39-Issue 2

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https://diglib.eg.org/handle/10.2312/2632881

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Browsing 39-Issue 2 by Subject "Computational geometry"

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    Invertible Paradoxic Loop Structures for Transformable Design
    (The Eurographics Association and John Wiley & Sons Ltd., 2020) Li, Zijia; Nawratil, Georg; Rist, Florian; Hensel, Michael; Panozzo, Daniele and Assarsson, Ulf
    We present an interactive tool compatible with existing software (Rhino/Grasshopper) to design ring structures with a paradoxic mobility, which are self-collision-free over the complete motion cycle. Our computational approach allows non-expert users to create these invertible paradoxic loops with six rotational joints by providing several interactions that facilitate design exploration. In a first step, a rational cubic motion is shaped either by means of a four pose interpolation procedure or a motion evolution algorithm. By using the representation of spatial displacements in terms of dual-quaternions, the associated motion polynomial of the resulting motion can be factored in several ways, each corresponding to a composition of three rotations. By combining two suitable factorizations, an arrangement of six rotary axes is achieved, which possesses a 1-parametric mobility. In the next step, these axes are connected by links in a way that the resulting linkage is collision-free over the complete motion cycle. Based on an algorithmic solution for this problem, collision-free design spaces of the individual links are generated in a post-processing step. The functionality of the developed design tool is demonstrated in the context of an architectural and artistic application studied in a master-level studio course. Two results of the performed design experiments were fabricated by the use of computer-controlled machines to achieve the necessary accuracy ensuring the mobility of the models.
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    Polygon Laplacian Made Simple
    (The Eurographics Association and John Wiley & Sons Ltd., 2020) Bunge, Astrid; Herholz, Philipp; Kazhdan, Misha; Botsch, Mario; Panozzo, Daniele and Assarsson, Ulf
    The discrete Laplace-Beltrami operator for surface meshes is a fundamental building block for many (if not most) geometry processing algorithms. While Laplacians on triangle meshes have been researched intensively, yielding the cotangent discretization as the de-facto standard, the case of general polygon meshes has received much less attention. We present a discretization of the Laplace operator which is consistent with its expression as the composition of divergence and gradient operators, and is applicable to general polygon meshes, including meshes with non-convex, and even non-planar, faces. By virtually inserting a carefully placed point we implicitly refine each polygon into a triangle fan, but then hide the refinement within the matrix assembly. The resulting operator generalizes the cotangent Laplacian, inherits its advantages, and is empirically shown to be on par or even better than the recent polygon Laplacian of Alexa and Wardetzky [AW11] - while being simpler to compute.