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Topic: Mean 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.
		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.
		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.
	en.wikipedia.org /wiki/Curvature (930 words)

	Mean curvature -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-16)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, mean curvature of a (The outer boundary of an artifact or a material layer constituting or resembling such a boundary) surface is a notion from (additional info and facts about differential geometry) differential geometry.
		For all such curvatures C of all the planes, there is a maximal and a minimal one.
		Their (A statistic describing the location of a distribution) average is the mean curvature at P of the surface.
	www.absoluteastronomy.com /encyclopedia/m/me/mean_curvature.htm (211 words)

	Encyclopedia: Curve (Site not responding. Last check: 2007-10-16)
		In particular, graph means the graphical representation of this collection, in the form of a curve or surface, together with axes, etc....
		On the other hand it is useful to be more general, in that (for example) it is possible to define the tangent vectors to X by means of this notion of curve.
		In mathematics, a manifold M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension.
	www.nationmaster.com /encyclopedia/Curve (3519 words)

	Preprint server: Mean Curvature Flow.
		Characterization of facet--breaking for nonsmooth mean curvature flow in the convex case, Interfaces and Free Boundaries, 3 (2001), 415--446.
		Numerical simulations of mean curvature flow in presence of a nonconvex anisotropy, Math.
		Anisotropic motion by mean curvature in the context of Finsler geometry, Hokkaido Math.
	www.dmf.bs.unicatt.it /~paolini/preprints/keymcm.html (539 words)

	Surface Optimization
		To approximate mean curvature at a vertex v, we use the formula from the paper by Hsu, Kusner, and Sullivan:
		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)

	Mean curvature (Site not responding. Last check: 2007-10-16)
		Mean curvature (H) is a half-sum of curvatures of any two mutually orthogonal normal section of the land surface.
		Mean curvature presents flow convergence and relative deceleration with equal weights.
		Mean curvature can be more representative topographic attribute than horizontal and vertical curvatures in relation to description of landscape processes.
	members.fortunecity.com /flor/h.htm (180 words)

	Karsten Grosse-Brauckmann: Research (Site not responding. Last check: 2007-10-16)
		Constant mean curvature surfaces appear in nature, in particular when the area of an interface is minimized under a volume constraint.
		The trousers decomposition of arbitrary embedded constant mean curvature surfaces makes our result significant for the general case: indeed, we can understand each trouser as a truncated triunduloid, and the trouser is approximately described by a complete triunduloid.
		As an example, I computed two one-parameter families of constant mean curvature IWP-surfaces, having one surface in common (an existence proof with the conjugate surface method should be straightforward).
	www.math.uni-bonn.de /people/kgb/Research/research.html (877 words)

	Soap Film
		The mean curvature of a surface is the average of the two prinicpal curvatures.
		There is a useful analogy to functions of two variables: A minimal surface corresponds to a local minimum of a function of two variables, and a surface of zero mean curvature corresponds to a critical point (a local minimum or a saddle point).
		Analogous to "every minimal surface has zero mean curvature" is the fact that every local minimum of a function of two variables is a critical point.
	oak.ucc.nau.edu /jws8/dpgraph/soap_film.html (1023 words)

	Citebase - Irreducible constant mean curvature 1 surfaces in hyperbolic space with positive genus (Site not responding. Last check: 2007-10-16)
		Irreducible constant mean curvature 1 surfaces in hyperbolic space with positive genus
		We present a global representation for surfaces in 3-dimensional hyperbolic space with constant mean curvature 1 (CMC-1 surfaces) in terms of holomorphic spinors.
		A complete surface of constant mean curvature 1 (CMC-1) in hyperbolic 3-space with constant curvature -1 has two natural notions of "total curvature"--- one is the total absolute curvature which is the integral over the surface of the absolute value of the Gaussian curvature, and the other is the...
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:dg-ga/9709008 (1162 words)

	Untitled Document (Site not responding. Last check: 2007-10-16)
		Measurement of the curvature of a phase requires deduction of the principal curvatures c1 and c2.
		This gradual change in the mean curvature is in agreement with figure 1.19 with phase transitions taking place in accordance with dramatic changes in curvature.
		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)

	Matthias Heil's CMC Tori picture page
		This is measured by two functions the Gaußian curvature and the mean curvature.
		If the mean curvature of a surface is identically zero one usually speaks of minimal surfaces and the study of these surfaces is a field for itself.
		If the mean curvature is constant but not zero the surfaces are called cmc-surfaces.
	www-sfb288.math.tu-berlin.de /~matt/cmc_tori.html (256 words)

