Spherical Polygons Research Articles

SU‐E‐I‐159: Delaunay Triangulation for Angular Interpolation and Single Stage Reconstruction of EPRI

Purpose: To employ Delaunay Triangulation of projection orientations in 3D electron paramagnetic resonance imaging (EPRI) for multidimensional angular interpolation and the duel tessellation (Voronoi diagram) for weighting coefficients enabling reconstruction by single stage inverse Radon Transform without the necessity of regridding data. Methods: Standard 3D pulsed EPR oxygen images were acquired of an OX063 phantom with projections in an approximately uniform but none rectilinear fashion. The projection orientation was mapped to a unit sphere, and the Delaunay triangulation was calculated for these points on the surface. Projections were interpolated to the incenter of each triangulation using a weighted mean of projections located at the vertices. The Voronoi diagram was then calculated, partitioning the surface of the sphere such that each projection was associated with a spherical polygon. The area of these polygons was used as a weighting coefficient for calculation of the discrete surface integral in the filtered backprojection (FBP) algorithm. Results: We were able to successfully reconstruct images using angular interpolation and the FBP algorithm without decomposition of the problem into stages of 1D interpolation and line integrals. Examination of images indicates improved ability to resolve oxygen concentration of phantoms Conclusions: Direct single stage reconstruction from non rectilinear sampling is an important progression in ERPI as it avoids the filtering effects of interpolation in regridding and multistage reconstruction. Proper analysis of the angular distributions of projections will allow for unrestricted sampling choices and potential optimization. Since sampling requirements of the discrete inverse Radon transform scale rapidly with higher dimension, angular projection interpolation is an important technique for artifact avoidance that must be reproduced in the move from single to multistage reconstruction. This work will form the basis for future investigation into the reconstruction of EPRI.

Searchable Sky Coverage of Astronomical Observations: Footprints and Exposures

Sky coverage is one of the most important pieces of information about astronomical observations. We discuss possible representations and present algorithms to create and manipulate shapes consisting of generalized spherical polygons with arbitrary complexity and size on the celestial sphere. This shape specification integrates well with our Hierarchical Triangular Mesh indexing toolbox, whose performance and capabilities are enhanced by the novel advanced features presented here. Our portable implementation of the relevant spherical geometry routines comes with wrapper functions for database queries, which are currently being used within several scientific catalog archives, including that of the Sloan Digital Sky Survey, the Galaxy Evolution Explorer, the UKIRT Infrared Deep Sky Survey, SuperCOSMOS, VISTA, Hubble Legacy Archive, and the Footprint Service of the Virtual Observatory.

Flattening single-vertex origami: The non-expansive case

A single-vertex origami is a piece of paper with straight-line rays called creases emanating from a fold vertex placed in its interior or on its boundary. The Single-Vertex Origami Flattening problem asks whether it is always possible to reconfigure the creased paper from any configuration compatible with the metric, to a flat, non-overlapping position, in such a way that the paper is not torn, stretched and, for rigid origami, not bent anywhere except along the given creases. Streinu and Whiteley showed how to reduce the problem to the carpenter's rule problem for spherical polygons. Using spherical expansive motions, they solved the cases of open < π and closed ⩽ 2 π spherical polygons. Here, we solve the case of open polygons with total length between [ π , 2 π ) , which requires non-expansive motions. Our motion planning algorithm works in a finite number of discrete steps, for which we give precise bounds depending on both the number of links and the angle deficit.

Some notes on maximal arc intersection of spherical polygons: its $\mathcal{NP}$ -hardness and approximation algorithms

Finding a sequence of workpiece orientations such that the number of setups is minimized is an important optimization problem in manufacturing industry. In this paper we present some interesting notes on this optimal workpiece setup problem. These notes show that (1) The greedy algorithm proposed in Comput. Aided Des. 35 (2003), pp. 1269–1285 for the optimal workpiece setup problem has the performance ratio bounded by O(ln n−ln ln n+0.78), where n is the number of spherical polygons in the ground set; (2) In addition to greedy heuristic, linear programming can also be used as a near-optimal approximation solution; (3) The performance ratio by linear programming is shown to be tighter than that of greedy heuristic in some cases.

