Fourier-Motzkin elimination is a fundamental operation in polyhedral geometry. It can be performed by several equivalent procedures, and can be regarded as an adaptation of Gaussian elimination to systems of linear inequalities. These procedures tend to generate large numbers of redundant inequalities. Efficiently detecting these redundancies is essential for obtaining software implementation of practical interest. In this paper, we propose a novel detection technique. We demonstrate its benefits over alternative approaches. A detailed experimentation is reported.

Fourier-Motzkin elimination is a fundamental operation in polyhedral geometry. It can be performed by several equivalent procedures, which can be regarded as an adaptation of Gaussian elimination to systems of linear inequalities. These procedures tend to generate large numbers of redundant inequalities. Efficiently detecting these redundancies is essential to obtain software implementation of practical interest. In this paper, we propose a detection technique. We demonstrate its benefits over alternative approaches. A detailed experimentation is reported.

We introduce IGF-Fit, a novel method for estimating surface normals from point clouds with varying noise and density. Unlike previous approaches that rely on point-wise weights and explicit representations, IGF-Fit employs a network that learns an implicit representation and uses derivatives to predict normals. The input patch serves as both a shape latent vector and query points for fitting the implicit representation. To handle noisy input, we introduce a novel noise transformation module with a training strategy for noise classification and latent vector bias prediction. Our experiments on synthetic and real-world scan datasets demonstrate the effectiveness of IGF-Fit, achieving state-of-the-art performance on both noise-free and density-varying data.

Tool-path planning is a crucial factor of computer-aided design (CAD) and computer-aided manufacturing (CAM). Previous path generation methods often transform the problem into local or global optimization methods to solve it, leading to a long computational time. With the development of modern industry, real-time path planning is becoming an urgent issue in advanced manufacturing. This article proposes an efficient neural network-based direct tool-path generation method on B-spline surface for subtractive end milling. In order to build the first corresponding dataset, adaptive iso-scallop height method is proposed, which can effectively avoid generating breakpoints at the boundary. B-Spline reparameterization is used to fit discrete tool paths to obtain regular control points data structure for further deep learning. After that, an intelligent neural network is proposed to learn the relationship between the input B-Spline surface and the reparameterized tool paths. Finally, experimental results and case study are provided to illustrate and clarify our method, which only needs a few microseconds of planning time while ensuring the quality of the generated paths. Due to its simple structure and low computational burden, this method can be easily applied to CAD/CAM software.

Smooth surfaces from an arbitrary topological control grid have been widely studied, which are mostly generalized from splines with uniform knot intervals. These methods fail to work well on extraordinary points (EPs) whose edges have varying knot intervals. This article presents a patching solution for arbitrary topological 2-manifold control grid with non-uniform knots that defines one bi-cubic Bézier patch per control grid face except those faces with EPs. Experimental results demonstrate that the new solution can improve the surface quality for non-uniform parameterization. Applications in surface reconstruction, arbitrary sharp features on the complex surface and tool path planning for the new surface representation are also provided in the paper.

Reconstructing a spline surface from a given arbitrary topological triangle mesh is a fundamental and challenging problem in computer-aided design and engineering. This article introduces a novel surface fitting method utilizing G-NURBS capable of handling control meshes with arbitrary topologies. This method employs adaptive control point adjustment, guided by the geometric attributes of the input model, ensuring precise representation of sharp features such as edges and corners. Two primary strategies are employed: A parameter correspondence approach designed for sharp features and a control mesh iterative refinement technique that incorporates geometrical feature information. The proposed method has been tested and evaluated on various CAD models to demonstrate its effectiveness. This method can achieve higher fitting accuracy while faithfully preserving the geometrical features with fewer control points.

