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Topic: Paraboloid

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Roman surface

Quadric

Ruled surface

Hyperboloid

Parabolic reflector

Parabola

Mean curvature

Minimal surface

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Hyperbola

Curvature

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Sphere

Riemann surface

Cartesian coordinates

	PlanetMath: generatrices of hyperbolic paraboloid
		It is easily seen that any point of the hyperbolic paraboloid is passed through by exactly two generatrices, one from the family (1) and the other from family (2).
		This is a parametric presentation of the hyperbolic paraboloid.
		This is version 7 of generatrices of hyperbolic paraboloid, born on 2007-09-11, modified 2007-09-15.
	planetmath.org /encyclopedia/GeneratricesOfHyperbolicParaboloid.html (300 words)

	A Simple Technique of Fabrication of Paraboloidal Concentrators
		Paraboloidal concentrators have been used for various applications, from cooking[1] and driving hot-air operated pumps for lifting water[2], to providing power for space-craft[3] through a mercury-vapor driven electric generator.
		The paraboloid was mounted on a horizontally on a framework such that it could be rotated around a central axis.
		The paraboloid was rotated and the extreme points on the graph illuminated by the reflected laser spot were marked off.
	ashokk_3.tripod.com /srinivasan.htm (1373 words)

	Inverted, doubly-curved umbrella, hyperbolic paraboloid shells with structurally integrated upper diaphragm - Patent ...
		A building unit which comprises a column adapted to be fitted to a foundation and supporting a plurality of identical, doubly-curved umbrella, hyperbolic paraboloid shells inverted so that their top edges form a plane figure, and a continuous, horizontal member integrated with said edges.
		Hyperbolic paraboloid roofs have aesthetic, structural and economic advantages for many purposes, as for example, inexpensive, attractive, large area structures free of intermediate columns, as for example, hangers, auditoriums and gymnasiums.
		Prior to the present invention, roofs constructed from hyperbolic paraboloid shells could not be used for any other purpose since a depth of approximately 20 percent of the span was required for the roof structure to become economical and efficient.
	www.freepatentsonline.com /4137679.html (1340 words)

	Paraboloid (Site not responding. Last check: )
		In mathematics, a paraboloid is a quadric, a type of surface in three dimensions, described by the equation:
		The hyperbolic paraboloid is shaped like a saddle and can have a critical point called a saddle point.
		With a = b an elliptic paraboloid is a paraboloid of revolution: a surface obtained by revolving a parabola around its axis.
	www.wikimoz.org /wiki/en/wikipedia/p/pa/paraboloid.html (166 words)

	Paraboloid (Site not responding. Last check: )
		The elliptic paraboloid is shaped like a cup and can have a maximum or minimum point.
		The hyperbolic paraboloid is shaped like a saddle and can have a critical point called a saddle point.
		With a = b an elliptic paraboloid is a paraboloid of revolution : a surface obtained by revolving a parabola around its axis.
	www.serebella.com /encyclopedia/article-Paraboloid.html (224 words)

	GameDev.net -- Dual Paraboloid Mapping in the Vertex Shader
		The name itself begs to be investigated, and the paraboloid mapping could be applied to environment maps as well as shadow maps so it seemed like a good idea to look into it.
		The original motivation behind paraboloid mapping was presented in the paper "View-Independent Environment Maps" by Heidrich and Seidel.
		Figure 2 shows both hemispheres reflecting rays into a parallel stream away from the point P. This is the key to the paraboloid mapping — environment mapping encodes the light surrounding a point into a grid of values to be stored in a map while shadow mapping does the same for depth values.
	www.gamedev.net /columns/hardcore/dualparaboloid (643 words)

	paraboloid - Search Results - MSN Encarta
		In mathematics, a paraboloid is a quadric, a type of surface in three dimensions, described by the equation: (elliptical paraboloid, opens upward)
		A parabolic reflector, known as a parabolic dish or a parabolic mirror, is a reflective device, commonly formed in the shape of a paraboloid of revolution
		The surface of revolution of the parabola which is the shape used in the reflectors of automobile headlights (Steinhaus 1999, p.
	encarta.msn.com /encnet/refpages/search.aspx?q=paraboloid (151 words)

	paraboloid - definition by dict.die.net
		paraboloid n : a surface having parabolic sections parallel to a single coordinate axis and elliptic sections perpendicular to that axis
		parabolo["i]de.] (Geom.) The solid generated by the rotation of a parabola about its axis; any surface of the second order whose sections by planes parallel to a given line are parabolas.
		Note: The term paraboloid has sometimes been applied also to the parabolas of the higher orders.
	dict.die.net /paraboloid (78 words)