	Theory of Curve and Surface Evolution (Site not responding. Last check: 2007-10-16)
		In particular, Grayson's theorem that every simple closed curve moving under its curvature collapses smoothly to a point is easy to demonstrate, as is the failure of this result in three dimensions.
		We apply the algorithms to a variety of complicated shapes, showing corner formation and breaking and merging, and conclude with a study of a dumbbell in #R sup 3# moving under its mean curvature.
		First we analyze the singularity produced by a dumbbell collapsing under its mean curvature and show that a multi-armed dumbbell develops a separate, residual closed interface at the center after the singularity forms.
	math.berkeley.edu /~sethian/Flow_chart/chart.applications.geometry.html (1303 words)

	1996 Building Publications - Numerical Methods for Computing Interfacial Mean Curvature. (Site not responding. Last check: 2007-10-16)
		A procedure is described for computing the mean curvature along condensed phase interfaces in two or three dimensions, without knowledge of the spatial derivatives of the interface.
		That portion of the template volume is shown to be linear in the mean curvature of the surface, relative to the phase lying on the opposite side of the interface, to within terms that can usually be made negligible.
		Application of the procedure to compute the mean curvature along a digitized surface is demonstrated.
	fire.nist.gov /bfrlpubs/build96/art084.html (192 words)

	A Curvature-Based Approach to Terrain Recognition (Site not responding. Last check: 2007-10-16)
		The authors describe an algorithm which uses a Gaussian and mean curvature profile for extracting special points on terrain and then use these points for recognition of particular regions of the terrain.
		In the Gaussian and mean curvature image, the points of maximum and minimum curvature are extracted and used for matching.
		Curvature values are calculated from the data by fitting a quadratic surface over a square window and calculating directional derivatives of this surface.
	csdl2.computer.org /persagen/DLAbsToc.jsp?resourcePath=/dl/trans/tp/&toc=comp/trans/tp/1989/11/iytoc.xml&DOI=10.1109/34.42859 (457 words)

	Atlas: How to use {\it MATHEMATICA} to find cyclic surfaces of constant curvature in Loretz-Minkowski space by Rafael ...
		Assume the mean curvature or Gauss curvature is constant.
		In the case that the mean curvature or Gauss curvature vanishes on the surface, we shall give a complete description of them.
		The computation of the mean curvature or Gauss curvature yields a polynomical equation in several variables.
	atlas-conferences.com /c/a/d/q/09.htm (798 words)

	[No title] (Site not responding. Last check: 2007-10-16)
		The minimum and maximum mean curvatures are calculated and displayed in two locator statistics boxes.
		Mean, Gaussian, Principal Min and Principal Max show the points on the surface with the minimum and maximum values of the currently-selected curvature type.
		Mean curvature is defined as the average between the two principal curvatures, approximating the average curvature through the point.
	www.cclabs.missouri.edu /things/instruction/aw/MinMaxCurve.html (283 words)

	OUP: Surfaces with Constant Mean Curvature: Kenmotsu
		The mean curvature of a surface is an extrinsic parameter measuring how the surface is curved in the three-dimensional space.
		A surface whose mean curvature is zero at each point is a minimal surface, and it is known that such surfaces are models for soap film.
		A surface whose mean curvature is constant but nonzero is obtained when we try to minimize the area of a closed surface without changing the volume it encloses.
	www.oup.co.uk /isbn/0-8218-3479-7 (374 words)

	ipedia.com: Curvature Article (Site not responding. Last check: 2007-10-16)
		The magnitude of curvature at points on physical curves can be measured in diopterss (alternative spelling: dioptre); a diopter is one per meter.
		A straight line has everywhere curvature 0; a circle of radius r has everywhere curvature of magnitude 1/r.
		This last formula also gives the mean curvature of an hypersurface in Euclidean space.
	www.ipedia.com /curvature.html (820 words)

	[No title] (Site not responding. Last check: 2007-10-16)
		First one estimates the mean curvature H of a minimizer by an application of the Gauss-Bonnet-Chern theorem with boundary.
		By Ònearly round,Ó we mean that rescalings to unit volume are smoothly close to the Euclidean sphere of unit volume.
		Even in dimension three, mean curvature flow may develop singularities, and worse the possibility of perimeter of higher topological type spoils the Gauss-Bonnet estimate on dP/dV as in 4.4(1).
	www.lehigh.edu /dlj0/yesterday/courses/IsopSph7=28=00.doc (3467 words)

	Constant scalar curvature hypersurfaces with spherical boundary in Euclidean space, Luis J. Alías, J. Miguel ...
		It is still an open question whether a compact embedded hypersurface in the Euclidean space with constant mean curvature and spherical boundary is necessarily a hyperplanar ball or a spherical cap, even in the simplest case of a compact constant mean curvature surface in $\mathbb{R}^3$ bounded by a circle.
		In this paper we prove that this is true for the case of the scalar curvature.
		Specifically we prove that the only compact embedded hypersurfaces in the Euclidean space with constant scalar curvature and spherical boundary are the hyperplanar round balls (with zero scalar curvature) and the spherical caps (with positive constant scalar curvature).
	projecteuclid.org /Dienst/UI/1.0/Display/euclid.rmi/1051544244 (252 words)

