Smooth quasi-developable surfaces bounded by smooth curves

Computing a quasi-developable strip surface bounded by design curves finds wide industrial applications. Existing methods compute discrete surfaces composed of developable lines connecting sampling points on input curves which are not adequate for generating smooth quasi-developable surfaces. We propose the first method which is capable of exploring the full solution space of continuous input curves to compute a smooth quasi-developable ruled surface with as large developability as possible. The resulting surface is exactly bounded by the input smooth curves and is guaranteed to have no self-intersections. The main contribution is a variational approach to compute a continuous mapping of parameters of input curves by minimizing a function evaluating surface developability. Moreover, we also present an algorithm to represent a resulting surface as a B-spline surface when input curves are B-spline curves.Comment: 18 page

Developable Quad Meshes

There are different ways to capture the property of a surface being developable, i.e., it can be mapped to a planar domain without stretching or tearing. Contributions range from special parametrizations to discrete-isometric mappings. So far, a local criterion expressing the developability of general quad meshes has been lacking. In this paper, we propose a new and efficient discrete developability criterion that is based on a property well-known from differential geometry, namely a rank-deficient second fundamental form. This criterion is expressed in terms of the canonical checkerboard patterns inscribed in a quad mesh which already was successful in describing discrete-isometric mappings. In combination with standard global optimization procedures, we are able to perform developable lofting, approximation, and design. The meshes we employ are combinatorially regular quad meshes with isolated singularities but are otherwise not required to follow any special curves. They are thus easily embedded into a design workflow involving standard operations like re-meshing, trimming, and merging operations

Forward and Inverse D-Form Modelling based on Optimisation

D-Form is a special piece-wise developable surface formed by aligning the boundaries of two planar domains. It has been widely used in different design scenarios. In this paper, we study how to computationally and intuitively model D-Forms. We present an optimisation-based framework that can efficiently generate D-Form shapes. Our framework can model D-Forms with two approaches based on two different user inputs, including the forward modelling from two given planar domains and, more importantly, the inverse modelling from a given space curve where the planar domains are no longer needed. Our optimisation is devised based on two critical characteristics of D-Forms. Firstly, the constituent developable surfaces of a D-Form are isometrically deformed from planar domains. Secondly, there is a close relationship between a D-Form and the convex hull of its seam. Through extensive evaluation, we demonstrate that our approach can model plausible D-Forms efficiently from various inputs with different geometric properties.<br/

Forward and Inverse D-Form Modelling based on Optimisation

D-Form is a special piece-wise developable surface formed by aligning the boundaries of two planar domains. It has been widely used in different design scenarios. In this paper, we study how to computationally and intuitively model D-Forms. We present an optimisation-based framework that can efficiently generate D-Form shapes. Our framework can model D-Forms with two approaches based on two different user inputs, including the forward modelling from two given planar domains and, more importantly, the inverse modelling from a given space curve where the planar domains are no longer needed. Our optimisation is devised based on two critical characteristics of D-Forms. Firstly, the constituent developable surfaces of a D-Form are isometrically deformed from planar domains. Secondly, there is a close relationship between a D-Form and the convex hull of its seam. Through extensive evaluation, we demonstrate that our approach can model plausible D-Forms efficiently from various inputs with different geometric properties.<br/

DA Wand: Distortion-Aware Selection using Neural Mesh Parameterization

We present a neural technique for learning to select a local sub-region around a point which can be used for mesh parameterization. The motivation for our framework is driven by interactive workflows used for decaling, texturing, or painting on surfaces. Our key idea is to incorporate segmentation probabilities as weights of a classical parameterization method, implemented as a novel differentiable parameterization layer within a neural network framework. We train a segmentation network to select 3D regions that are parameterized into 2D and penalized by the resulting distortion, giving rise to segmentations which are distortion-aware. Following training, a user can use our system to interactively select a point on the mesh and obtain a large, meaningful region around the selection which induces a low-distortion parameterization. Our code and project page are currently available.Comment: Project page: https://threedle.github.io/DA-Wand/ Code: https://github.com/threedle/DA-Wan

How to use parametric curved folding design methods- a case study and comparison

Designs based on developable surfaces can be convenient for many reasons, however designing developable patterns that make use of curved creases is a challenge. Many studies propose new methods to tackle the problem but sometimes these methods do not generate a parametric model which is easily modifiable by changing the input parameters. Furthermore, the known methods are applicable only to certain families of curved folded models, because there is no generalized method for curved folding yet. Thus, sometimes, it is hard for designers to decide which method is more suitable for their needs. This paper shows how to use different well-known and newer approaches to produce parametric curved folded designs. The potentialities and criticalities of three approaches are compared by applying them to the same case study, namely the “curved folded tripod”. The aim, thus, is to make the design of curved folded geometries more accessible to designers without a background in origami theory

Applying dynamic relaxation techniques to form-find and manufacture curve-crease folded panels

The research incorporated in the paper stems from the design and fabrication of a self-supporting, multi-panel installation for the Venice Biennale 2012 and operates against the backdrop of the exciting potentials that the field of curved-crease folding offers in the development of curved surfaces that can be manufactured from sheet material. The two main challenges were developing an intuitive design strategy and production of information adhering to manufacturing constraints. The essential contribution of the paper is a proposed interactive form-finding method for curve-crease geometries that could negotiate the multiple objectives of ease of use in exploratory design, and manufacturing constraints of their architectural-scale assemblies

Developability Approximation for Neural Implicits through Rank Minimization

Developability refers to the process of creating a surface without any tearing or shearing from a two-dimensional plane. It finds practical applications in the fabrication industry. An essential characteristic of a developable 3D surface is its zero Gaussian curvature, which means that either one or both of the principal curvatures are zero. This paper introduces a method for reconstructing an approximate developable surface from a neural implicit surface. The central idea of our method involves incorporating a regularization term that operates on the second-order derivatives of the neural implicits, effectively promoting zero Gaussian curvature. Implicit surfaces offer the advantage of smoother deformation with infinite resolution, overcoming the high polygonal constraints of state-of-the-art methods using discrete representations. We draw inspiration from the properties of surface curvature and employ rank minimization techniques derived from compressed sensing. Experimental results on both developable and non-developable surfaces, including those affected by noise, validate the generalizability of our method