Freeform surfaces from single curved panels

Planar quadrilateral (PQ) mesh generation is a key process in computer-aided design, particularly for architectural applications where the goal is to discretize a freeform surface using planar quad faces. The conjugate direction field (CDF) defined on the freeform surface plays a significant role in generating a PQ mesh, as it largely determines the PQ mesh layout. Conventionally, a CDF is obtained by solving a complex non-linear optimization problem that incorporates user preferences, i.e., aligning the CDF with user-specified strokes on the surface. This often requires a large number of iterations that are computationally expensive, preventing the interactive CDF design process for a desirable PQ mesh. To address this challenge, we propose a data-driven approach based on neural networks for controlled CDF generation. Our approach can effectively learn and fuse features from the freeform surface and the user strokes, and efficiently generate quality CDF respecting user guidance. To enable training and testing, we also present a dataset composed of 50000+ freeform surfaces with ground-truth CDFs, as well as a set of metrics for quantitative evaluation. The effectiveness and efficiency of our work are demonstrated by extensive experiments using testing data, architectural surfaces, and general 3D shapes.

Mesh surfaces with planar hexagonal faces, what we refer to as PH meshes, offer an elegant way of paneling freeform architectural surfaces due to their node simplicity (i.e., valence-3 nodes) and naturally appealing layout. We investigate PH meshes to understand how the shape, size, and pattern of PH faces are constrained by surface geometry. This understanding enables us to develop an effective method for paneling freeform architectural surfaces with PH meshes. Our method first constructs an ideal triangulation of a given smooth surface, guided by surface geometry. We show that such an ideal triangulation leads to a Dupin-regular PH mesh via tangent duality on the surface. We have developed several novel and effective techniques for improving undesirable mesh layouts caused by singular behaviors of surface curvature. We compute support structures associated with PH meshes, including exact vertex offsets and approximate edge offsets, as demanded in panel manufacturing. The efficacy of our method is validated by a number of architectural examples.

Constructing smooth freeform surfaces of arbitrary topology with higher order continuity is one of the most fundamental problems in shape and solid modeling. This paper articulates a novel method to construct C∞ smooth surfaces with negative Euler numbers based on hyperbolic geometry and discrete curvature flow. According to Riemann uniformization theorem, every surface with negative Euler number has a unique conformal Riemannian metric, which induces Gaussian curvature of --1 everywhere. Hence, the surface admits hyperbolic geometry. Such uniformization metric can be computed using the discrete curvature flow method: hyperbolic Ricci flow. Consequently, the basis function for each control point can be naturally defined over a hyperbolic disk, and through the use of partition-of-unity, we build a freeform surface directly over hyperbolic domains while having C∞ property. The use of radial, exponential basis functions gives rise to a true meshless method for modeling freeform surfaces with greatest flexibilities, without worrying about control point connectivity. Our algorithm is general for arbitrary surfaces with negative Euler characteristic. Furthermore, it is C∞ continuous everywhere across the entire hyperbolic domain without singularities. Our experimental results demonstrate the efficiency and efficacy of the proposed new approach for shape and solid modeling.

Solar radiation pressure (SRP) is an extremely critical perturbative force that affects the GNSS satellites’ precise orbit determination (POD). Its imperfect modelling is one of the main error sources of POD, whose magnitude is even to10−9 m/s2. The shadow factor (i.e., eclipse factor) is one crucial parameter of SRP, generally estimated by the cylindrical model, the conical model, or shadow models considering the Earth’s oblateness and the atmospheric effect, such as the Perspective Projection Method atmosphere (PPMatm) model and Solar radiation pressure with Oblateness and Lower Atmospheric Absorption, Refraction, and Scattering Curve Fit (SOLAARS-CF) model. This paper applies the former four shadow models to determine the corresponding precise orbit using BeiDou satellites’ ground-based observation, and then compared and assessed the orbit accuracy through Satellite Laser Ranging (SLR) validation and Inter-Satellite Link (ISL) check. The results show that the PPMatm model’s accuracy is equivalent to the SOLAARS-CF model. Compared with the conical shadow model, SLR validations show the orbit accuracy from the PPMatm and SOLAARS-CF model can be generally improved by 2–10 mm; ISL range check shows that the Root Mean Square (RMS) can be decreased by 2–7 mm. These results show that the shadow model in GNSS POD should fully consider the Earth’s oblateness and the atmospheric effect, especially for the perturbative acceleration higher than 10–10 m/s2.Graphical

Grid generation on free-form surface using guide line advancing and surface flattening method

We define triangulated piecewise linear constant mean curvature surfaces using a variational characterization. These surfaces are critical for area amongst continuous piecewise linear variations which preserve the boundary conditions, the simplicial structures, and (in the nonminimal case) the volume to one side of the surfaces. We then find explicit formulas for complete examples, such as discrete minimal catenoids and helicoids. We use these discrete surfaces to study the index of unstable minimal surfaces, by numerically evaluating the spectra of their Jacobi operators. Our numerical estimates confirm known results on the index of some smooth minimal surfaces, and provide additional information regarding their area-reducing variations. The approach here deviates from other numerical investigations in that we add geometric interpretation to the discrete surfaces.