	Deployable offset paraboloid antenna - Patent 4030103
		An antenna with a paraboloid shape which is fed from a feed in a position offset from the focal point of the paraboloid, said antenna deployable from a retractable stowed position to an extended deployed position, comprising:
		The preformed contour of the mesh is an offset paraboloid, and it is held in this offset paraboloid shape by attachment to the preformed parabolic shape of the radial ribs.
		A deployable offset paraboloid antenna i.e., an antenna with a paraboloid shape which is fed from a feed in a position offset from the focal point, of the present invention is generally indicated at 2 in FIGS.
	www.freepatentsonline.com /4030103.html (1295 words)

	Prefabricated hyperbolic paraboloid roof - Patent Review 3958375
		A hyperbolic paraboloid roof shell section made in prefabricated portions each of which has a joining edge parallel to the joining edge of another prefabricated portion and is adapted to extend between two beams of the roof frame.
		A hyperbolic paraboloid roof is a configuration which has esthetic, structural and economic advantages for many purposes as, for example, inexpensive, attractive, large area structures free of intermediate columns, as, for example, hangers, auditoriums and gymnasiums.
		Prior to the present invention, hyperbolic paraboloid roof structures were customarily assembled piece by piece on the site by cutting, fitting and assembling the various portions of the roof.
	www.wikipatents.com /3958375.html (3778 words)

	Omnipelagos.com ~ article "Paraboloid" (Site not responding. Last check: )
		A daily life example of a hyperbolic paraboloid is the shape of a Pringles potato chip.
		The elliptic paraboloid, parametrized simply as has Gaussian curvature and mean curvature which are both always positive, have their maximum at the origin, become smaller as a point on the surface moves further away from the origin, and tend asymptotically to zero as the said point moves infinitely away from the origin.
		The two paraboloidal functions and are harmonic conjugates, and together form the analytic function which is the analytic continuation of the parabolic function
	www.omnipelagos.com /entry?n=paraboloid (382 words)

	Spherical Mirror Null
		As before a visible error is a defect in the paraboloid surface and thus the wavefront undergoing the knife edge test.
		The distance from the paraboloid to knife edge falls between the radius and focal length.
		A null test of a F/5 paraboloid can be made at focus using a flat to return the projected light back to the paraboloid for a return to focus.
	www.bbastrodesigns.com /waineo.html (1217 words)

	GameDev.net - Dual Paraboloid Mapping in the Vertex Shader
		Now that we have generated the paraboloid map, we need to be able to access it to put it to good use.
		The process of calculating the texture coordinates to access the paraboloid maps is quite similar to the method used to generate the map.
		This is because we apply the paraboloid warping to the vertices of a model when generating the map.
	www.gamedev.net /reference/articles/article2308.asp (2496 words)

	Ivars Peterson's MathTrek - Floating Bodies
		Two factors determine the way in which the paraboloid floats: its shape (the ratio of its height to the radius of its circular base) and its density (specific gravity).
		One intriguing situation involves a paraboloid with a relative density of 0.9.
		Rorres's paraboloid model suggests that, if an iceberg with a roughly paraboloidal shape melts so that it gets narrower, the block of ice can suddenly tip over once the height-to-radius ratio reaches a critical value and its base dips into the water.
	www.maa.org /mathland/mathtrek_08_23_04.html (937 words)

	Surfaces in 3 Space
		This is the paraboloid, the shape of the parabolic mirror in your flashlight, and in the Space Telescope.
		This plane intersects the paraboloid in a parabola, which reflects the light ray towards the focus as described earlier.
		Thus the paraboloid has the same optical properties as the parabola, but in 3 dimensions.
	www.mathreference.com /la-qf,3d.html (991 words)

	Reflector antennas
		Energy which is not directed toward the paraboloid has a wide-beam characteristic which will destroy the narrow pattern of the parabolic reflector.
		The truncated paraboloid may be used in target height-finding systems if the reflector is rotated 90 degrees, as shown in view (B) of figure 3-5.
		The cylindrical paraboloid reflector is fed either by a linear array of dipoles, a slit in the side of a waveguide, or by a thin waveguide radiator.
	www.tpub.com /neets/book11/46a.htm (1029 words)

	On Floating Bodies (Book II)
		A profile view of the paraboloid is shown together with a plot of the energy arm vs. the tilt angle.
		An oblique view of the paraboloid is shown together with a plot of the energy arm vs. the tilt angle.
		The top one is a sphere atop a cylinder illustrating the so-called “tombstone” theorem that Archimedes considered his most important, that the volume and surface area of a sphere are both two-thirds, respectively, of the volume and surface area of its circumscribing cylinder.
	www.math.nyu.edu /~crorres/Archimedes/Floating/floating.html (1409 words)

