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Topic: Gaussian curvature

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	Curvature - Wikipedia, the free encyclopedia
		For a plane curve C, the curvature at a given point P has a magnitude equal to the reciprocal of the radius of an osculating circle (a circle that "kisses" or closely touches the curve at the given point), and is a vector pointing in the direction of that circle's center.
		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.
		Curvature form for the appropriate notion of curvature for vector bundles and principal bundles with connection.
	en.wikipedia.org /wiki/Curvature (930 words)

	Curvature -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		The magnitude of curvature at points on physical curves can be measured in (A unit of measurement of the refractive power of a lens which is equal to the reciprocal of the focal length measured in meters) diopters (also spelled dioptre); a diopter has the dimension one per meter.
		Unlike Gauss curvature, the mean curvature depends on the embedding, for instance, a (A surface generated by rotating a parallel line around a fixed line) cylinder and a plane are locally (A line connecting isometric points) isometric but the mean curvature of a plane is zero while that of a cylinder is nonzero.
		A space without curvature is called a "flat space" or (A space in which Euclid's axioms and definitions apply; a metric space that is linear and finite-dimensional) Euclidean space.
	www.absoluteastronomy.com /encyclopedia/c/cu/curvature.htm (1037 words)

	Gaussian Curvature (Site not responding. Last check: 2007-10-21)
		To understand what the Gaussian curvature of a point on a surface is, you must first know what the curvature of curve is. At any point on a curve in the plane, the line best approximating the curve that passes through this point is the tangent line.
		A positive Gaussian curvature value means the surface is locally either a peak or a valley.
		One way of shading an image with Gaussian curvature (which is a scalar) is to use the curvature to vary the Hue in an HSV color system.
	www.cgl.uwaterloo.ca /~smann/Research/gaussiancurvature.html (451 words)

	Curvature Article, Curvature Information (Site not responding. Last check: 2007-10-21)
		For a plane curve C, the curvature at a given point P has magnitude equal to the reciprocal of the radius of an osculating circle (a circle that "kisses" or closely touches the curve atthe given point), and is a vector pointing in the direction of that circle's center.
		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.
		Curvature vector and geodesic curvature for appropriate notions of curvature of curves in Riemannian manifolds, ofany dimension.
	www.anoca.org /surface/plane/curvature.html (805 words)

	Constrained Controlled Coverage \| Projects \| Sensor Based Planning Lab
		High geodesic curvature of the start curve typically leads to self-intersections of the resultant offset curves; these self-intersections are undesirable because they have a drastically adverse effect on the resultant uniformity of material deposition.
		However, the offsets of a geodesic curve on a surface with non-zero Gaussian curvature (a measure of how much a surface bends in two directions at a given point on the surface) are not geodesics.
		The integral of geodesic curvature on the offset curve is the summation of the integral of geodesic curvature of the offset curve and the integral of Gaussian curvature of the surface region bounded between the two curves
	voronoi.sbp.ri.cmu.edu /projects/prj_constrained.html (1516 words)

	CURVATURE FACTS AND INFORMATION (Site not responding. Last check: 2007-10-21)
		For a plane_curve ''C'', the curvature at a given point ''P'' has a magnitude equal to the ''reciprocal'' of the radius of an osculating_circle (a circle that "kisses" or closely touches the curve at the given point), and is a vector pointing in the direction of that circle's center.
		The integral of the Gaussian curvature over the whole surface is closely related to the surface's Euler_characteristic; see the Gauss-Bonnet_theorem.
		Mean curvature is closely related to the first variation of surface_area, in particular a minimal_surface like a soap_film has mean curvature zero and soap_bubble has constant mean curvature.
	www.witwib.com /?s=curvature (842 words)

	Theorema Egregium - Wikipedia, the free encyclopedia
		The Theorema Egregium ('Remarkable Theorem') is an important theorem of Carl Friedrich Gauss concerning the curvature of surfaces.
		Informally, the theorem says that the curvature of a surface can be determined entirely by measuring angles and distances on the surface, that is, it does not depend on how the surface might be embedded in (3-dimensional) space.
		The Gaussian curvature of a surface is invariant under local isometry.
	en.wikipedia.org /wiki/Theorema_egregrium (361 words)

	Surface Optimization
		To approximate Gaussian curvature at a vertex, we take the sum of the angles incident to the vertex, and subtract this sum from 360 degrees.
		To measure "overall" curvature, we consider a combination of the Gaussian curvature and the mean curvature.
		In this scheme, to calculate the curvature "penalty" at a vertex v, we take the simple average of the curvature measures of the neighbors of v and compare the average with the curvature measure at v; the "penalty" is simply the absolute value of the difference.
	www.ocf.berkeley.edu /~ryot/slide/optimization/surfaces/surfaces.html (979 words)

