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Topic: Gauss map

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	Encyclopedia: Carl Friedrich Gauss
		Gauss was a child prodigy of the highest order, of whom there are many almost unbelievable anecdotes pertaining to his astounding precocity while a mere toddler, and made his first ground-breaking mathematical discoveries while still a teenager.
		Gauss was so pleased by this result that he requested that a regular heptadecagon be inscribed on his tombstone.
		Gauss predicted correctly the position at which it could be found again, and it was rediscovered by Franz Xaver von Zach on December 31, 1801 in Gotha, and one day later by Heinrich Olbers in Bremen.
	www.nationmaster.com /encyclopedia/Carl-Friedrich-Gauss (6117 words)

	Gauss map -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-09-17)
		The Gauss map can be defined the same way for (additional info and facts about hypersurface) hypersurfaces in, this way we get a map from a hypersurface to the unit sphere.
		Finally, the notion of Gauss map can be generalized to an oriented submanifold of dimension in an oriented ambient (additional info and facts about Riemannian manifold) Riemannian manifold of dimension.
		The target space for the Gauss map is a Grassmann bundle built on the tangent bundle.
	www.absoluteastronomy.com /encyclopedia/g/ga/gauss_map.htm (327 words)

	Gauss map - Encyclopedia, History, Geography and Biography
		The Gauss map can be defined the same way for hypersurfaces in $\mathbb{R}^n$ , this way we get a map from a hypersurface to the unit sphere $S^{n-1}\in \mathbb{R}^n$ .
		For a general oriented k-submanifold of $\mathbb{R}^n$ the Gauss map can be also be defined, and its target space is the oriented Grassmannian $\tilde{G}_{k,n}$ , i.e.
		Finally, the notion of Gauss map can be generalized to an oriented submanifold $S$ of dimension $k$ in an oriented ambient Riemannian manifold $M$ of dimension $n$ .
	www.arikah.com /encyclopedia/Gauss_map (361 words)

	Curvature - Wikipedia, the free encyclopedia
		This is Gauss' celebrated Theorema Egregium, which he found while concerned with geographic surveys and mapmaking.
		Unlike Gauss curvature, the mean curvature depends on the embedding, for instance, a cylinder and a plane are locally isometric but the mean curvature of a plane is zero while that of a cylinder is nonzero.
		Gauss map for more geometric properties of Gauss curvature.
	en.wikipedia.org /wiki/Curvature (930 words)

	Gauss map Info - Bored Net - Boredom (Site not responding. Last check: 2007-09-17)
		In mathematics, the Gauss map is a construction in differential geometry: for a surface lying in 3-space, it associates to any point of the unit (normal) vector that is orthogonal to the tangent plane to at.
		Hence is a map from to the unit sphere.
		The notion of Gauss map can be generalized to a submanifold of dimension in an ambient manifold of dimension.
	www.borednet.com /e/n/encyclopedia/g/ga/gauss_map.html (250 words)

	Gauss map (Site not responding. Last check: 2007-09-17)
		The Jacobian of the Gauss map is equal to Gausscurvature, the differential of theGauss map is called shape operator.
		Finally, the notion of Gauss map can be generalized to an oriented submanifold S ofdimension k in an oriented ambient Riemannian manifold M of dimension n.
		The targetspace for the Gauss map N is a Grassmann bundle built on the tangent bundle TM.
	www.therfcc.org /gauss-map-215161.html (226 words)

	Cusps of Gauss Mappings: Gauss Mappings of Surfaces
		Gauss thought of his mapping N as assigning to each point of a surface a point on the sphere at infinity, analogous to the celestial sphere used in navigation and surveying [Ba, p.
		In particular, the Gauss mapping of an immersed hypersurface of Euclidean space is a Lagrange mapping, since it is the catastrophe map of the family of projections to lines (Chpater 5).
		The Gauss map is the Lagrangian map associated with the Lagrangian immersion N, i.e.
	www.emis.de /monographs/CGM/6.html (1667 words)

	Gauss map
		Gauss map can be defined the same way for hypersurfaces in, this way we get a map from a hypersurface to the unit sphere.
		For general oriented k-submanifold of the Gauss map can be also be defined, its target space is oriented Grassmannian, i.e.
		Finally, the notion of Gauss map can be generalized to an oriented submanifold of dimension in an oriented ambient Riemannian manifold of dimension.
	www.faqfolio.com /faqfolio/g/ga/gauss_map.html (229 words)

	Gauss map - InfoSearchPoint.com (Site not responding. Last check: 2007-09-17)
		In mathematics, the Gauss map is a construction in differential geometry: for a surface $S$ lying in 3-space, it associates to any point $p$ of $S$ the unit (normal) vector $N=N(p)$ that is orthogonal to the tangent plane to $S$ at $p$ .
		Hence $N$ is a map from $S$ to the unit sphere $S^2$ .
		The notion of Gauss map can be generalized to a submanifold $S$ of dimension $k$ in an ambient manifold $M$ of dimension $n$ .
	www.infosearchpoint.com /display/Gauss_map (301 words)

