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Topic: Hyperboloid of two sheets

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Hyperboloid

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	Hyperboloid - Wikipedia, the free encyclopedia
		Alternatively, a hyperboloid of two sheets of axis AB is obtained as the set of points P such that AP-BP is a constant, AP being the distance between A and P. A and B are then called the foci of the hyperboloid.
		A hyperboloid of two sheets can be obtained by revolving a hyperbola around its focal axis.
		A hyperboloid of one sheet is a ruled surface; if it is a hyperboloid of revolution, it can also be obtained by revolving a line about a skew line.
	en.wikipedia.org /wiki/Hyperboloid (178 words)

	hyperboloid
		A quadratic surface of which there are two basic forms: a hyperboloid of one sheet, generated by spinning a hyperbola around its conjugate axis, and a hyperboloid of two sheets produced by rotating a hyperbola about its transverse axis.
		This fact enables a close approximation to a hyperboloid to be made in the form of a string model.
		Hyperboloids are seen in the manmade world in the form of cooling towers at power stations and, most strikingly, in the shape of the McDonnell Planetarium in St. Louis, Missouri (shown above).
	www.daviddarling.info /encyclopedia/H/hyperboloid.html (327 words)

	MA 225 Class 11 (Site not responding. Last check: 2007-11-06)
		The hyperboloid of one sheet and its horizontal traces
		The hyperboloid of one sheet and its vertical traces
		The hyperboloid of two sheets and its horizontal traces
	math.bu.edu /people/paul/225/spring01/class11.html (45 words)

	The Hyperboloid
		The hyperboloid is a well-known quadratic surface that comes in two varieties: the hyperboloid of one sheet (above) and the hyperboloid of two sheets (below).
		It is interesting to note that the hyperboloid of one sheet is asymptotic to a cone, as shown below.
		The hyperboloid of one sheet is also a ruled surface.
	math.hmc.edu /faculty/gu/curves_and_surfaces/surfaces/hyperboloid.html (129 words)

	MTH015 Hyperboloids: Family of Hyperbolids (Site not responding. Last check: 2007-11-06)
		, the level surface is a hyperboloid of two sheets.
		, the surface is a hyperboloid of one sheet.
		The change from two sheets to one sheet is a significant change in shape of the level surface.
	www.math.union.edu /~dpvc/math/courses/MTH015/Hyperboloids/welcome.html (160 words)

	Index: Quadratic Surfaces (1-14)
		A quadratic (or quadric) surface is a surface in three-space defined by an equation of degree two.
		Thus it is the three-dimensional analog of a conic section, which is a curve in two-space defined by an equation of degree two.
		A hyperbola gives rise to different surfaces of revolution, depending on whether it is revolved about the conjugate axis (which passes between the two branches of the hyperbola) or the transverse axis (which crosses the two branches).
	math.arizona.edu /~models/QuadraticSurfaces (241 words)

	SlicePlot3D: Hyperboloids (Site not responding. Last check: 2007-11-06)
		Hyperboloids all have hyperbolic traces in two directions, and elliptical traces in the third direction.
		Here again are two pictures of a pair of models displaying the same hyperboloidal surface.
		The blue and white slices show the region between the two sheets, and the pink and white slices show the two regions within each sheet.
	people.cornellcollege.edu /pnichols/miscmath/sliceforms/hyperboloids.html (184 words)

	IGQS: Hyperboloid of One Sheet
		The hyperboloid of one sheet is possibly the most complicated of all the quadric surfaces.
		For one thing, its equation is very similar to that of a hyperboloid of two sheets, which is confusing.
		A hyperboloid of one sheet looks an awful lot like a cooling tower at the Springfield Nuclear Power Plant.
	www.math.umn.edu /~rogness/quadrics/hyper1.shtml (489 words)