We present a novel implicit neural representation to reconstruct CAD models from point clouds with high quality. Our method first extracts edge points from input points by an edge detection network and then recovers implicit surfaces while preserving the sharp features of ground-truth models. The edge detection network uses a U-Net structure for feature encoding, and the attention module is introduced to improve the accuracy in the CAD models. This detection network is light weighted and runs fast. Afterward, we propose an MLP-based network to train an implicit representation from input points with its extracted edge points. A two-stage training process is proposed, and loss functions are designed for each stage to ensure that the sharp features of the input points are learned while the surface details are fitted. Comparing our method with other SOTA methods in the ABC dataset, our method is significantly superior to the existing nonlearning and learning 3D reconstruction methods in terms of surface approximation quality and sharp feature preservation. Moreover, we can gain spline representations from learned shapes for CAM as the application of our method.

We introduce a neural network, MixNet, for learning implicit representations of 3D subtle models with large smooth areas and exact shape details in the form of interpolation of two different implicit functions. Our network takes a point cloud as input and uses conventional MLP networks and SIREN networks to predict different implicit fields. We use a learnable interpolation function to combine the implicit values of these two networks and achieve the respective advantages of them. The network is self-supervised with only reconstruction loss, leading to faithful 3D reconstructions with smooth planes, correct details, and plausible spatial partition without any ground-truth segmentation. We evaluate our method on ABC, the largest and most diverse CAD dataset, and some typical shapes to test in terms of geometric correctness and surface smoothness to demonstrate superiority over current alternatives suitable for shape reconstruction.

Let$H(X_{t})$be the differential entropy of an$n$-dimensional random vector$X_{t}$introduced by Costa. Cheng and Geng conjectured that$C_{1}(m, n): (-1)^{m+1}(\mathrm{d}^{m}/\mathrm{d}^{m}t)H(X_{t})\geq 0$. McKean conjectured that$C_{1}(m, n): (-1)^{m+1}(\mathrm{d}^{m}/\mathrm{d}^{m}t)H(X_{t})\geq 0 (-1)^{m+1}(\mathrm{d}^{m}/\mathrm{d}^{m}t)H(X_{Gt})$. McKean's conjecture was only considered in the univariate case before:$C_{2}(1,1)$and$C_{2}(2,1)$were proved by McKean and$C_{2}(i, 1), i=3,4,5$were proved by Zhang-Anantharam-Geng under the log-concave condition. In this paper, we prove$C_{2}(1, n),\ C_{2}(2, n)$and observe that McKean's conjecture might not be true for$n\ > \ 1$and$m > 2$. We further propose a weaker conjecture$C_{3}(m, n): (-1)^{m+1}(\mathrm{d}^{m}/\mathrm{d}^{m}t)H(X_{t}) \ \geq\ (-1)^{m+1}\frac{1}{n}(\mathrm{d}^{m}/\mathrm{d}^{m}t)H(X_{Gt})$and prove$C_{3}(3,2), C_{3}(3,3), C_{3}(3,4)$under the log-concave condition. A systematic procedure to prove$C_{l}(m, n)$is proposed and the results mentioned above are proved using this procedure.

In this paper, the concept of sparse difference resultant for a Laurent transformally essential system of Laurent difference polynomials is introduced and its properties are proved. In particular, order and degree bounds for the sparse difference resultant are given. Based on these bounds, an algorithm to compute the sparse difference resultant is proposed, which is single exponential in terms of the number of variables, the Jacobi number, and the size of the system. Also, the precise order, degree, a determinant representation, and a Poisson-type product formula for the difference resultant are given.

In this paper, the concept of sparse differential resultant for a differentially essential system of differential polynomials is introduced and its properties are proved. In particular, a degree bound for the sparse differential resultant is given. Based on the degree bound, an algorithm to compute the sparse differential resultant is proposed, which is single exponential in terms of the order, the number of variables, and the size of the differentially essential system.

In this paper, a new theory of resolvent systems is developed for prime difference ideals and difference ideals defined by coherent and proper irreducible ascending chains. Algorithms to compute such resolvent systems are also given. As a consequence, we prove that any irreducible difference variety is birationally equivalent to an irreducible difference variety of codimension one. As a preparation to the resolvent theory, we also prove that the saturation ideal of a coherent and proper ascending chain is unmixed in the sense that all its prime components have the same dimension and order.