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Topic: Orientable

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	Orientability - Wikipedia, the free encyclopedia
		Orientability, for surfaces, is easily defined, regardless of whether the surface is embedded in an ambient space or not.
		Note that whether the surface is orientable is independent of triangulation; this fact is not obvious, but a standard exercise.
		Formally, a n-dimensional differentiable manifold is called orientable if it possesses a differential form ω of degree n which is nonzero at every point on the manifold.
	en.wikipedia.org /wiki/Orientability (842 words)

	Orientability article - Orientability geometry topology sphere plane torus Möbius strip real projective - ... (Site not responding. Last check: 2007-10-20)
		In geometry and topology, a surface in $\mathbb{R}^3 is called non-orientable, if a figure such as the letter "R" can be moved about on the surface so that it becomes mirror-reversed.$
		In general, the property of being orientable is not equivalent to being two-sided; however, this holds when the ambient space (such as $\mathbb{R}^3 above) is$ orientable.
		Formally, a $n -dimensional differentiable manifold is called$ orientable if it possesses a differential form $\omega of degree n which is nonzero at every point on the manifold .$
	www.what-means.com /encyclopedia/Oriented (722 words)

	Embedding Graphs on Surfaces
		The orientable (nonorientable) genus of an embedding H~ of a graph H is equal to the minimum g such that a picture of H which respects the clockwise order of H~ can be embedded on an orientable (nonorientable) surface of genus g without edges crossing.
		The orientable (nonorientable) genus of a graph H is equal to the minimum g such that H has an orientable (nonorientable) combinatorial embedding of genus g.
		For an orientable surface with g handles, a simple graph which is embeddable on this surface has at most 3n -6 + 6g edges and for a nonorientable surface with g crosscaps, the maximum is 3n - 6 + 3g.
	www.csr.uvic.ca /~wendym/torus/embedding.html (1986 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		In case you meant this, note that it is also true that if the base space is orientable, the total space of any principal S1 bundle will also be orientable, and here is the proof: at every point in the n-dimensional base space M, we have an n-form \omega that defines the orientation.
		The base space is the 2-torus, which of course is orientable with H2(M) nontrivial, and in the dimension range you want.
		In 3 dimensions, we can let M be any orientable closed manifold other than a rational homology sphere.
	www.lehigh.edu /~dmd1/ki97.txt (605 words)

	[No title]
		The noti* on of orientability* is straightforward and unambiguous.
		We shall define an orientation of an orientable representation (E ; R) to be * a map of representations from it to the "universal orientable* representation" (S R; S* *).
		This implies that the identity functor of UG (n) is itself th* e uni- versal orientable* complex representation.
	hopf.math.purdue.edu /Costenoble-May-Waner/CMWFinal.txt (16893 words)

	A Plethora of Useless Information (Site not responding. Last check: 2007-10-20)
		Orientability is a global property of a surface.
		Surfaces are always locally orientable, making the distinction of local and global orientability silly.
		Otherwise, there are two distinct sides to the surface and the surface is deemed orientable.
	www.eden.rutgers.edu /~alfare/home.htm (1021 words)

	The Surface Normal (Site not responding. Last check: 2007-10-20)
		Intuitively, orientability means that if the initial points of the unit surface normals n are placed on the surface, then their terminal points are all on the same side of the surface.
		If a surface is closed (such as a sphere), then we assume its orientation to be with all normal vectors pointing toward the outside of the surface.
		Similarly, a parametric surface is orientable if n(u,v) varies continuously across the surface and if n(u,v) defines only one surface normal at each point on the surface.
	math.etsu.edu /multicalc/chap3/chap3-6/part4.htm (262 words)

	Glossary: Orientable (Site not responding. Last check: 2007-10-20)
		A surface is orientable if it does not contain any Möbius bands.
		The orientable surfaces are the sphere, the torus, and the tori of higher genus.
		The orientable surfaces all have even Euler Characteristic.
	www.geom.uiuc.edu /docs/research/RP2-handle/Glossary/Orientable.html (71 words)