Automated scanning of dental impressions

Presented in this paper is an automated scanning procedure of a dental impression that utilizes an SLS (Structured Light System). The proposed procedure identifies missing areas from the initial scanning data from default orientations and attempts to locate additional scanning orientations to fill the missing areas. The core part of the proposed procedure is an algorithm that computes additional scanning orientations using the information of the missing areas. The algorithm was designed carefully by considering three major technological requirements of the problem: dual visibility, reliability, and efficiency. To satisfy the dual visibility requirement, the proposed algorithm uses the concept of a visibility map as well as the diameter of a spherical polygon. Once dual visibility is satisfied, the proposed algorithm attempts to locate the optimal scanning orientation to maximize the reliability (accuracy) of the scanning data by finding the centroid of a convex spherical polygon. As the proposed procedure is based on well-known 2D geometric algorithms, it is very efficient. The applications of the proposed procedure include not only dental impressions but also other small objects that do not have CAD models.

Computational geometry column 51

Can a simple spherical polygon always be triangulated? The answer depends on the definitions of "polygon" and "triangulate". Define a spherical polygon to be a simple, closed curve on a sphere S composed of a finite number of great circle arcs (also known as geodesic arcs) meeting at vertices. Can every spherical polygon be triangulated? Figure 1 shows an example of what is intended. 1 The planar analog is well-known and a cornerstone of computational geometry: the interior of every planar simple polygon can be triangulated (and efficiently so). The situation for spherical polygons is not so straightforward. There are three complications.

Methods for rapidly processing angular masks of next-generation galaxy surveys

As galaxy surveys become larger and more complex, keeping track of the completeness, magnitude limit and other survey parameters as a function of direction on the sky becomes an increasingly challenging computational task. For example, typical angular masks of the Sloan Digital Sky Survey contain about N= 300 000 distinct spherical polygons. Managing masks with such large numbers of polygons becomes intractably slow, particularly for tasks that run in time with a naive algorithm, such as finding which polygons overlap each other. Here we present a ‘divide-and-conquer’ solution to this challenge: we first split the angular mask into pre-defined regions called ‘pixels’, such that each polygon is in only one pixel, and then perform further computations, such as checking for overlap, on the polygons within each pixel separately. This reduces tasks to ⁠, and also reduces the important task of determining in which polygon(s) a point on the sky lies from to ⁠, resulting in significant computational speedup. Additionally, we present a method to efficiently convert any angular mask to and from the popular healpix format. This method can be generically applied to convert to and from any desired spherical pixelization. We have implemented these techniques in a new version of the mangle software package, which is freely available at http://space.mit.edu/home/tegmark/mangle/, along with complete documentation and example applications. These new methods should prove quite useful to the astronomical community, and since mangle is a generic tool for managing angular masks on a sphere, it has the potential to benefit terrestrial mapmaking applications as well.

Poles of inaccessibility: A calculation algorithm for the remotest places on earth

An algorithm is presented to calculate the point on the surface of a sphere maximising the great-circle distance to a given spherical polygon. This is used to calculate the spots furthest from the sea in major land masses, also known as Poles of Inaccessibility (PIA), a concept that has raised the interest of explorers. For the Eurasian pole of inaccessibility (EPIA), the results reveal a misplacement in previous calculations ranging from 156 to 435 km. Although in general there is only one pole for a given coastline, the present calculations show that, within the error inherent to the definition of the coastline, two locations are candidates for EPIA, one equidistant from Gulf of Ob, Gulf of Bengal and Arabian Sea, and the other equidistant from Gulf of Ob, Gulf of Bengal and Gulf of Bohai, both poles being located in the north westernmost Chinese province of Xinjiang. The distance to the sea at these locations is 2510 and 2514 km respectively, about 120 km closer than generally thought.