	Curvature of surfaces
		Gaussian curvature is an intrinsic geometric property: it stays the same no matter how a surface is bent, as long as it is not distorted, neither stretched or compressed.
		Gaussian curvature is a numerical quantity associated to an area of a surface, very closely related to angle defect.
		The total Gaussian curvature of a region on a surface is the angle by which its boundary opens up, when laid out in the plane.
	www.geom.uiuc.edu /docs/doyle/mpls/handouts/node21.html (1137 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Surf curvature determines inconsistent curvature on surfaces by producing a quick-rendered image that evaluates the curvature at points on a surface and assigns colors based on the value.
		GAUSSIAN curvature is less effective for finding very small amounts of curvature, because the small curvatures compound each other when multiplied.
		GAUSSIAN curvature is most useful for seeing the total character of a surface's curvature.
	www.cclabs.missouri.edu /things/instruction/aw/Evaluatesurface.html (2530 words)

	Curvature - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-21)
		For a plane curve C, the curvature at a given point P has magnitude equal to the reciprocal of the radius of an osculating?
		The magnitude of curvature at points on physical curves can be measured in diopters (alternative spelling: dioptre); a diopter is one per meter.
		This last formula also gives the mean curvature of an hypersurface in Euclidean space.
	xahlee.org /_p/wiki/Gaussian_curvature.html (847 words)

	Contribution of Gaussian Curvature to Strain Energy of Plates (Site not responding. Last check: 2007-10-21)
		It is shown that when a plate with a smooth boundary, on every point of which either the displacement or slope is zero, there is, in general, a contribution to the flexural strain energy from the Gaussian curvature term.
		When the plate boundary is approximated by a polygon with a large, but finite, number of sides, it is shown that the contribution of the Gaussian curvature, converted to a line integral around the boundary, is zero both along the sides and on corners where the displacement gradient is continuous.
		The singular behavior is shown to be the result of discontinuities of the Gaussian curvature term as a functional of the boundary contour.
	www.pubs.asce.org /WWWdisplay.cgi?8902710 (147 words)

	Mathematica
		Gauss's theorema egregium states that the Gaussian curvature of a surface embedded in three-space may be understood intrinsically to that surface.
		In particular, Gaussian curvature can be measured by checking how closely the arc lengths of circles of small radii correspond to what they should be in Euclidean space,.
		Gauss (effectively) expressed the theorema egregium by saying that the Gaussian curvature at a point is given by -R(v,w)v,w where R is the Riemann tensor, and v and w are an orthonormal basis for the tangent space.
	www.siyulian.blogspot.com (505 words)

	Re: Gaussian Curvature and General Relativity
		> Gaussian curvature is defined in terms of the first fundamental form, > which is the derivative of the normal map.
		While the Gaussian curvature was initially defined in terms of the "first fundamental form", Gauss showed, "Theorem Egregium", that it was an intrinsic property of the manifold, and hence independent of any (isometric) embedding.
		Of course, calculating the "principle curvatures" (eigenvalues) is this case is probably a meaningless exercise.
	www.lns.cornell.edu /spr/2004-02/msg0058753.html (359 words)

	Omega Art - Gaussian curvature and the Gauss-Bonnet theorem
		Gaussian curvature is a measure for the "curvedness" of a surface.
		Gaussian curvature and the related Gauss-Bonnet theorem are concerned with general surfaces and general curves: the curves may be non-smooth, and the surface may have holes, sharp edges, or anything you like.
		Gaussian curvature is a differential-geometric concept, while the Euler characteristic is topological in nature and has nothing to do with curves.
	www.omega-art.com /math/gauss.html (2761 words)

	Untitled Document
		Measurement of the curvature of a phase requires deduction of the principal curvatures c1 and c2.
		In this instance, the variables Dc1 and Dc2 are the departures of the principal curvatures c1 and c2 from the equilibrium curvatures cr of the system (e.g.
		In the case of a cylinder it is easy enough to show that the mean curvature is ±1/2R but harder in case of the inverse hexagonal phase where R is undefined at discontinuities.
	www.ch.ic.ac.uk /liquid_crystal/curvature.htm (1043 words)

	Gaussian curvature and the equilibrium among bilayer cylinders, spheres, and discs -- Jung et al. 99 (24): 15318 -- ...
		Gaussian curvature and the equilibrium among bilayer cylinders, spheres, and discs -- Jung et al.
		Gaussian curvature and the equilibrium among bilayer cylinders, spheres, and discs
		curvature is zero (12, 13), which is not the case here.
	www.pnas.org /cgi/content/full/99/24/15318 (2841 words)

