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Topic: Catenoid

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	Catenoid Soap Films
		For a separation/diameter ratio under 0.66274, there are a thick-necked stable catenoid and a thin-necked unstable catenoid.
		For a catenoid, the desired volume can easily be calculated, but one could also adjust the volume until the pressure is zero.
		Having a disk across one of the catenoids spoils the symmetry around the diagonals, the film becomes tangent to the insides of the angles, although that level of detail is far too small to be seen in this image.
	www.susqu.edu /facstaff/b/brakke/evolver/examples/cat/catenoids.html (445 words)

	Catenoid - Biocrawler (Site not responding. Last check: )
		A catenoid is a three-dimensional shape made by rotating a catenary curve around the x axis.
		A catenoid is one of several types of minimal surfaces.
		A physical model of a catenoid can be formed by dipping 2 circles into a soap solution, (popping any film in the center of the circle), and slowly drawing the circles apart.
	www.biocrawler.com /encyclopedia/Catenoid (100 words)

	catenoid
		The catenoid was first described by Leonhard Euler in 1740 and is the oldest known minimal surface (a shape of least area when bounded by a given closed space).
		It is the minimal surface connecting two parallel circles of unequal diameter on the same axis; soap film between two circular rings takes this form (see bubbles).
		The catenoid is the only known minimal surface that is also a surface of revolution, and is one of only four minimal surfaces that have the topological properties of being unbounded, embedded, and non-periodic; the others are the simple plane, the helicoid, and Costa's surface.
	www.daviddarling.info /encyclopedia/C/catenoid.html (140 words)

	Catenoid
		A catenoid is a three-dimensional shape made by rotating a catenary curve around an axis.
		A surface swept out by a line rotating with uniform velocity around an axis perpendicular to the line and simultaneously moving along the axis with uniform velocity is called a helicoid.
		A catenoid and a helicoid are representative minimal surfaces.
	www.teachersparadise.com /ency/en/wikipedia/c/ca/catenoid.html (181 words)

	Catenoid
		These catenoids are the first explicitly known discrete versions of a complete minimal surface beside the trivial plane which are solutions of the variational problem for the discrete surface area.
		The catenoids of the 4-parameter family are embedded and complete discrete minimal catenoids with dihedral rotational symmetry and planar meridians.
		The planar trapezoids of the catenoid may be triangulated independently of each other, that is, one can choose either of the two choices for the diagonal edge across each planar trapezoid.
	www.eg-models.de /models/Surfaces/Minimal_Surfaces/2000.05.002/_preview.html (494 words)

	NationMaster - Encyclopedia: Catenoid
		Verrill Minimal Surface In mathematics, a minimal surface is a surface with a mean curvature of zero.
		A circle, in Euclidean geometry, is the set of all points at a fixed distance, called the radius, from a fixed point, the centre.
		In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output.
	www.nationmaster.com /encyclopedia/Catenoid (312 words)

	Discr. Catenoid (Site not responding. Last check: )
		The catenoids are complete surfaces without boundary although the applet restricts to displaying a finite portion.
		The smooth catenoid is the simplest example of a minimal surface, and it is used as sample surface in many constructions.
		This so-called discrete catenoid is a critical point of the area functional applied to the set of piecewise linear, triangulated surfaces.
	www.vismath.de /vgp/content/minimal/PaCatenoid.html (192 words)

	Platonic Solids
		For example, the catenoid curve is a surface of revolution of the trigonometric cosh() function and can be demonstrated by the suspended film between 2 facing circles.
		A large catenoid canopy is lightweight and stable in high winds.
		A solid catenoid is stronger than a flat or convex surface.
	mcs.une.edu.au /~cwatson7/PlatonicSolids.htm (631 words)

	The Catenoid
		The surface of revolution of a catenary, called a catenoid, has the property that its mean curvature is everywhere zero; we say that it is a minimal surface.
		Although the catenoid looks substantially like the hyperboloid, there are substantial differences in their values away from the plane z = 0, as well as in their properties.
		The catenoid also has the fascinating property that it can be deformed into a helicoid in such a way that every surface along the way is a minimal surface which is locally isometric to the helicoid.
	www.math.hmc.edu /~gu/curves_and_surfaces/surfaces/catenoid.html (114 words)