	Tight Non-Orientable Surfaces
		The non-orientable surfaces are divided into two families, one formed by adding handles to the Klein bottle, the other by adding handles to the real projective plane (just as all the orientable surfaces can be formed by adding handles to a sphere).
		The surfaces based on the Klein bottle have even Euler characteristic, and those based on the projective plane have odd Euler characteristic.
		The middle surface is a torus of genus 2 (a 2-handled torus), but the right-hand surface is non-orientable.
	www.geom.uiuc.edu /docs/dpvc/Tight-non-orientable.html (576 words)

	Robotics Institute: Exact Cellular Decomposition of Closed Orientable Surfaces Embedded in R^3
		For applications such as paint deposition, the effector (the paint atomizer) does not explicitly cover the target surface, but instead covers an offset surface - a surface that is a fixed distance away from the target surface.
		The main contribution of this work is a method to construct unknown offset surfaces using a procedure, also developed in this paper, to detect critical points.
		Rizzi, and E. Acar, "Exact Cellular Decomposition of Closed Orientable Surfaces Embedded in R^3," Proceedings of the 2001 IEEE International Conference on Robotics and Automation (ICRA '01), May, 2001.
	www.ri.cmu.edu /pubs/pub_4455.html (320 words)

	Rooted Maps, Functional Equations and Continued Fractions
		It is proved that every closed connected orientable surface is homeomorphic to one of them.
		What makes it nonorientable is that if a 2× 2 coordinate system specifying a forward direction and a right direction is translated in the forward direction once around the center of the band, then the orientation of the right direction is reversed.
		Two rooted maps with the same genus are isomorphic if there exists an homeomorphism of the associated surface, preserving its orientation, mapping the vertices, edges, faces and the root half-edge on the first map respectively on those of the second one.
	algo.inria.fr /seminars/sem98-99/beraud.html (1200 words)

	[No title]
		AB- A polyhedron on S, a compact 2-manifold, is called a triangulation if each face is a triangle with 3 distinct vertices and the intersection of any two distinct triangles is either empty, a single vertex, or a single edge.
		A routine application of Euler's formula shows that if S is an orientable surface of genus p, then delta (S) >= 4(p - 1)+2((7+{radlin}1+48p))/2).
		AB- Let F be a closed, orientable surface of genus g, and let e(g) be the minimal number of vertices of a triangulation of F by an embedding polyhedron.
	www.math.niu.edu /~rusin/known-math/95/polyhed.min (1539 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		writes: >Note that the surface doesn't have to be orientable (eg Mobius band).
		Certainly a (tame) simple closed curve in R^3 always bounds both orientable and non-orientable surfaces.
		"The surface" produced by Seifert's algorithm is always orientable, and I think that good usage requires "Seifert surface" always to refer to an orientable (preferably, an oriented) surface (with no closed components).
	www.math.niu.edu /~rusin/known-math/00_incoming/seifert (305 words)

	M103 Notes 4-20-04
		Orientable or Non-orientable: we considered the moebius band and the Klein Bottle as examples of non-orientable surfaces.
		Can be realized (imbedded) in a plane, in 3 space, in 4 space.
		If the surface is orientable, it is a sphere with n handles, so V-E+R = 1 - 2n +1 = 2-2n.
	www.humboldt.edu /~mef2/Courses/m103n4_20_04.html (284 words)

	Computing a Canonical Polygonal Schema of an Orientable Triangulated Surface - Lazarus, Pocchiola, Vegter, Verroust ... (Site not responding. Last check: 2007-10-20)
		Abstract: A closed orientable surface of genus g can be obtained by appropriate identi cation of pairs of edges of a 4ggon (the polygonal schema).
		The identi ed edges form 2g loops on the surface, that are disjoint except for their common end-point.
		Computing a canonical polygonal schema of an orientable triangulated surface.
	citeseer.ist.psu.edu /lazarus01computing.html (476 words)

	Citations: Computing the orientable genus of projective graphs - Fiedler, Huneke, Richter, Robertson (ResearchIndex) (Site not responding. Last check: 2007-10-20)
		....by a linear function of the genus g(G) On the other hand, Auslander, Brown, and Youngs [1] proved that there are graphs embeddable in the projective plane whose orientable genus is arbitrarily large.
		In the orientable case take a graph with large planar crossing number but with a drawing where all crossings involve a fixed edge e; in the non orientable case take a projective planar graph with a large face width (see
		Joseph Fiedler, J. Philip Huneke, R. Bruce Richter, and Neil Robertson, Computing the orientable genus of projective graphs, J. of Graph Theory 20(1995)297-308.
	citeseer.ist.psu.edu /context/591740/0 (1235 words)

