No Arabic abstract
Origami structures enabled by folding and unfolding can create complex 3D shapes. However, even a small 3D shape can have large 2D unfoldings. The huge initial dimension of the 2D flattened structure makes fabrication difficult, and defeats the main purpose, namely compactness, of many origami-inspired engineering. In this work, we propose a novel algorithmic kirigami method that provides super compaction of an arbitrary 3D shape with non-negligible surface thickness called algorithmic stacking. Our approach computationally finds a way of cutting the thick surface of the shape into a strip. This strip forms a Hamiltonian cycle that covers the entire surface and can realize transformation between two target shapes: from a super compact stacked shape to the input 3D shape. Depending on the surface thickness, the stacked structure takes merely 0.001% to 6% of the original volume. This super compacted structure not only can be manufactured in a workspace that is significantly smaller than the provided 3D shape, but also makes packing and transportation easier for a deployable application. We further demonstrate that, the proposed stackable structure also provides high pluripotency and can transform into multiple 3D target shapes if these 3D shapes can be dissected in specific ways and form a common stacked structure. In contrast to many designs of origami structure that usually target at a particular shape, our results provide a universal platform for pluripotent 3D transformable structures.
We investigate discrete spin transformations, a geometric framework to manipulate surface meshes by controlling mean curvature. Applications include surface fairing -- flowing a mesh onto say, a reference sphere -- and mesh extrusion -- e.g., rebuilding a complex shape from a reference sphere and curvature specification. Because they operate in curvature space, these operations can be conducted very stably across large deformations with no need for remeshing. Spin transformations add to the algorithmic toolbox for pose-invariant shape analysis. Mathematically speaking, mean curvature is a shape invariant and in general fully characterizes closed shapes (together with the metric). Computationally speaking, spin transformations make that relationship explicit. Our work expands on a discrete formulation of spin transformations. Like their smooth counterpart, discrete spin transformations are naturally close to conformal (angle-preserving). This quasi-conformality can nevertheless be relaxed to satisfy the desired trade-off between area distortion and angle preservation. We derive such constraints and propose a formulation in which they can be efficiently incorporated. The approach is showcased on subcortical structures.
Drawing network maps automatically comprises two challenging steps, namely laying out the map and placing non-overlapping labels. In this paper we tackle the problem of labeling an already existing network map considering the application of metro maps. We present a flexible and versatile labeling model. Despite its simplicity, we prove that it is NP-complete to label a single line of the network. For a restricted variant of that model, we then introduce an efficient algorithm that optimally labels a single line with respect to a given weighting function. Based on that algorithm, we present a general and sophisticated workflow for multiple metro lines, which is experimentally evaluated on real-world metro maps.
We desgin a novel fully convolutional network architecture for shapes, denoted by Shape Fully Convolutional Networks (SFCN). 3D shapes are represented as graph structures in the SFCN architecture, based on novel graph convolution and pooling operations, which are similar to convolution and pooling operations used on images. Meanwhile, to build our SFCN architecture in the original image segmentation fully convolutional network (FCN) architecture, we also design and implement a generating operation} with bridging function. This ensures that the convolution and pooling operation we have designed can be successfully applied in the original FCN architecture. In this paper, we also present a new shape segmentation approach based on SFCN. Furthermore, we allow more general and challenging input, such as mixed datasets of different categories of shapes} which can prove the ability of our generalisation. In our approach, SFCNs are trained triangles-to-triangles by using three low-level geometric features as input. Finally, the feature voting-based multi-label graph cuts is adopted to optimise the segmentation results obtained by SFCN prediction. The experiment results show that our method can effectively learn and predict mixed shape datasets of either similar or different characteristics, and achieve excellent segmentation results.
In this paper I present several novel, efficient, algorithmic techniques for solving some multidimensional geometric data management and analysis problems. The techniques are based on several data structures from computational geometry (e.g. segment tree and range tree) and on the well-known sweep-line method.
We describe a new data structure for dynamic nearest neighbor queries in the plane with respect to a general family of distance functions. These include $L_p$-norms and additively weighted Euclidean distances. Our data structure supports general (convex, pairwise disjoint) sites that have constant description complexity (e.g., points, line segments, disks, etc.). Our structure uses $O(n log^3 n)$ storage, and requires polylogarithmic update and query time, improving an earlier data structure of Agarwal, Efrat and Sharir that required $O(n^varepsilon)$ time for an update and $O(log n)$ time for a query [SICOMP, 1999]. Our data structure has numerous applications. In all of them, it gives faster algorithms, typically reducing an $O(n^varepsilon)$ factor in the previous bounds to polylogarithmic. In addition, we give here two new applications: an efficient construction of a spanner in a disk intersection graph, and a data structure for efficient connectivity queries in a dynamic disk graph.