板材ユニットを用いた木造ドームの構法開発

The design and composition methods for wooden dome are described. Wooden spheric dome is transformed into spherical polygon in order to put standard sized lumbers to use. The surface of spherical polygon is replaced with a combination of units, which are consisted of lumbers and shaped six-pointed star. Each unit has wedges, which keep the adhesive angle among units, on its angular points. And the wooden dome is composed of units and wedges by adhesives. The coefficient of construction availability is devised for checking up on measurements of lumbers and rise-to-span ratio of wooden dome in the range of construction availability. It was found that the structural performance of wooden dome is not deteriorated with constructional convenience in the coefficient of construction availability, from 0 to 0.9.

Recursive mode star identification algorithms

Star identification can be accomplished by several different available algorithms that identify the stars observed by a star tracker. However, efficiency and reliability remain key issues and the availability of new active pixel cameras requires new approaches. Two novel algorithms for recursive mode star identification are presented here. The first approach is derived by the spherical polygon search (SP-search) algorithm, it was used to access all the cataloged stars observed by the sensor field-of-view (FOV) and recursively add/remove candidate cataloged stars according to the predicted image motion induced by camera attitude dynamics. Star identification is then accomplished by a star pattern matching technique which identifies the observed stars in the reference catalog. The second method uses star neighborhood information and a catalog neighborhood pointer matrix to access the star catalog. In the recursive star identification process, and under the assumption of "slow" attitude dynamics, only the stars in the neighborhood of previously identified stars are considered for star identification in the succeeding frames. Numerical tests are performed to validate the absolute and relative efficiency of the proposed methods.

An efficient solid angle sampling technique using bounding volume approach

Solid angle sampling is an important variance‐reduced technique when solving global illumination problems by Monte Carlo methods. In this paper, we present an efficient solid angle sampling technique where a paraboloidal luminaire is taken as a case study. The efficiency of our technique is due to the fact that we employ a tight bounding volume to approximate the solid angle. Our technique includes three processes. The construction process builds a bounding volume. It is a convex, frustum‐like polyhedron with a compromise between the tightness and the vertex numbers. The projection process approximates the solid angle as a convex spherical polygon on a unit hemisphere. Finally, the triangulation process triangulates the convex spherical polygon into spherical triangles for stratified sampling. We analyzed our technique in Monte Carlo direct lighting and Monte Carlo path tracing rendering algorithms. The results show that our technique provides up to 90% sampling efficiency. The significance of the proposed technique is that solid angle sampling, from being not possible for the paraboloidal luminaire, is now feasible. In addition, this technique is efficient for sampling, and it is also applicable to other types of luminaires, such as cylindrical and conic luminaires.

A Simple Derivation of the Gauss-Bonet Theorem

A simple derivation of the Gauss-Bonet theorem is presented based on the representation of spherical polygons by Euler angles and Rodrigues transposition theorem. This leads to a derivation of the theorem which avoids completely the explicit evaluation of rotation matrices.

On the symplectic volume of the moduli space of spherical and Euclidean polygons

In this paper, we study the symplectic volume of the moduli space of polygons by using Witten's formula. We propose to use this volume as a measure for the flexibility of a polygon with fixed side-lengths. The main result of our is that among all the polygons with fixed perimeter in S3 or E3 the regular one is the most flexible and that among all the spherical polygons the regular one with side-length π/2 is the most flexible.

Approximating centroids for the maximum intersection of spherical polygons

This paper considers the problem of investigating the spherical regions owned by the maximum number of spherical polygons. We present a practical O( n( v+ I)) time algorithm for finding the approximating centroids for the maximum intersection of spherical polygons, where n, v, and I are, respectively, the numbers of polygons, all vertices, and intersection points. In order to elude topological errors and handle geometric degeneracies, our algorithm takes the approach of edge-based partitioning of the sphere. Furthermore, the numerical complexity is avoided since the algorithm is completely spherical.

Tessellating the Sphere with Regular Polygons

Spherical geometry can be used to entice students into a deeper understanding of Euclidean geometry. Determining which regular spherical polygons tessellate the sphere is another motivating topic that is accessible to high school students. The most recognizable tessellations of the sphere are found on balls, such as soccer balls, volleyballs, and golf balls. Even Spaceship Earth at Epcot Center in Disney World involves tessellation.

