Polyhedral Model

1.Introduction The following exercise is based on experiments conducted in circular Origami. This type of paper folding allows for a completely different geometry than the square type since it lends itself very easily to the creation of shapes based on 30-60-90 degree angles. This allows for experimentation with shapes made up of equilateral triangles such as deltahedra. The results of this research were used in Annenberg sponsored activities conducted in a progressive middle school in Houston TX, as well as a workshop presented at the 1999 Bridges Conference in Winfield, KS. Not including preparatory and follow up work by the teacher, the activities in Houston were composed of two main parts, the collaborative construction of a three-yard-across, eighty-faced regular deltahedron (the EndoPentakis Icosi-dodecahedron) and the following exercise. The barn-raising was presented last year in Winfield, and the paper folding is the topic of this paper. Circular Origami introduces a whole ...

The article addresses the genesis and visualization of the capstone image to Kepler's polyhedral hypothesis of the planetary intervals from his first major work, Mysterium Cosmographicum (1596). The contention is that the famous Tabula III was directed less by Kepler than it was an initiative spearheaded by Georg Gruppenbach, the printer of Mysterium, and Kepler's mentor Michael Mäistlin, who sought to produce a marketable broadsheet that would appeal to the contemporary German fashion for illustrations of polyhedral geometry. More generally, the article seeks to redefine the key role played by the printing workshop and the decorative arts in the theory's inception and ultimate graphic manifestation.

A long running program often spends most of its time in nested loops. The polyhedral model pro-vides powerful abstractions to optimize loop nests with regular accesses for parallel execution. Affine transformations in this model capture a complex sequence of execution-reordering loop transforma-tions that improve performance by parallelization as well as better locality. Although a significant amount of research has addressed affine scheduling and partitioning, the problem of automatically finding good affine transforms for communication-optimized coarse-grained parallelization along with locality optimization for the general case of arbitrarily-nested loop sequences remains a challenging problem -most frameworks do not treat parallelization and locality optimization in an integrated manner, and/or do not optimize across a sequence of producer-consumer loops. In this paper, we develop an approach to communication minimization and locality optimization in tiling of arbitrarily nested...

We present a system for rendering novel viewpoints from a set of calibrated and silhouette-segmented images using the visual hull together with multi-view stereo. The visual hull predicted from the object silhouettes is used to restrict the search range of the multi-view stereo. This reduces redundant computation and the possibility of incorrect matches. Unlike previous visual hull approaches, we do

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A long-standing open problem in the study of tight surfaces centered around a question posed by Nicolaas Kuiper asking whether the surface with Euler characteristic −1 (a real projective plane with one handle) could be tightly immersed in three-space [8]. Kuiper had established that all other surfaces

We introduce a novel "sensation preserving" simplification algorithm for faster collision queries between two polyhedral objects in haptic rendering. Given a polyhedral model, we construct a multiresolution hierarchy using " filtered edge collapse", subject to constraints imposed by collision detection. The resulting hierarchy is then used to compute fast contact response for haptic display. The computation model is inspired by human tactual perception of contact information. We have successfully applied and demonstrated the algorithm on a time-critical collision query framework for haptically displaying complex object-object interaction. Compared to existing exact contact query algorithms, we observe noticeable performance improvement in update rates with little degradation in the haptic perception of contacts.

We aim to reconstruct three-dimensional polyhedra from axonometric line drawings. Existence of mirror symmetry in polyhedra can assist the reconstruction process. We present a new approach for determining planes of mirror symmetry of such polyhedral objects based on prior detection of their planar faces and any axes of symmetry of these faces. The axes are obtained from skewed facial symmetries, for which we also give a new method of determination.

We present a unified mathematical framework for analyzing the tradeoffs between parallelism and storage allocation within a parallelizing compiler. Using this framework, we show how to find a good storage mapping for a given schedule, a good schedule for a given storage mapping, and a good storage mapping that is valid for all legal (one-dimensional affine) schedules. We consider storage mappings that collapse one dimension of a multidimensional array, and programs that are in a single assignment form and accept a one-dimensional affine schedule. Our method combines affine scheduling techniques with occupancy vector analysis and incorporates general affine dependences across statements and loop nests. We formulate the constraints imposed by the data dependences and storage mappings as a set of linear inequalities, and apply numerical programming techniques to solve for the shortest occupancy vector. We consider our method to be a first step towards automating a procedure that finds ...

The polyhedral model provides powerful abstractions to optimize loop nests with regular accesses. Affine transformations in this model capture a complex sequence of execution-reordering loop transformations that can improve performance by parallelization as well as locality enhancement. Although a significant body of research has addressed affine scheduling and partitioning, the problem of automatically finding good affine transforms for communication-optimized coarse-grained parallelization together with locality optimization for the general case of arbitrarily-nested loop sequences remains a challenging problem. We propose an automatic transformation framework to optimize arbitrarily-nested loop sequences with affine dependences for parallelism and locality simultaneously. The approach finds good tiling hyperplanes by embedding a powerful and versatile cost function into an Integer Linear Programming formulation. These tiling hyperplanes are used for communication-minimized coarse-grained parallelization as well as for locality optimization. The approach enables the minimization of inter-tile communication volume in the processor space, and minimization of reuse distances for local execution at each node. Programs requiring one-dimensional versus multi-dimensional time schedules (with scheduling-based approaches) are all handled with the same algorithm. Synchronization-free parallelism, permutable loops or pipelined parallelism at various levels can be detected. Preliminary studies of the framework show promising results.

The following exercise is based on experiments conducted in circular Origami. This type of paper folding allows for a completely
different geometry than the square type since it lends itself very easily to the creation of shapes based on 30-60-90 degree angles. This allows for experimentation with shapes made up of equilateral triangles such as deltahedra. The results of this research were used in Annenberg sponsored activities conducted in a progressive middle school in Houston TX, as well as a workshop presented at the 1999 Bridges Conference in Winfield, KS. Not including preparatory and follow up work by the teacher, the activities in Houston were composed of two main parts, the collaborative construction of a three-yard-across, eighty-faced regular deltahedron (the Endo-Pentakis Icosi-dodecahedron) and the following exercise. The barn-raising was presented last year in Winfield, and the paper folding is the topic of this paper.

We present a novel and fast algorithm to compute penetration depth (PD) between two polyhedral models for physically-based animation. Given two overlapping polyhedra, it computes the minimal translation distance to separate them using a combination of object-space and image-space techniques. The algorithm computes pairwise Minkowski sums of decomposed convex pieces and performs a closest point query using rasterization hardware. It uses bounding volume hierarchies, object-space and image-space culling ...

The recent discovery that there is a tight polyhedral immersion of the projective plane with one handle, while there is no smooth tight immersion of the same surface, provides a rare example in low dimensions of a signicant dierence between smooth and polyhedral surfaces. In this paper the author shows that the obstruction to smoothing the polyhedral model is not local in nature, and describes some of the ways in which the proof of the nonexistence of the smooth tight surface does not carry over to the polyhedral case.