Polygonal Schema Research Articles

Topology for Substrate Routing in Semiconductor Package Design

In this work, we propose a new signal routing method for solving routing problems that occur in the design process of semiconductor package substrates. Our work uses a topological transformation of the layers of the package substrate in order to simplify the routing problem into a problem of connecting points on a circle with non-intersecting straight line segments. The circle, which we call the Circular Frame, is a polygonal schema, which is originally used in topology to study the topological structure of 2-manifolds. We show through experiments that our new routing method based on the Circular Frame competes with certain grid-based routing algorithms.

Scalable Mesh Refinement for Canonical Polygonal Schemas of Extremely High Genus Shapes.

Any closed manifold of genus g can be cut open to form a topological disk and then mapped to a regular polygon with 4g sides. This construction is called the canonical polygonal schema of the manifold, and is a key ingredient for many applications in graphics and engineering, where a parameterization between two shapes with same topology is often needed. The sides of the 4g-gon define on the manifold a system of loops, which all intersect at a single point and are disjoint elsewhere. Computing a shortest system of loops of this kind is NP-hard. A computationally tractable alternative consists of computing a set of shortest loops that are not fully disjoint in polynomial time using the greedy homotopy basis algorithm proposed by Erickson and Whittlesey and then detach them in post processing via mesh refinement. Despite this operation is conceptually simple, known refinement strategies do not scale well for high genus shapes, triggering a mesh growth that may exceed the amount of memory available in modern computers, leading to failures. In this article we study various local refinement operators to detach cycles in a system of loops, and show that there are important differences between them, both in terms of mesh complexity and preservation of the original surface. We ultimately propose two novel refinement approaches: the former greatly reduces the number of new elements in the mesh, possibly at the cost of a deviation from the input geometry. The latter allows to trade mesh complexity for geometric accuracy, bounding deviation from the input surface. Both strategies are trivial to implement, and experiments confirm that they allow to realize canonical polygonal schemas even for extremely high genus shapes where previous methods fail.

Drawing Graphs in the Plane with a Prescribed Outer Face and Polynomial Area

We study the classic graph drawing problem of drawing a planar graph using straight-line edges with a prescribed convex polygon as the outer face. Unlike previous algorithms for this problem, which may produce drawings with exponential area, our method produces drawings with polynomial area. In addition, we allow for collinear points on the boundary, provided such vertices do not create overlapping edges. Thus, we solve an open problem of Duncan et al., which, when combined with their work, implies that we can produce a planar straight-line drawing of a combinatorially-embedded genus-g graph with the graph's canonical polygonal schema drawn as a convex polygonal external face.

Planar Drawings of Higher-Genus Graphs

In this paper, we give polynomial-time algorithms that can take a graph G with a given combinatorial embedding on an orientable surface S of genus g and produce a planar drawing of G in R^2, with a bounding face defined by a polygonal schema P for S. Our drawings are planar, but they allow for multiple copies of vertices and edges on P's boundary, which is a common way of visualizing higher-genus graphs in the plane. Our drawings can be defined with respect to either a canonical polygonal schema or a polygonal cutset schema, which provides an interesting tradeoff, since canonical schemas have fewer sides, and have a nice topological structure, but they can have many more repeated vertices and edges than general polygonal cutsets. As a side note, we show that it is NP-complete to determine whether a given graph embedded in a genus-g surface has a set of 2g fundamental cycles with vertex-disjoint interiors, which would be desirable from a graph-drawing perspective.

Computing Canonical Polygonal Schemata with Generalized Maps

This paper shows that a well-known algorithm proposed to compute the canonical polygonal schema of a surface can be transferred onto a 2-dimensional generalized map. We show that transformation rules on polygonal schemata can be achieved in O(1) with generalized maps, which can help optimizing existing algorithms.

Families of cut-graphs for bordered meshes with arbitrary genus

Given a triangulated surface M with arbitrary genus, the set of its cut-graphs depends on the underlying topology and the selection of a specific one should be guided by the surface geometry and targeted applications. Most of the previous work on this topic uses mesh traversal techniques for the evaluation of the geodesic metric, and therefore the cut-graphs are influenced by the mesh connectivity. Our solution is to build up the cut-graph on the iso-contours of a function f : M → R , that cut the topological handles of M , and on the completion of the cut-graph on the planar domain. In the planar domain, geodesic curves are defined by line segments whose counterparts on M , with respect to a diffeomorphism ϕ : M → R 2 , are smooth approximations of geodesic paths. Our method defines a family of cut-graphs of M which can target different applications, such as global parameterization with respect to different criteria (e.g., minimal length, minimization of the parameterization distortion, or interpolation of points as required by remeshing and texture mapping) or the calculation of polygonal schemes for surface classification. The proposed approach finds a cut-graph of an arbitrary triangle mesh M with n vertices and b boundary components in O(( b − 1) n) time if M has 0-genus, and O( n(log( n) + 2 g + b − 1)) time if g ⩾ 1. The associated polygonal schema is reduced if g = 0, and it has a constant number of redundant edges otherwise.