	A General Theory for Developing Projection Plotting Equations (Gauss's Theory)
		On the map on the bottom of Figure 1, the same region shows as a true rectangle, with sides composed of straight lines and opposing sides parallel to one another.
		In fact, Gauss proved that mathematically, if the pseudorectangle becomes infinitely small (which is physically impossible, but is quite possible in the crazy world of abstract mathematics), this process gets carried to an extreme and the pseudorectangle in fact becomes a true rectangle.
		Gauss's Theory is to use this technique of shrinking pseudorectangles to equate infinitely small pseudorectangles on the surface of a sphere to their corresponding regions on a flat Cartesian surface (like a map), and then construct mathematical equations that allow you to translate coordinates from one surface to the other.
	www.cnr.colostate.edu /class_info/nr502/lg4/projection_mathematics/plotting_equations.html (847 words)

	Cusps of Gauss Mappings: Gauss Mappings of Surfaces
		Thus the Gauss map of the perturbed monkey saddle is stable for all
		This unfolding of the Gauss map of the monkey saddle is identical with the unfolding of the complex squaring map described by Arnold [A1, pp.
		This family of Gauss maps is the same as the unfolding of the quarter folded handerchief described by Arnold [A1] and Callahan [18].
	www.maa.org /cvm/1998/01/cgm/article/2b.html (357 words)

	Gauss Map Computation for Free--Form Surfaces - Smith, Farouki (ResearchIndex) (Site not responding. Last check: 2007-09-17)
		Abstract: The Gauss map of a smooth doubly--curved surface characterizes the range of variation of the surface normal as an area on the unit sphere.
		Boundary segments of the Gauss map correspond to variations of the normal along the patch boundary or the parabolic lines (loci of vanishing Gaussian...
		2 Cusps of Gauss Mappings (context) - Banchoff, Gaffney et al.
	citeseer.ist.psu.edu /468177.html (649 words)

	Gauss map from LiveJournal (Site not responding. Last check: 2007-09-17)
		The Gaussian curvature and the mean curvature are determined by the differential of the Gauss map of the underlying surface.
		The Gauss map assigns to each point in the surface the unit normal vector of the tangent plane to the surface at this point.
		Gauss was very prolific, as was Euler, of course." Even in his seventies there were years when Erdös published fifty papers, which is more than most good mathematicians write in a lifetime.
	www.ljseek.com /search/Gauss%20map (723 words)

	ICMS 2002 (Cgm): The Gauss Map, A Dynamical Approach
		We show the linear interpolation between the surface and its Gauss spherical image so that the singularities of the Gauss map are expressed as limits of singularities of homothetic images of parallel surfaces of the original surface.
		and indicate the behavior of the asymptotic vectors in a neighborhood of a cusp of the Gauss mapping.
		Various characterizations of the singularities of the Gauss map in terms of lines of curvature, ridges, and double tangencies are included in "Cusps of Gauss Mapping".
	www.math.union.edu /~dpvc/TFB/ICMS-poster/cgm/welcome.html (389 words)

	Cusps of Gauss Mappings: Gauss Mappings of Surfaces
		In the terminology of Whitney [Wh], the Gauss mapping N is good if the gradient of K is never zero on the parabolic set.
		We begin our investigation of the singularities of the Gauss mapping with a collection of key examples which exhibit all of the geometric phenomena which we shall associate with these singularities.
		Thus the Gauss map is stable, with a simple fold along the parabolic curve.
	www.zblmath.fiz-karlsruhe.de /exx/monographs/CGM/2.html (536 words)

	Read about Gauss map at WorldVillage Encyclopedia. Research Gauss map and learn about Gauss map here! (Site not responding. Last check: 2007-09-17)
		Jacobian of the Gauss map is equal to
		differential of the Gauss map is called shape operator.
		Finally, the notion of Gauss map can be generalized to an oriented submanifold S of dimension k in an oriented ambient
	encyclopedia.worldvillage.com /s/b/Gauss_map (303 words)

	Normal Distribution Mapping
		The close view of the bump map was modeled as a height field, and random rays (modulated by the close scale BRDF) were traced to produce a histogram representation of the distant scale BRDF for the entire bump map.
		When the mapping is applied to a particular surface, the result is called the Gauss map of the surface (see Figure 4).
		It is analogous to the 3D Gauss map, which maps a surface onto the unit sphere.
	www.cs.unc.edu /~olano/papers/ndm (4368 words)

	Periodic and Fixed Points of the Gauss Map
		Corollary: The periodic points of the Gauss map are the reciprocals of the reduced quadratic irrationals.
		There are general results in the theory of chaotic dynamical systems, with which we could hope to establish the character of the set of periodic points of the Gauss map [24,27,17].
		The number of iterations of the Gauss map required to reach zero from this initial point is, by the speed of the Euclidean algorithm,
	www.cecm.sfu.ca /organics/papers/corless/confrac/html/node6-an.shtml (543 words)