Efficient optimal design of smooth optical freeform surfaces using ray targeting

Flexible Kokotsakis Meshes with Skew Faces: Generalization of the Orthodiagonal Involutive Type

The development of various construction robots has transformed the architectural landscape, facilitating the design and construction of more unique structures through the implementation of diverse designs.This study explores 13 architectural building applications constructed using the additive construction method, comparing them with traditional construction methods as alternatives.Initial steps involved establishing critical criteria for the construction process through extensive literature review and expert consultations.The significance of these criteria was determined using fuzzy Shannon's Entropy Based on Alpha Level, identifying considerations paramount during the construction phase.Subsequently, the importance degrees of these criteria were instrumental in evaluating potential construction methods for each architectural application using the Fuzzy TOPSIS method, renowned for its efficacy.Findings prioritize building size, construction time, and cost for the architectural applications under review.The comparative analysis revealed that 6 out of the 13 applications constructed with the additive method would benefit more from alternative methods, with the Wood Skeleton method emerging as the superior alternative, closely followed by Concrete Pouring with Mold.Incorporating Fuzzy Shannon's Entropy and the Fuzzy TOPSIS method, it offers a more nuanced and comprehensive assessment of various construction techniques.Furthermore, the study's focus on key criteria such as building size, construction time, and cost aligns closely with the industry's evolving priorities, emphasizing efficiency and sustainability.The findings underscore the need for a more adaptive and selective application of construction technologies and setting a new benchmark for future studies in construction method optimization that blend traditional techniques with cuttingedge decision-making tools.

Toolpath generation for ultraprecision machining: Freeform surface generation and extrapolation

We present new interaction and visualization paradigms relying on free form surfaces for studying and exploring human anatomy. We propose an interface for building three-dimensional anatomical scenes incorporating 3D anatomical organ models, freely orientated slices and free form surfaces extracted from the Visible Human dataset. Compared with planar slices, free form surfaces allow to follow curved anatomic structures such as the aorta tree or the pelvis. In the present paper, we describe in detail 3D interaction techniques for creating free form surfaces. The interactive placement of surface boundary curves relies on the combination of an interactive slice navigator and of a 3D visualization interface integrated within a single java applet. Surface boundary curve control points are placed with the mouse at the desired locations within the selected slices. The corresponding boundary curves are displayed in the 3D visualization interface as fat 3D cubic spline curves, which provide immediate feedback. Boundary curves may be easily duplicated, translated and modified. The specified boundary curves are interpolated by Coons patches, yielding a perfectly smooth surface. That surface may be visualized in combination with semi-transparent organ models. It may also be flattened and shown in a separate window. The presented application is available online as a Java applet (http://visiblehuman.epfl.ch).

A new method of experimental buckling of circular cylindrical roof shell models with free edges subjected to external pressure

Conjugate parametrizations of surfaces were first discretized almost a century ago as quad meshes with planar faces. With the recent development of discrete differential geometry, two discretizations of principal curvature line parametrizations were discovered: circular nets and conical nets, both of which are special cases of discrete conjugate nets. Subsequently, circular and conical nets were given a unified description as isotropic line congruences in the Lie quadric. We propose a generalization by considering polar pairs of line congruences in the ambient space of the Lie quadric. These correspond to pairs of discrete conjugate nets with orthogonal edges, which we call principal binets, a new and more general discretization of principal curvature line parametrizations. We also introduce two new discretizations of orthogonal and Gauß-orthogonal parametrizations. All our discretizations are subject to the transformation group principle, which means that they satisfy the corresponding Lie, Möbius, or Laguerre invariance respectively, in analogy to the smooth theory.

This paper studies a Laplace operator on semi-discrete surfaces. A semi-discrete surface is represented by a mapping into three-dimensional Euclidean space possessing one discrete variable and one continuous variable. It can be seen as a limit case of a quadrilateral mesh, or as a semi-discretization of a smooth surface. Laplace operators on both smooth and discrete surfaces have been an object of interest for a long time, also from the viewpoint of applications. There are a wealth of geometric objects available immediately once a Laplacian is defined, e.g., the mean curvature normal. We define our semi-discrete Laplace operator to be the limit of a discrete Laplacian on a quadrilateral mesh, which converges to the semi-discrete surface. The main result of this paper is that this limit exists under very mild regularity assumptions. Moreover, we show that the semi-discrete Laplace operator inherits several important properties from its discrete counterpart, like symmetry, positive semi-definiteness, and linear precision. We also prove consistency of the semi-discrete Laplacian, meaning that it converges pointwise to the Laplace---Beltrami operator, when the semi-discrete surface converges to a smooth one. This result particularly implies consistency of the corresponding discrete scheme.

Electroforming is a nonconventional manufacturing process of metallic parts by using electrodeposition on a pre-shaped cathodic mandrel. It can produce a metallic structure and builds the material by atom level. Smooth free surface and normal edge deposition are important to the electroforming of complex-shaped parts. A hybrid electroforming process combining mechanical particle polishing and auxiliary cathode was proposed to improve the quality of the free surface and inhibit the over-deposition on the mandrel edge. The smooth effect of particle polishing on the growing deposit surface was theoretical analyzed. The effect of the auxiliary cathode for a complex model with freeform surface was simulated. Contrast electroforming tests were performed in a bath without any addition agents under different conditions. The results positively proved the smooth effect and the overgrowth inhibition effect of the hybrid process. A nickel complex sample with smooth free surface and inconspicuous edge overgrowth was successfully electroformed.