	CS 294 Project 1
		The map from colors to rough curvature values is displayed to the left of each row, with values ranging from large negative numbers through zero to large positive numbers.
		In addition to coloring vertices according to various types of curvature, line textures showing the normal, tangent direction, or principle curvature directions at each point on the surface can be applied via keyboard commands.
		Next, the min and max curvatures at the vertex are averaged, and the 'curvature contribution' is stored.
	www.eecs.berkeley.edu /~steve0/cs294proj1 (1450 words)

	Mathematics Colloquium (Site not responding. Last check: 2007-10-16)
		Recovering the Shape of a Surface from the Mean Curvature.
		It is conjectured that the shape of a closed Riemannian surface is uniquely determined by the mean curvature.
		Our rigidity results are based on the analysis of the foliations of curvature lines of the isometries and their relation to the principal stretch foliations of $D$.
	math.dartmouth.edu /~colloq/w96/kamberov.html (294 words)

	[No title]
		During this past decade the theory of formation of singularities was developed for Ricci flow and mean curvature flow, which has had a large impact on other geometric flows.
		Mean curvature and inverse mean curvature flows have been used to solve long-standing problems in general relativity (such as the Riemannian Penrose conjecture).
		For example, surface tension along moving interfaces in fluids and materials is proportional to mean curvature: mean curvature flow and affine mean curvature flow are useful for morphological image processing.
	www.ipam.ucla.edu /programs/gf2004 (715 words)

	The Generalized Monkey Saddle
		The hyperbolic paraboloid and monkey saddle are really special cases of this general class of surfaces.
		It is somewhat easier to compute a general formula for Gaussian and mean curvature from the polar form of the generalized monkey saddle (see second equations below.) Because there are no complex numbers involved, differentiation is straightforward.
		Mean curvature can also be computed from the polar representation:
	www.math.hmc.edu /faculty/gu/math142/mellon/curves_and_surfaces/surfaces/genmonkey.html (156 words)

	Diffusion-Generated Motion by Mean Curvature (Site not responding. Last check: 2007-10-16)
		In two dimensions, curvature is the rate of change in the direction of a curve per unit length.
		Mean curvature is a way of extending the idea of curvature to three or more dimensions.
		By defining the speed of the surface equal to its mean curvature, we arrive at an extension for three or more dimensions.
	www.cs.ubc.ca /nest/scv/group-info/alumnis/ruuth/intro.html (234 words)

	TOTAL MEAN CURVATURE AND SUBMANIFOLDS OF FINITE TYPE
		The purpose of this book is to introduce the reader to two interesting topics in geometry which have developed over the last fifteen years, namely, total mean curvature and submanifolds of finite type.
		The theory of total mean curvature is the study of the integral of the n-th power of the mean curvature of a compact n-dimensional submanifold in a Euclidean m-space and its applications to other branches of mathematics.
		The relation of total mean curvature to analysis, geometry and topology are discussed in detail.
	www.worldscibooks.com /mathematics/0065.html (179 words)

	The Eight Surface (Site not responding. Last check: 2007-10-16)
		The surface comes to a point at its very center, which causes problems in the VRML version of the surface (you may receive a series of warnings when you view it).
		Note also that the mean curvature becomes infinitely large at the center of the surface, and the surface can thus not be colored by mean curvature.
		Note that, although the numerator of the mean curvature function contains the imaginary number i, the function is always real or infinite because cos(4v)-1 is less than or equal to zero, so its square root is always imaginary or zero.
	www.math.hmc.edu /~gu/math142/mellon/curves_and_surfaces/surfaces/eightsurf.html (139 words)

	Notes on surfaces (Site not responding. Last check: 2007-10-16)
		Then the Gaussian curvature of a surface is k1k2 and the Mean* curvature of a surface is (k1+k2)/2.
		An interesting result of constant Gaussian curvature is that an inelastic net that fits the surface in one area fits it everywhere.
		What it really means is that if you follow any "straight" line far enough you will come back to where you started.
	www.uib.no /People/nfytn/notes.htm (486 words)

	VTK: vtkCurvatures Class Reference (Site not responding. Last check: 2007-10-16)
		NB: dihedral_angle is the ORIENTED angle between -PI and PI, this means that the surface is assumed to be orientable the computation creates the orientation The units of Mean Curvature are [1/m]
		Excepting spherical and planar surfaces which have equal principal curvatures, the curvature at a point on a surface varies with the direction one "sets off" from the point.
		If a given mesh produces curvatures of opposite senses then the flag InvertMeanCurvature can be set and the Curvature reported by the Mean calculation will be inverted.
	www.vtk.org /doc/nightly/html/classvtkCurvatures.html (471 words)

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