	Paraboloid Reflector
		A reflector which is a portion of the surface of a paraboloid, generally of revolution.
		Specialty definitions using "paraboloid reflector": antenna with cylindrical parabolic reflector, antenna with paraboloidal reflector &diams; parabolic aerial, parabolic antenna, paraboloidal reflector.
		Parabolreflektor (dish, dish reflector, parabolic reflector, paraboloidal deflector, paraboloidal dish reflector, paraboloidal reflector), Paraboloidreflektor (parabolic reflector, paraboloidal reflector).
	www.websters-online-dictionary.org /definition/english/pa/paraboloid+reflector.html (216 words)

	Lgarde - Inflatable space structural systems - Papers
		The off-axis paraboloids may be approximated by a circular section along any cut by a plane perpendicular to the plane of the ellipse.
		The functional shape of the deviation as a function of position across the paraboloid has been shown to be the same for 1 and 3 meter diameter paraboloids.
		The implication is that the paraboloid slope error will be related to the absolute error by a constant divided by the paraboloid diameter.
	www.lgarde.com /papers/scaling.htm (2448 words)

	Paraboloid Reflector
		A reflector which is a portion of the surface of a paraboloid, generally of revolution.
		Specialty definitions using "paraboloid reflector": antenna with cylindrical parabolic reflector, antenna with paraboloidal reflector &diams; parabolic aerial, parabolic antenna, paraboloidal reflector.
		Parabolreflektor (dish, dish reflector, parabolic reflector, paraboloidal deflector, paraboloidal dish reflector, paraboloidal reflector), Paraboloidreflektor (parabolic reflector, paraboloidal reflector).
	www.websters-online-dictionary.com /definition/english/pa/paraboloid+reflector.html (216 words)

	December 1998 Newsletter
		I usually allow the edge of the mirror to be considered perfect and alter the remaining glass to a paraboloid in reference to the edge.
		The sphere has the same radius of curvature throughout its surface, but the paraboloid has a shorter radius of curvature near the center, longer towards the edge, and a continuously smooth rate of change of curvature as you move from the center to the edge.
		If the mirror is a paraboloid, its radius of curvature changes from longer on the outer regions to shorter on the inner regions.
	astro.sci.uop.edu /~sas/Newsletter/TTN_Parabolizing2.html (779 words)

	Muse: Advanced Floating
		He was especially fascinated by paraboloids, objects that look like rocket nosecones.
		Archimedes looked only at cases where the paraboloid's flat base is completely above or below the liquid's surface, because that is all the math of his day could handle.
		A paraboloid almost as dense as the liquid would float point down, with its flat base sticking up out of the liquid a little.
	www.sciencenewsforkids.org /pages/puzzlezone/muse/muse0105.asp (442 words)

	Alignment and Assembly
		The optical alignment sensor used for this purpose illuminates the paraboloid from near its focus; light passes through the paraboloid, reflects from the flat, and returns to a quad cell detector near the paraboloid focus.
		dependence upon azimuth angle; proper axial and lateral alignment is achieved when the paraboloid focus is coincident with the center of a sphere which is part of the optical reference assembly mounted on the center aperture plate.
		The next paraboloid (P3) then will be aligned so that its focus is coincident with that of P4; then H4 will be added so that its focus is coincident with that of H6, and so forth through P1, H3, and H1.
	cxc.harvard.edu /newsletters/news_02/subsection3_4_5.html (650 words)

	Rating Mirrors Weighted RMS
		I started out looking at this while puzzling over the problem of how to turn my polished f/3.9 mirror into a paraboloid, which by a naive calculation was going to require something like 6 waves of total correction (at the surface) to make.
		All of this analysis assumes the target is a paraboloid of freely adjustable focal length.
		In the latter case I suspect what's happening is that the use of local polishers is resulting in lots of low amplitude, small scale errors, with a few much larger departures thrown in.
	www.bbastrodesigns.com /weighted-rms/paraboloid.html (831 words)

	11.2.2.1 The Mathematics of Dual-Paraboloid Maps
		The principle that underlies paraboloid maps is the same principle that underlies a parabolic lens or satellite dish.
		Figure 74 shows how two paraboloids can focus the entire environment surrounding a point into two images.
		Unlike the sphere mapping approach that the encodes the entire environment in a single texture, the dual-paraboloid mapping scheme requires two textures to store the environment, one texture for the ``front'' environment and another texture for the ``back'' environment.
	www.opengl.org /resources/code/samples/sig99/advanced99/notes/node185.html (266 words)

	paraboloids.html
		The contours are ellipses in the xy plane; that's why it's called an elliptic paraboloid.
		It's hard to see unless you rotate this graph, but this surface only opens up, never down, but there is one direction in which the height of the surface stays constant.
		This paraboloid is balanced perfectly between the elliptic and hyperbolic cases.
	www-math.bgsu.edu /~zirbel/calc3/paraboloids.html (153 words)

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