	Atlas: The Gauss Map and Resistive Networks by Hei-Chi Chan (Site not responding. Last check: 2007-09-17)
		Gauss obtained some partial results but he was not too satisfied with the outcome, as some crucial questions were still unanswered at that time.
		This map, which is now known as the Gauss map, is related to the metrical theory of continued fraction.
		To this day the Gauss map has fascinated researchers from various branches of mathematics and science: it is ergodic with respect to a known invariant measure and it has applications in computer science, in cosmology and in chaos theory.
	atlas-conferences.com /cgi-bin/abstract/cakr-42 (181 words)

	THE GAUSS MAP (Site not responding. Last check: 2007-09-17)
		This mapping can be thought of as moving the normal of a point on a surface to a congruent parallel vector at the origin.
		Instead of merely mapping a point to the sphere, one can place a weight on the sphere proportional to the surface area of the polygon.
		Despite this, the mapping for convex objects is injective.
	erie.nlm.nih.gov /~blowek1/igi/node2.html (565 words)

	Gauss map - Definition up Erdmond.Com (Site not responding. Last check: 2007-09-17)
		The Jacobian of the Gauss map is equal to Gauss Curvature, the differential of the Gauss map is called shape_operator.
		Gauss map can be defined the same way for hypersurfaces in \mathbb{R}^n, this way we get a map from a hypersurface to the unit sphere S^{n-1}\in \mathbb{R}^n.
		For general oriented ''k''-submanifold of \mathbb{R}^n the Gauss map can be also be defined, its target space is ''oriented'' Grassmannian \tilde{G}_{k,n}, i.e.
	www.erdmond.com /Gauss_map.html (297 words)

	Example III: The Gauss Map
		The following explorations are meant to convince you that the claims made about the Gauss map in this paper are in fact true.
		This code simply generates the first N iterates of the orbit orb(x0) of the Gauss map.
		of the Lyapunov exponent of an orbit orb(x0) of the Gauss map.
	www.cecm.sfu.ca /organics/papers/corless/lyap/html/node4.html (323 words)

	List of topics named after Carl Friedrich Gauss - Wikipedia, the free encyclopedia
		Carl Friedrich Gauss (1777 - 1855) is the eponym of all of the topics listed below.
		The divergence theorem is also known as Gauss' theorem or the Ostrogradsky-Gauss theorem
		The Gaussian probability distribution, known to probabilists and statisticians as the normal distribution, but called the Gaussian distribution by machine learning practitioners, physicists and engineers.
	www.wikipedia.org /wiki/Gaussian (123 words)

	Cusps of Gauss Mappings: Characterizations of Gaussian Cusps (Site not responding. Last check: 2007-09-17)
		Conversely, if the Gauss map is stable, and the parabolic image curve has nonzero curvature at P, and (i) holds, then P is a cusp of the Gauss map.
		Conversely, if the Gauss map is stable, and the asymptotic direction map is regular at P, and (j) holds, then P is a cusp of the Gauss map.
		If P is a cusp of the Gauss map of X, let l be the line of curvature of X which is tangent to the parabolic curve at P.
	www.mat.ub.es /EMIS/monographs/CGM/3.html (1454 words)

	LMS Proceedings Abstract, paper PLMS 1385 (Site not responding. Last check: 2007-09-17)
		We define the hyperbolic Gauss map and the hyperbolic Gauss indicatrix of a hypersurface in hyperbolic space.
		In the study of the singularities of the hyperbolic Gauss map (indicatrix), we find that the hyperbolic Gauss indicatrix is much easier to calculate.
		We introduce the notion of hyperbolic Gauss--Kronecker curvature whose zero sets correspond to the singular set of the hyperbolic Gauss map (indicatrix).
	www.lms.ac.uk /publications/proceedings/abstracts/p1385a.html (156 words)

	An Estimate For The Gauss Curvature Of Minimal Surfaces In R^m Whose Gauss Map Omits A Set Of Hyperplanes - Osserman, ... (Site not responding. Last check: 2007-09-17)
		Alternate document: Details An Estimate For The Gauss Curvature Of Minimal Surfaces In R^m Whose Gauss Map Omits A Set Of Hyperplanes (97) Robert Osserman, Min Ru An Estimate For The Gauss Curvature Of Minimal Surfaces In R^m Whose Gauss Map Omits A Set Of Hyperplanes (1997)
		We give an estimate of the Gauss curvature for minimal surfaces in R m whose Gauss map omits more than m(m + 1)=2 hyperplanes in P m\Gamma1 (C).
		Osserman, R. and Ru, M. "An estimate for the Gauss curvature of minimal surfaces in R m whose Gauss map omits a set of hyperplanes." J. Differential Geometry 46 (1997) 578-593.
	citeseer.ist.psu.edu /osserman97estimate.html (552 words)

	Smale's paradox - MindSharer Article Archive (Site not responding. Last check: 2007-09-17)
		Indeed, the degree of the Gauss map must be preserved in such "turning" — in particular it follows that there is no such turning of $S^1 in R^2 .$
		But the degree of the Gauss map for the embeddings $f and -f in R^3 are both equal to 1.$
		In fact the degree of the Gauss map of all immersions of a 2-sphere in $R^3 is 1; so there is in fact no obstacle.$
	articles.mindsharer.com /html/Smale%27s_paradox (228 words)

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