	Surface Evolver catenoid example
		The catenoid is the minimal surface formed between two rings not too far apart.
		The initial radius given is the minimum for which a catenoid can exist for the given separation of the rings.
		Exercise for the reader: Get the Surface Evolver to display an unstable catenoid by declaring the catenoid facets to be the boundary of a body, and adjusting the body volume with the b command to get zero pressure.
	www.susqu.edu /brakke/evolver/workshop/doc/catenoid.htm (585 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal
		A catenoid is a three-dimensional shape made by rotating a catenary curve around the
		A physical model of a catenoid can be formed by dipping two circles into a soap solution and slowly drawing the circles apart.
		In other words, one can make a continuous and isometric deformation of a catenoid to a helicoid such that every member of the deformation family is minimal.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=catenoid (167 words)

	Fluctuating Vesicles
		Noting that the shape of the necks was similar to that of a catenoid, we tried to understand the energetics of parallel membranes connected by catenoidal passages.
		The matching of a catenoidal shape with an almost flat membrane is a very complicated problem, which can only be solved either numerically or with approximations.
		Our model of a periodic stack of connected membranes can be extended to more than two membranes: the electrostatics analogy remains valid, but one has to distinguish between positive and negative charges, depending on the direction of the neck with respect to the reference membrane.
	www.chem.ucla.edu /~michalet/papers/lvmh/e-fluctuations.html (895 words)

	vgp.discrete.catenoid (JavaView Reference Manual)
		Discrete catenoid is a critical point for the discrete area functional, and its profile curve is given by an explicit formula.
		Project for computing discrete catenoid from explicit formulas.
		Control panel of project discrete catenoid shows info panels of catenoid and harmonic map.
	www.javaview.de /doc/reference/vgp/discrete/catenoid/package-summary.html (52 words)

	Deformation of the Catenoid to the Helicoid
		We also can make a surface swept out by a line rotating with uniform velocity around an axis perpendicular to the line and simultaneously moving along the axis with uniform velocity.
		A catenoid and a helicoid are representative minimal surfaces.
		It is very interesting that we can make a continuous and isometric deformation of a catenoid to a helicoid such that every member of the deformation family is minimal.
	mathmuse.sci.ibaraki.ac.jp /deform/DeformationE.html (135 words)

	GRAPE - Minimal Surface Library - catenoid (Site not responding. Last check: )
		The catenoid is the eldest known minimal surface.
		It is the only minimal surface which is also a surface of rotation.
		The input file name of this surface is catenoid.am.
	www-sfb256.iam.uni-bonn.de /grape/EXAMPLES/AMANDUS/catenoid.html (68 words)

	Andart: Knotted minimal surfaces
		While playing around with surface evolver I got the idea of attaching a catenoid handle to it, producing this rather neat surface.
		An obvious generalisation is to add a catenoid to the top too, producing a three-ended minimal surface.
		The width of the neck between them depends on their distance like in an ordinary catenoid; beyond a certain point the minmal surface pinches off.
	www.aleph.se /andart/archives/2006/11/knotted_minimal_surfaces.html (192 words)

	[No title]
		Like the catenary, the catenoid is able to maintain its stable shape, because the tension exerted at every point is equalized by the changing effect of the principle of least action.
		Or, inversely, the shape of the catenary is the unique form which equalizes the tension exerted along the chain, by the effort of the chain to support its weight under the effect of gravity.
		Following Gauss, Riemann recognized that in the type of least action physical manifold exemplified by the catenoid or Gauss's potential surfaces, the curves of maximum and minimum curvature are harmonically related, which means that their mutual curvatures change at the same rate, in perpendicular directions.
	www.wlym.com /antidummies/part60.html (3651 words)

	New Surfaces of Konrad Polthier
		This catenoid is a complete discrete minimal surface whose vertices are given by explicit formulas.
		This surface belongs to a 4-parameter family of complete discrete catenoids from which a 2-parameter subfamily interpolates the smooth catenoid.
		The discrete catenoids are the first explicitly known discrete versions of a complete minimal surface beside the trivial plane which are critical points of the variational problem for minimizing the discrete surface area.
	page.mi.fu-berlin.de /polthier/images/NewSurfaces/NewSurfaces.html (688 words)