	Non (Site not responding. Last check: 2007-10-20)
		NPR and CP-16 orientable viewfinders can be adapted to the ACL, with some slight modification to the finder at the point where it attaches to the camera, machining the appropriate mount for the camera, and the introduction of a small diopter to bring the finder into the correct range for the ACL.
		But this is how my own NPR finder was made to work an ACL, yielding an image identical to the ACL 1.5 Angenieux orientable.
		PHOTO 1: The rear of the NPR Finder needs metal filed away to fit in an ACL-type adapter, (an adapter very similar to the adapter used on the ACL 1.5).
	members.aol.com /Super16ACL/misc5.htm (650 words)

	Rigidity for Orientable Functors, by Ivan A. Panin and Serge A. Yagunov (Site not responding. Last check: 2007-10-20)
		Rigidity for Orientable Functors, by Ivan A. Panin and Serge A. Yagunov
		In this paper we introduce the notion of orientable cohomology theory on the category of projective smooth schemes, define a family of transfer maps and, as a consequence of these constructions, prove that taken with finite coefficients such cohomology doesn't change after an extension of algebraically closed fields.
		Besides K-theory, we treat the following examples of orientable theories: Etale Cohomology, Motivic Cohomology, Algebraic Cobordism.
	www.math.uiuc.edu /K-theory/0489 (125 words)

	Amazon.ca: Books: All Compact Orientable Three Dimensional Manifolds Admit Total Foliations (Site not responding. Last check: 2007-10-20)
		Amazon.ca: Books: All Compact Orientable Three Dimensional Manifolds Admit Total Foliations
		All Compact Orientable Three Dimensional Manifolds Admit Total Foliations
		We will notify you within 2-3 weeks if we have trouble obtaining this title.
	www.amazon.ca /exec/obidos/ASIN/0821822330 (140 words)

	Atlas: Drawings of projective graphs in orientable surfaces by Gelasio Salazar (Site not responding. Last check: 2007-10-20)
		Atlas: Drawings of projective graphs in orientable surfaces by Gelasio Salazar
		A couple of years ago we proved that for each orientable surface S, there is a constant c
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafp-37.
	atlas-conferences.com /c/a/f/p/37.htm (341 words)

	Víctor Núñez (Site not responding. Last check: 2007-10-20)
		It was recently shown by Gómez, González, and Hoste that if M is a non-orientable closed 3-manifold, then M can be represented as the union of three orientable handlebodies with pairwise disjoint interiors, M=H
		If M is a non-orientable Seifert manifold such that M cannot be expressed as an S
		-bundle structure with fiber a closed orientable surface, then M has tri-genus of the form (0,g,g), with g a very big number.
	www.utm.edu /staff/jschomme/topology/c/a/a/k/48.htm (218 words)

	20. : Orientable closed surface construction from volume data (Site not responding. Last check: 2007-10-20)
		: Orientable closed surface construction from volume data
		In the present paper, we propose a new method that produces a set of triangular patches from a given volume data.
		The fact that the set of patches has no holes, no redundancy, no self-intersection, and has orientable closed surface topology is shown.
	home.att.net /~edgrenda/p93/p93_20.htm (184 words)

	A Prototype System for Robust, Interactive and User-Friendly Modeling of Orientable 2-Manifold Meshes (Site not responding. Last check: 2007-10-20)
		In this paper, we present a prototype system for robust, interactive and user friendly modeling of orientable 2-manifold meshes.
		To develop the system we introduce new topological entities for effectively manipulating 2-manifold mesh structures.
		"A Prototype System for Robust, Interactive and User-Friendly Modeling of Orientable 2-Manifold Meshes," smi, p.
	csdl2.computer.org /persagen/DLAbsToc.jsp?resourcePath=/dl/proceedings/&toc=comp/proceedings/smi/2002/1546/00/1546toc.xml&DOI=10.1109/SMA.2002.1003527 (294 words)

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