a new observation: ruled surfaces constructed by vertical extrusion from planar smooth curves that approach the input shape's outer contour lines without collisions can effectively remove material during rough machining. Accordingly, we develop an iterative algorithm that alternates in each iteration between computing a viewpoint to determine an outer contour line and optimizing a smooth curve to approximate that contour line under the collision-free constraint. Specifically, a view selection approach based on a genetic algorithm is used to optimize the viewpoint for removing materials as much as possible, and an adaptive fitting algorithm is presented to find the constrained curves. The feasibility and practicability of our method are demonstrated through 10 physical examples. Compared with manual designs, our method obtains lower errors with the same number of cuts.

Abstract We propose a novel method for generating symmetric piecewise developable approximations for shapes in approximately global reflectional or rotational symmetry. Given a shape and its symmetry constraint, the algorithm contains two crucial steps: (i) a symmetric deformation to achieve a nearly developable model and (ii) a symmetric segmentation aided by the deformed shape. The key to the deformation step is the use of the symmetric implicit neural representations of the shape and the deformation field. A new mesh extraction from the implicit function is introduced to construct a strictly symmetric mesh for the subsequent segmentation. The symmetry constraint is carefully integrated into the partition to achieve the symmetric piecewise developable approximation. We demonstrate the effectiveness of our algorithm over various meshes.

We propose a novel and automatic method to model shapes using a small set of discrete developable patches. Central to our approach is using implicit neural shape representation that makes our algorithm independent of tessellation and allows us to obtain the Gaussian curvature of each point analytically. With this powerful representation, we first deform the input shape to be an almost developable shape with clear and sparse salient feature curves. Then, we convert the deformed implicit field to a triangle mesh, which is further cut to disk topology along parts of the sparse feature curves. Finally, we achieve the resulting piecewise developable mesh by alternatingly optimizing discrete developability, enforcing manufacturability constraints, and merging patches. The feasibility and practicability of our method are demonstrated over various shapes. Compared to the state-of-the-art methods, our method achieves a better tradeoff between the number of developable patches and the approximation error.

We propose a practical method to construct sparse integer-constrained cone singularities with low distortion constraints for conformal parameterizations. Our solution for this combinatorial problem is a two-stage procedure that first enhances sparsity for generating an initialization and then optimizes to reduce the number of cones and the parameterization distortion. Central to the first stage is a progressive process to determine the combinatorial variables, i.e., numbers, locations, and angles of cones. The second stage iteratively conducts adaptive cone relocations and merges close cones for optimization. We extensively test our method on a data set containing 3885 models, demonstrating practical robustness and performance. Our method achieves fewer cone singularities and lower parameterization distortion than state-of-the-art methods.

We propose an efficient method to construct sparse cone singularities under distortion-bounded constraints for conformal parameterizations. Central to our algorithm is using the technique of shape derivatives to move cones for distortion reduction without changing the number of cones. In particular, the supernodal sparse Cholesky update significantly accelerates this movement process. To satisfy the distortion-bounded constraint, we alternately move cones and add cones. The capability and feasibility of our approach are demonstrated over a data set containing 3885 models. Compared with the state-of-the-art method, we achieve an average acceleration of 15 times and slightly fewer cones for the same amount of distortion.

We propose a novel method to compute high-quality piecewise developable approximations for triangular meshes. Central to our approach is an evolutionary genetic algorithm for optimizing the combinatorial and discontinuous fitness function, including the approximation error, the number of patches, the patch boundary length, and the penalty for small patches and narrow regions within patches. The genetic algorithm's operations (i.e., initialization, selection, mutation, and crossover) are explicitly designed to minimize the fitness function. The main challenge is evaluating the fitness function's approximation error as it requires developable patches, which are difficult or time-consuming to obtain. Resolving the challenge is based on a critical observation: the approximation error and the mapping distortion between an input surface and its developable approximation are positively correlated empirically. To efficiently measure distortion without explicitly generating developable shapes, we creatively use conformal mapping techniques. Then, we control the mapping distortion at a relatively low level to achieve high shape similarity in the genetic algorithm. The feasibility and effectiveness of our method are demonstrated over 240 complex examples. Compared with the state-of-the-art methods, our results have much smaller approximation errors, fewer patches, shorter patch boundaries, and fewer small patches and narrow regions.

We propose a novel method to compute a piecewise mesh with a few developable patches and a small approximation error for an input triangular mesh. Our key observation is that a deformed mesh after enforcing discrete developability is easily partitioned into nearly developable patches. To obtain the nearly developable mesh, we present a new edge-oriented notion of discrete developability to define a developability-encouraged deformation energy, which is further optimized by the block nonlinear Gauss-Seidel method. The key to successfully applying this optimizer is three types of auxiliary variables. Then, a coarse-to-fine segmentation technique is developed to partition the deformed mesh into a small set of nearly discrete developable patches. Finally, we refine the segmented mesh to reduce the discrete Gaussian curvature while keeping the patches smooth and the approximation error small. In practice, our algorithm achieves a favorable tradeoff between the number of developable patches and the approximation error. We demonstrate the feasibility and practicability of our method over various examples, including seventeen physical manufacturing models with paper.