	Animated Demonstrations for Multivariable Calculus
		Like the hyperboloid of one sheet, the traces in certain planes are hyperbolas that change orientation after a point.
		Given a real-valued function of two real variables, one way to understand the nature of its surface is to make a contour map, a plot in the plane of various level curves.
		whose level surface, f (x, y, z) = k, is a hyperboloid of two sheets when k < 0, a hyperboloid of one sheet when k > 0, and an elliptic cone when k = 0.
	archives.math.utk.edu /ICTCM/EP-10/C9/html/paper.html (1055 words)

	Surface -- Hyperboloid of Two Sheets (Site not responding. Last check: 2007-11-06)
		The axis of rotation of a hyperboloid of two sheets is coincident with the x' -axis.
		values are given is not important--the volume exists at all points between the two.
		Example of a two-sheet hyperboloid HYP2 with a-, b-, and c-parameters of.5,.35, and.35 respectively.
	www-phys.llnl.gov /N_Div/COG/Manual/Surface/HyperboloidOfTwoSheets.html (149 words)

	IGQS: Hyperboloid of Two Sheets
		The hyperboloid of two sheets looks an awful lot like two (elliptic) paraboloids facing each other.
		It's a complicated surface, mainly because it comes in two pieces.
		If they exist, then it's a hyperboloid of one sheet.
	www.math.umn.edu /~rogness/quadrics/hyper2.shtml (309 words)

	DocM's Calculus III WebSite
		Thus, we suspect that this surface is be a hyperboloid of two sheets.
		(All variables positive would have given an ellipsoid; only one negative would be a hyperboloid of one sheet.) Since x is the positive variable, the hyperboloid will be symmetric with respect to the x-axis (i.e., it will "lie along" the x-axis).
		Since two kinds of traces are hyperbolas and one kind is circles, we know that we have a hyperboloid of two sheets -- just as we thought when we examined the equation.
	www.math.ou.edu /~tjmurphy/Teaching/courseSupplements/2433quadric/2433quadric.html (450 words)

	Quadratic Surfaces (1-14) (Site not responding. Last check: 2007-11-06)
		An elliptic hyperboloid is a solid or surface, some of whose plane sections are hyperbolas, while others are ellipses.
		If no elliptical cross-section goes around the conjugate axis of any hyperbolic cross-section, the hyperboloid has two sheets.
		The other sheet would be an inverted model placed symmetrically above this one.
	math.arizona.edu /~models/QuadraticSurfaces/source/5.html (89 words)

	Quadric Surfaces -- Identification (Site not responding. Last check: 2007-11-06)
		The name of the shape is derived from the shape of the graph in two dimensions.
		A signal of this shape is the presence of two minus signs.
		Equations for elliptic paraboloids and hyperbolic paraboloids contain two quadratic terms and one linear term.
	amath.colorado.edu /student/chartier/Calc3/Quad/quadric.html (321 words)

	16 Quadrics
		A surface defined by an algebraic equation of degree two is called a quadric.
		Figure 2: Left: hyperboloid of one sheet (3).
		When two of the semiaxes are the same, we can also write the area of the ellipsoid in closed form.
	www.geom.uiuc.edu /docs/reference/CRC-formulas/node61.html (363 words)

	lesson09.html
		To any equation in two variables f(x,y)=0, you can assign the set of all points in the plane whose coordinates (x,y) satisfy this equation.
		ellipsoid, hyperboloid of one sheet, hyperboloid of two sheets, elliptic paraboloid, hyperbolic paraboloid, conic surface, elliptic cylinder, hyperbolic cylinder, parabolic cylinder, 2 intersecting planes, 2 parallel planes, 1 plane, 1 point and empty place.
		The first 9 types are called non-degenerate, and they represent genuine curvilinear surfaces; the last 5 types are made up of straight-line objects.
	www.botik.ru /~duzhin/maple/lesson091.html (923 words)

	Quadric Surfaces
		Quadric surfaces are sometimes difficult to draw in two dimensions.
		The general form for the equation of the hyperboloid of one sheet is
		The general form for the equation of the hyperboloid of two sheets is
	www2.norwich.edu /frey/LiveGraphics3D/QuadricSurfaces.htm (228 words)