	Catenoid Applet
		curve associated with catenoid to axis of rotation.
		the parameters of the discrete catenoid are automatically chosen such that its vertices interpolate the smooth catenoid with radius 1.
		the discretization of the catenoid is zig-zag, i.e.
	www.zib.de /vgp/content/minimal/PaCatenoid_Applet.html (78 words)

	diff geo images page
		On the left side, the catenoid, the surface of revolution with profile curve the catenary {v, cosh[v]}, is shown with a
		This curve obtained by mapping a spiral centered at the origin of the u-v plane to the catenoid by the parameterization catenoid[u,v] = {cos[u]*cosh[v],sin[u]cosh[v],v}.
		This contrasts with the previous example of the Gauss map of the catenoid.
	www.math.uiowa.edu /~wseaman/DGImage53100.htm (3122 words)

	Alteration
		The bidenoid is a well-controlled modification of the catenoid by Hermann Karcher.
		Luquesio P. Jorge of Ceara Federal University (Brazil) and William M. Meeks III of the University of Massachusetts at Amherst (USA), were the first who-starting from the imagined shape of the surface to be newly constructed-undertook the controlled alteration of a minimal surface out to infinity.
		From this approximate representation, along with Osserman' s theory, they concluded in 1980 with the Weierstraß representation formula for a new minimal surface: the trinoid, in which three ends are fitted together.
	page.mi.fu-berlin.de /polthier/booklet/alteration.html (385 words)

	Catenoid - Qwika
		Catenoid A catenoid A catenoid is a three-dimensional shape made by...
		Catenoid Catenoid - surface, formed by the rotation the catenary...
		The helicoid and the catenoid are parts of a family of helicoid-catenoid minimal surfaces.
	www.qwika.com /find/Catenoid (297 words)

	Soap Film
		For any perturbation, the change in area is proportioal to the square of the strength of the perturbation (this is a consequence of the zero mean curvature of the surface.)
		To understand this figure, it is useful to consider the one-parameter family of soap films that can span the two rings (of radius r = 1) at a fixed value of z*:
		Only at a = 0.1934 and a = 0.8828 are these films true catenoids, since 1/z* acosh(1/a) = 1/a for these values of a when z* = 0.45.
	oak.ucc.nau.edu /jws8/dpgraph/soap_film.html (1023 words)

	3D-XplorMath Surface Gallery (Site not responding. Last check: )
		About the Schoen No-Go Theorem H. Karcher R. Schoen's characterization of the catenoid says: Any finite total curvature complete embedded minimal surface which has TWO ends, is the catenoid.
		The fundamental piece is similar to that of the catenoid fence, except that the handle does not go outward to the neighbouring catenoid but goes inward to meet its other half.
		However a gap remains and as one tries to close it (by morphing with the modulus, aa, of the underlying rectangular Torus) the surface degenerates to look almost like two catenoids which move farther apart as one tries to close the gap.
	rsp.math.brandeis.edu /3d-xplormath/Surface/schoen_no-go_thm/schoen_no-go_thm.html (180 words)

	David Hoffman colloquium talk (Site not responding. Last check: )
		Not counting the plane, the first minimal surfaces (those that locally minimize area) were discovered in the 18th Century: the catenoid by Euler in 1744 and the helicoid by Meusnier in 1776.
		These clasical minimal surfaces are topologically the sphere with a point (two for the catenoid) removed.
		For instance, it is now known that if such a surface has more than one topological end, it must have finite total curvature, and each end must be asymptotic to either a plane or a catenoid.
	math.berkeley.edu /~ribet/Colloquium/hoffman.html (191 words)

	Maple Program
		You can look at a helicoid, a catenoid, and the intermediate surfaces.
		Helicat(x,y,0) is the helicoid, Helicat(x,y,1) is the catenoid, and Helicat(x,y,t) wheret is between
		We recommend using the patch style and one of the lighting schemes.
	personal.cityu.edu.hk /~ma4527/cg_gallery.htm (359 words)

	GRAPE - Minimal Surface Library - deformation of the catenoid into the fournoid (Site not responding. Last check: )
		If the parameter varys, the fournoid of Jorge-Meeks is deformed into the catenoid.
		The "hexanoid" (minimal surfaces which has all symmetries of a cube and six catenoid ends) is contained in this family (see the third postscript file).
		The deformation can be continued by turning the four horizontal catenoid ends inwards (see the last postscript file).
	www.iam.uni-bonn.de /grape/EXAMPLES/AMANDUS/cat4noid.html (108 words)

	Catenoid
		catenoid is a valid word in this word list.
		The word "catenoid" uses 8 letters: A C D E I N O T.
		Words within catenoid not shown as it has more than seven letters.
	www.morewords.com /word/catenoid (216 words)

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