Morphing Planar Graph Drawings Optimally

We provide an algorithm for computing a planar morph between any two planar straight-line drawings of any $n$-vertex plane graph in $O(n)$ morphing steps, thus improving upon the previously best known $O(n^2)$ upper bound. Further, we prove that our algorithm is optimal, that is, we show that there exist two planar straight-line drawings $\Gamma_s$ and $\Gamma_t$ of an $n$-vertex plane graph $G$ such that any planar morph between $\Gamma_s$ and $\Gamma_t$ requires $\Omega(n)$ morphing steps

Fast exact and approximate geodesics on meshes

The computation of geodesic paths and distances on triangle meshes is a common operation in many computer graphics applications. We present several practical algorithms for computing such geodesics from a source point to one or all other points efficiently. First, we describe an implementation of the exact "single source, all destination" algorithm presented by Mitchell, Mount, and Papadimitriou (MMP). We show that the algorithm runs much faster in practice than suggested by worst case analysis. Next, we extend the algorithm with a merging operation to obtain computationally efficient and accurate approximations with bounded error. Finally, to compute the shortest path between two given points, we use a lower-bound property of our approximate geodesic algorithm to efficiently prune the frontier of the MMP algorithm. thereby obtaining an exact solution even more quickly.Engineering and Applied Science

Fast and Memory-Efficient Voronoi Diagram Construction on Triangle Meshes

© 2017 The Author(s) Computer Graphics Forum © 2017 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd. Geodesic based Voronoi diagrams play an important role in many applications of computer graphics. Constructing such Voronoi diagrams usually resorts to exact geodesics. However, exact geodesic computation always consumes lots of time and memory, which has become the bottleneck of constructing geodesic based Voronoi diagrams. In this paper, we propose the window-VTP algorithm, which can effectively reduce redundant computation and save memory. As a result, constructing Voronoi diagrams using the proposed window-VTP algorithm runs 3–8 times faster than Liu et al.'s method [LCT11] , 1.2 times faster than its FWP-MMP variant and more importantly uses 10–70 times less memory than both of them

The compensation of Gaussian curvature in developable cones is local

In this paper we use the angular deficit scheme [V. Borrelli, F. Cazals, and J.-M. Morvan, {\sl Computer Aided Geometric Design} {\bf 20}, 319 (2003)] to determine the distribution of Gaussian curvature in developable cones (d-cones) [E. Cerda, S. Chaieb, F. Melo, and L. Mahadevan, {\sl Nature} {\bf 401}, 46 (1999)] numerically. These d-cones are formed by pushing a thin elastic sheet into a circular container. Negative Gaussian curvatures are identified at the rim where the sheet touches the container. Around the rim there are two narrow bands with positive Gaussian curvatures. The integral of the (negative) Gaussian curvature near the rim is almost completely compensated by that of the two adjacent bands. This suggests that the Gauss-Bonnet theorem which constrains the integral of Gaussian curvature globally does not explain the spontaneous curvature cancellation phenomenon [T. Liang and T. A. Witten, {\sl Phys. Rev. E} {\bf 73}, 046604 (2006)]. The locality of the compensation seems to increase for decreasing d-cone thickness. The angular deficit scheme also provides a new way to confirm the curvature cancellation phenomenon.Comment: 12 pages; 5 figure

Essential techniques for laparoscopic surgery simulation

Laparoscopic surgery is a complex minimum invasive operation that requires long learning curve for the new trainees to have adequate experience to become a qualified surgeon. With the development of virtual reality technology, virtual reality-based surgery simulation is playing an increasingly important role in the surgery training. The simulation of laparoscopic surgery is challenging because it involves large non-linear soft tissue deformation, frequent surgical tool interaction and complex anatomical environment. Current researches mostly focus on very specific topics (such as deformation and collision detection) rather than a consistent and efficient framework. The direct use of the existing methods cannot achieve high visual/haptic quality and a satisfactory refreshing rate at the same time, especially for complex surgery simulation. In this paper, we proposed a set of tailored key technologies for laparoscopic surgery simulation, ranging from the simulation of soft tissues with different properties, to the interactions between surgical tools and soft tissues to the rendering of complex anatomical environment. Compared with the current methods, our tailored algorithms aimed at improving the performance from accuracy, stability and efficiency perspectives. We also abstract and design a set of intuitive parameters that can provide developers with high flexibility to develop their own simulators

Morphing Schnyder drawings of planar triangulations

We consider the problem of morphing between two planar drawings of the same triangulated graph, maintaining straight-line planarity. A paper in SODA 2013 gave a morph that consists of $O(n^2)$ steps where each step is a linear morph that moves each of the $n$ vertices in a straight line at uniform speed. However, their method imitates edge contractions so the grid size of the intermediate drawings is not bounded and the morphs are not good for visualization purposes. Using Schnyder embeddings, we are able to morph in $O(n^2)$ linear morphing steps and improve the grid size to $O(n)\times O(n)$ for a significant class of drawings of triangulations, namely the class of weighted Schnyder drawings. The morphs are visually attractive. Our method involves implementing the basic "flip" operations of Schnyder woods as linear morphs.Comment: 23 pages, 8 figure

Curvature-adapted Remeshing of CAD Surfaces

A common representation of surfaces with complicated topology and geometry is through composite parametric surfaces as is the case for most CAD modelers. A challenging problem is how to generate a mesh of such a surface that well approximates the geometry of the surface, preserves its topology and important geometric features, and contains nicely shaped elements. In this work, we present an optimization-based surface remeshing method that is able to satisfy many of these requirements simultaneously. This method is inspired by the recent work of Levy \ub4 and Bonneel (Proc. 21th International Meshing Roundtable, October 2012), which embeds a smooth surface into a high-dimensional space and remesh it uniformly in that embedding space. Our method works directly in the 3d spaces and uses an embedding space in R6 to evaluate mesh size and mesh quality. It generates a curvatureadapted anisotropic surface mesh that well represents the geometry of the surface with a low number of elements. We illustrate our approach through various examples

INTRINSIC CURVATURE: A MARKER OF MILLIMETER-SCALE TANGENTIAL CORTICO-CORTICAL CONNECTIVITY?

In this paper, we draw a link between cortical intrinsic curvature and the distributions of tangential connection lengths. We suggest that differential rates of surface expansion not only lead to intrinsic curvature of the cortical sheet, but also to differential inter-neuronal spacing. We propose that there follows a consequential change in the profile of neuronal connections: specifically an enhancement of the tendency towards proportionately more short connections. Thus, the degree of cortical intrinsic curvature may have implications for short-range connectivity