	Common Transversals and Tangents (Site not responding. Last check: 2007-11-06)
		Our approach is to fix two (skew) lines and a quadric, and then describe the common transversals and tangents in this case.
		For example, the picture on the right shows the same sphere, common transversals and tangents, and red line, and it shows a hyperboloid of two sheets tangent to every one of the common transversals and tangent to the sphere.
		Briefly, fixing the two lines and the first quadric, the space of second quadrics for which there are infinitely many common transversals and tangents is a curve in the P
	www.math.tamu.edu /~sottile/pages/2l2s (293 words)

	Hyperboloid (Site not responding. Last check: 2007-11-06)
		Support Function of the hyperboloid of one sheet
		Support Function of the hyperboloid of two sheets
		Gray, A. ``The Hyperboloid of Revolution.'' §18.5 in
	www.itu.dk /edu/documentation/mathworks/math/math/h/h434.htm (40 words)

	Topic: hyperboloid (Site not responding. Last check: 2007-11-06)
		The surface created by rotating a hyperbola about one of its axes of symmetry.
		Rotation about the conjugate axis produces a hyperboloid of one sheet.
		Rotation about the transverse axis creates a hyperboloid of two sheets.
	www.elko.k12.nv.us /webapps/vmd/full/h/hyperboloid2.htm (36 words)

	Animated algebraic surfaces (Site not responding. Last check: 2007-11-06)
		Deformation of hyperboloid, which deforms from one sheet into two sheets, with a cone at the middle of the deformation (when the two sheets touch each other).
		The same sheet with a zoom, to let you peek into its inside.
		If you still find it hard to imaging what happens inside this strange bottle for the above surface, here is it again, but with two caps cut off.
	wims.unice.fr /~xiao/gallery (312 words)

	[No title]
		C is called the generating curve (or directrix) of the cylinder, and the parallel lines are called rulings.
		The equation of a cylinder whose rulings are parallel to one of the coordinate axes contains only the variable corresponding to the other two axes.
		Quadric Surface The equation of a quadric surface in space is a second-degree equation of the form EMBED Equation.DSMT4 There are six basic quadric surfaces Ellipsoid Hyperboloid of One Sheet Hyperboloid of Two Sheets Elliptic Cone Elliptic Paraboloid Hyperbolic Paraboloid See pages 765-766 for standard form for each quadric surface and plane traces.
	www.pc.maricopa.edu /Mathematics/faculty/jsousa/Documents/MAT240/Ch10Notes/10.6Notes.doc (261 words)

	Math 22 S04
		It will cover what we did in Chapters 12 and 13.
		You will get a review sheet with practice problems.
		i) Moving from one sheet through a cone to two two sheets.
	count.ucsc.edu /~bauerle/math_22_s04.htm (228 words)

	Math 281 Animations
		Animation to illustrate the limiting process involved in the definition of the derivative: control h using the slider
		a graph of a function with two critical points
		this example has two critical points---one local max and one saddle
	www.grossmont.net /jennyvandeneynden/archive/M281_links.htm (823 words)

	Equation of a Hyperboloid of Two Sheets - EquationSheet
		Equation of a Hyperboloid of Two Sheets - EquationSheet
		We have a new Forum, please start posting away!
		Pasting the above "text" into MathType 5.0 will enter the above equation and make it possible to edit, save, and use in applications such as MS Word.
	www.equationsheet.com /eqninfo/Equation-344.html (70 words)

	Quadrics for the HP-41
		XEQ "QUADRIC" the HP-41 displays the successive contents of register R04 (which tend to zero)
		and eventually displays " 2-SH HYP" (Q) is an hyperboloid of 2 sheets
		-The signs may be + - - ; - + - or - - + for a 2-sheeted hyperboloid.
	www.hpmuseum.org /software/41/41qdrics.htm (593 words)

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