	DARPA Image Understanding Workshop - Abstract #59 (Site not responding. Last check: 2007-10-21)
		Gaussian curvature is an intrinsic local shape characteristic of a smooth object surface that is invariant to orientation of the object in 3-D space and viewpoint.
		Accurate determination of the sign of Gaussian curvature at each point on a smooth object surface (i.e., the identification of {\it hyperbolic}, {\it elliptical} and {\it parabolic} points) can provide very important information for both recognition of objects in automated vision tasks and manipulation of objects by a robot.
		In comparison with photometric stereo, this new technique determines the sign of Gaussian curvature directly from image features without having to derive local surface orientation, is invariant to incident orientation errors of two of three light sources, and is invariant to the relative strength of incident radiance with respect to each of these light sources.
	www.cc.gatech.edu /conferences/iuw/abstracts/abstract59.html (251 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Consider a surface whose total Gaussian curvature is small, where the total curvature is defined to be the integral of the absolute value of Gaussian curvature.
		One now asks if every surface with small total curvature is actually a small perturbation of a developing surface (as opposed to the situation where there is no developing surface close to the original one).
		Establishing relations between intrinsic geometry of a surface and properties of this surface as a subset of the ambient space is one of the most classic areas of differential geometry.
	comet.lehman.cuny.edu /sormani/ams/krat.html (295 words)

	Riemannian Gaussian curvature Riemann Modern Relativity modernrelativity special general black hole mass energy ...
		Limiting ourselves to the consideration of space curvature the Gaussian curvature K of a two dimensional cross section of space is defined as
		Spacetime curvature, or a nonzero Riemann tensor, is always present where there is any kind of matter, whether or not the affine connection has been transformed away.
		Whether we have a vacuum field solution or not, whether we have Riemannian space-time curvature or not, all the inertial forces or all the gravitational forces are directly due to nonzero affine connections.
	www.geocities.com /zcphysicsms/chap6.htm (3998 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		The Gaussian curvature is computed by finding at each point the principal curvatures, and then multiplying all of them together.
		Also a cylinder has zero Gaussian curvature, because one of its principal curvatures (at every point) is always zero.
		That's why it is possible to put any plane figure on a cylinder, but it is impossible to lay a plane-figure on a sphere, becuase the Gaussian curvatures do not agree.
	www.math.niu.edu /~rusin/known-math/99/egregium (188 words)

	[No title]
		Examples: Spheres Tori Cones Minimum variation surfaces are surfaces that minimize the variation in curvature, specifically the integral of the square of the magnitude of the derivative of the curvature.
		In the figures, the principal direction corresponding to the most positive normal curvature is represented by a cyan line, while the principal direction corresponding to the least positive normal curvature is represented by a yellow line.
		Gaussian Curvature Gaussian curvature is the product of the minimum and maximum values of curvature at a given point.
	www.ocf.berkeley.edu /~ryot/X/poster.doc (341 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		As is well known, the Gaussian curvature of a minimal surface (zero mean curvature surface) immersed into a Riemannian space of constant curvature is bounded from above by the constant curvature of the ambient space, with equality precisely at the totally geodesic points of the surface.
		In contrast to this, when the dimension of the ambient Lorentzian space is greater than 3 (codimension greater than 1), one cannot deduce any \textit{a priori} regularity on the behaviour of the curvature of spacelike zero mean curvature surfaces, due to the fact that the normal bundle is indefinite.
		In recent years, we have studied in depth the global behaviour of the curvature of spacelike zero mean curvature surfaces in 3-dimensional and 4-dimensional Lorentzian spaces.
	www.mat.uc.pt /~geomfis/datos/alias.html (295 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		In this paper, we study some properties related to the second Gaussian curvature of ruled surfaces in a 3-dimensional Euclidean space.
		Furthermore, we characterize ruled surfaces in a 3-dimensional Euclidean space in terms of the second Gaussian curvature, the mean curvature and the Gaussian curvature.
		In particular, we show that the only ruled surface with vanishing second Gaussian curvature is a helicoid.
	www.pphmj.com /abstracts/jpgt/vol2issue2/ab-3.htm (91 words)

	Gaussian curvature (Site not responding. Last check: 2007-10-21)
		Gaussian curvature (K) is a product of maximum and minimum land surface curvatures.
		According to Teorema egregium by Gauss, K of a surface retains its values after bending the surface without changing lengths of curves on the surface, that is, the surface is bent without breaking, stretching, and compressing.
		Gaussian curvature is used in geological studies to describe geological surfaces and structures.
	members.fortunecity.com /flor/k.htm (175 words)

	Maxsurf News November 1999 (Site not responding. Last check: 2007-10-21)
		Positive Gaussian curvature occurs when the major and minor curvatures are in the same direction such as on a round bilge yacht.
		Negative Gaussian curvature occurs when the direction of the two curvatures is opposite so that the plate is twisted.
		You may choose from the direction of minimum principal curvature at the centre of the plate or the local plate longitudinal axis.
	www.formsys.com /Maxsurf/MSNewsletters/MSNewsNov99.html (1147 words)

	Analyze Faces
		If the lines have a sharp angle on the surface transition, then the adjacent surfaces are tangent continuous, but not curvature continuous.
		For a fixed vector V and fixed angle theta, isophotes on a face are the set of points having a normal vector that makes an angle theta with the vector V. This set is usually a curve.
		The Mean Curvature display mode analyzes the selected faces and overlays a Mean (average) curvature plot for both U and V directions.
	www.vx.com /help/0290.htm (662 words)

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