Approximating Centroids for the Maximum Intersection of Spherical Polygons

AbstractThis paper considers the problem of investigating the spherical regions owned by the maximum number of spherical polygons. We present a practical O(n(v+I)) time algorithm for finding the approximating centroids for the maximum intersection of spherical polygons, where n, v, and I are, respectively, the numbers of polygons, all vertices, and intersection points. In order to elude topological errors and handle geometric degeneracies, our algorithm takes the approach of edge-based partitioning of the sphere. Furthermore, the numerical complexity is avoided since the algorithm is completely spherical.

Topological phase in classical mechanics

The historical development of the concept of the topological phase in classical mechanics from the mid-19th century to the present is discussed. There are three stages to be recognized in this period. The first, the mid-19th century stage, is concerned with studying the kinematics of rigid body rotation and includes such milestone developments as the Euler theorem on finite rotation of rigids, Gauss formula for the sum excess of the angles of a spherical polygon, Rodrigues's proof of the noncommutativity property of two finite rotations, and, finally, Hamilton's Lectures on Quaternions where the solid angle theorem is formulated and proved. The experimental discovery of the nonholonomic error of gyroscopes and its exhaustive explanation by A Yu Ishlinskii represent the second stage. The third stage, which started in the 1980s, has witnessed the rediscovery of the nonholonomic effect in the framework of Hamiltonian formalism and is dominated by the study of how the topological phase — or an additional angle — forms in a mechanical system being treated in action–angle variables.

Maximal intersection of spherical polygons by an arc with applications to 4-axis machining

Many geometric optimization problems in CAD/CAM can be reduced to a maximal intersection problem on the sphere: given a set of N simple spherical polygons on the unit sphere and a real number constant L≤2π, find an arc of length L on the unit sphere that intersects as many spherical polygons as possible. Past results can only solve this maximization problem for two very restricted special cases: the arc must be either a great circle or a semi-great circle. In this paper, a simple and deterministic algorithm based on domain partitioning is presented for solving this maximal arc intersection problem in the general case when the number L is arbitrary. The algorithm is made possible by reducing the domain of the arcs to a continuous sub-space in R 2 and then establishing a quotient space partitioning in this sub-space based on a congruence relation. The number of the constituting congruent sub-regions in this quotient space partitioning is shown to have an upper-bound O( E 3), where E is the total number of edges on the polygons. The proposed algorithm has a worst-case upper bound O( ME) on its running time, where M is an output-sensitive number and is bounded by O( E 3). Examples including two realistic tests for 4-axis NC machining are presented.

Reducing feasible contacts between polyhedral models to red–blue intersections on the sphere

Orientation-related problems in geometric design can be naturally expressed on the spherical surface S 2. A wide subset of such problems can be solved directly on the sphere by adapting well-known planar data structures and algorithms. This paper shows that the detection of feasible contacts between two translating polyhedral models can be formulated as a problem of this type. First, a dual spherical representation of polyhedra is introduced, which reduces the contact detection above to finding intersections between two sets of spherical polygons. Next, the red–blue blocks plane sweep algorithm is adapted to obtain both edge intersections and point-in-polygon inclusions in the spherical setting. An experimental comparison of this algorithm against a naı̈ve one shows an increasing advantage of the former as the complexity of the setting grows. The obtained edge–edge and vertex–face polyhedral contacts provide the relevant feature pairs to be tested for interference, leading to considerable savings in collision detection between polyhedral models, as shown in the experimental test performed.

Fast and Easy Reach-Cone Joint Limits

We describe a simple and fast method of creating and using joint sinus reach cones to limit the range of motion on universal and ball-and-socket joints. Reach cones can be created interactively or based on results from biomechanics. A reach cone is specified as a spherical polygon on the surface of a reach sphere that defines all positions the longitudinal segment axis beyond the joint can take. Reach polygons can be concave or convex, and cover more than one hemisphere. Longitudinal axis rotation limits can also be defined for locations in the reach cone. Calculations to determine whether a segment is within the reach cone are based on the relation of the segment longitudinal axis to the planes defining the reach cone and are done in the local coordinate space of the segment. The calculations have no discernible effect on the speed of interactive display or animation.