In this paper, we tackle the problem of tiling a domain with a set of deformable templates. A valid solution to this problem completely covers the domain with templates such that the templates do not overlap. We generalize existing specialized solutions and formulate a general layout problem by modeling important constraints and admissible template deformations. Our main idea is to break the layout algorithm into two steps: a discrete step to lay out the approximate template positions and a continuous step to refine the template shapes. Our approach is suitable for a large class of applications, including floorplans, urban layouts, and arts and design.

We present a framework for generating street networks and parcel layouts. Our goal is the generation of high-quality layouts that can be used for urban planning and virtual environments. We propose a solution based on hierarchical domain splitting using two splitting types: streamline-based splitting, which splits a region along one or multiple streamlines of a cross field, and template-based splitting, which warps pre-designed templates to a region and uses the interior geometry of the template as the splitting lines. We combine these two splitting approaches into a hierarchical framework, providing automatic and interactive tools to explore the design space.

We present a general computational framework to locally characterize any shape space of meshes implicitly prescribed by a collection of non-linear constraints. We computationally access such manifolds, typically of high dimension and co-dimension, through first and second order approximants, namely tangent spaces and quadratically parameterized osculant surfaces. Exploration and navigation of desirable subspaces of the shape space with regard to application specific quality measures are enabled using approximants that are intrinsic to the underlying manifold and directly computable in the parameter space of the osculant surface. We demonstrate our framework on shape spaces of planar quad (PQ) meshes, where each mesh face is constrained to be (nearly) planar, and circular meshes, where each face has a circumcircle. We evaluate our framework for navigation and design exploration on a variety of inputs, while keeping context specific properties such as fairness, proximity to a reference surface, etc.

Automatic reconstruction of 3D objects from 2D orthographic views has been a major research issue in CAD/CAM. In this paper, two accelerating techniques to improve the eÆciency of reconstruction are presented. First, some pseudo elements are removed by depth and topology information as soon as the wire-frame is constructed, which reduces the searching space. Second, the proposed algorithm does not establish all possible surfaces in the process of generating 3D faces. The surfaces and edge loops are generated by using the relationship between the boundaries of 3D faces and their projections. This avoids the growth in combinational complexity of previous methods that have to check all possible pairs of 3D candidate edges.

Solving geometric constraint systems in 3-D is much more complicated than that in 2-D because the number of variables is larger and some of the results valid in 2-D cannot be extended for 3-D. In this paper, we propose a new DOF-based graph constructive method to geometric constraint systems solving that can efficiently handle well-, over- and under-constrained systems based on the dependence analysis. The basic idea is that the solutions of some geometric elements depend on some others because of the constraints between them. If some geometric elements depend on each other, they must be solved together. In our approach, we first identify all structurally redundant constraints, then we add some constraints to well constrain the system. And we prove that the order of a constraint system after processing under-constrained cases is not more than that of the original system multiplied by 5. After that, we apply a recursive searching process to identify all the clusters, which is shown to be capable of getting the minimum order-reduction result of a well-constrained system. We also briefly describe the constraint evaluation phase and show the implementation results of our method.