Optimal System of Loops on an Orientable Surface

Every compact orientable boundaryless surface M can be cut along simple loops with a common point v0, pairwise disjoint except at v0, so that the resulting surface is a topological disk; such a set of loops is called a {\it system of loops} for M. The resulting disk may be viewed as a polygon in which the sides are pairwise identified on the surface; it is called a polygonal schema. Assuming that M is a combinatorial surface, and that each edge has a given length, we are interested in a shortest (or optimal) system of loops homotopic to a given one, drawn on the vertex-edge graph of M. We prove that each loop of such an optimal system is a shortest loop among all simple loops in its homotopy class. We give an algorithm to build such a system, which has polynomial running time if the lengths of the edges are uniform. As a byproduct, we get an algorithm with the same running time to compute a shortest simple loop homotopic to a given simple loop.

Topological quadrangulations of closed triangulated surfaces using the Reeb graph

Although surfaces are more and more often represented by dense triangulations, it can be useful to convert them to B-spline surface patches, lying on quadrangles. This paper presents a method for the construction of coarse topological quadrangulations of closed triangulated surfaces, based on Morse theory. In order to construct on the surface a quadrangulation of its canonical polygonal schema, we compute first a Reeb graph then a canonical set of generators embedded on the surface. Some results are shown on different surfaces.

A new technique to compute polygonal schema for 2-manifolds with application to null-homotopy detection

We provide a new technique for deriving optimal-sized polygonal schema for triangulated compact 2-manifolds without boundary inO(n) time, wheren is the size of the given triangulationT. We first derive a polygonal schemaP embedded inT using Seifert-Van Kampen's theorem. A reduced polygonal schemaQ of optimal size is computed fromP, where a surjective map from the vertices ofP is retained to the vertices ofQ. This helps detecting null-homotopic (contractible to a point) cycles. Given a cycle of lengthk, we determine if it is null-homotopic inO(n+k logg) time and in θ(n+k) space, whereg is the genus of the given 2-manifold. The actual contraction for a null-homotopic cycle can be computed in θ(nk) time and space. This is a considerable improvement over the previous best-known algorithm for this problem.

Recent Literature on Paramnesia

Paramnesia, whether of the simple or reduplicative variety, has recently received a remarkable impetus, both in studies of the phenomena in normal individuals and its occurrences in various psychopathic states, psychasthenias and mental disorders. Recent text books, however, have given it but scant notice, but the current literature is rich in the reports of cases and the formulation of various theories. It bears a certain analogy to those peculiar losses of the feeling of reality, which are transient alterations in the feeling of recognition (or the so-called of familiarity) or, on the other hand, to the feelings of strangeness, the phenomena of the as opposed to the sense of the called by Janet, psycholeptic crises. The purpose of this brief review is to gather together and summarize the recent literature on the subject. Outside of the strictly scientific contributions on the subject, the late Lafcadio Hearn, in a popular vein, describes the phenomena of paramnesia with a rare philosophic insight. (Kokoro, Hints and Echoes of Japanese Inner Life-Boston, I896.) In discussing Shintoism and Buddhism, he raises the question whether the ego is the concentrated sum of the creative thinking of previous lives, and then adds-To the same psychological category possibly belongs likewise a peculiar feeling which troubled men's minds long before the time of Cicero, and troubles them even more betimes in our own generation,-the feeling of having already seen a place really visited for the first time. Some strange air of familiarity about the streets of a foreign town, or the forms of a foreign landscape, comes to the mind with a sort of soft, weird shock and leaves one vainly ransacking memory for interpretations. Grasset (La Sensation du 'Deja vu.' Journal de Psychologie normale et pathologique, Jan.-Feb., I904), again essays to apply his geometrical polygonal scheme, to an explanation of these complex phenomena. He asserts that there are two equally essential elements: I. The recognition of an image, of an emotion, of a psychic state, that one has the consciousness of never experiencing. 2. The ignorance of the origin of the first impression (visual or auditory image), and emotion previously acquired by the In the second case it is necessary that there be in the psychisme of the subject an image or an impression, which shares in it, which has been deposited or formed unknown to the subject. The patient shows a stupefying anxiety, when he ascertains the presence in his mind of an image or a precise impression, whereof he is unable to say when or how it came into his brain. Perhaps unconsciously or unknown to him, he acquires this psychic recognition, which he is able to later utilize in different conscious intellectual operations, without even recalling either the moment or the circumstances of the acquisition of this psychic recognition. The phenomenon is explained by double psychisme and the necessary separation of psychic centres, either superior or inferior (the polygonal schema); the superior being conscious, while the inferior is unconscious or subconscious. The two centres collaborate in an inextricable manner, but they are also able, in certain psychological or extra-physiological conditions, to disaggregate and functionate alone (disagregation subpolygonale). In these states of subpolygonal disaggregation, the inferior psychic states are able to acquire diverse impressions, without the knowledge of the 550