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# Topic: Hyperbolic spiral

 Xah: Special Plane Curves: Archimedean Spiral Hyperbolic spiral is also called reciprocal spiral, because it is the inverse curve of Archemedes' spiral, with inversion center at the center. The inverse curve of Archimedes' spiral with inversion circle of radius 1 at pole is the reciprocal spiral. The inverse curve of Fermat's spiral with inversion circle of radius 1 at pole is the lituus. www.xahlee.org /SpecialPlaneCurves_dir/ArchimedeanSpiral_dir/archimedeanSpiral.html   (390 words)

 Archimedean spiral - Wikipedia, the free encyclopedia This Archimedean spiral is distinguished from the logarithmic spiral by the fact that successive turnings of the spiral have a constant separation distance (equal to 2πb if θ is measured in radians), while in a logarithmic spiral these distances form a geometric progression. Virtually all static spirals appearing in nature are logarithmic spirals, not Archimedean ones. Many dynamic spirals (such as the Parker spiral of the solar wind, or the pattern made by a St. en.wikipedia.org /wiki/Archimedean_spiral   (267 words)

 Spiral - TheBestLinks.com - Longitude, Mathematics, Polar coordinates, Vangelis, ...   (Site not responding. Last check: 2007-10-22) In mathematics, a spiral is a curve which turns around some central point or axis, getting progressively closer to or farther from it, depending on which way you follow the curve. A two-dimensional spiral may be described using polar coordinates by saying that r is a continuous monotonic function of θ. A spherical spiral is the curve on a sphere traced by a ship traveling from one pole to the other while keeping a fixed angle (but not a right angle) with respect to the meridians of longitude (cf. www.thebestlinks.com /Spiral.html   (291 words)

 Hyperbolic The hyperbolic spiral originated with Pierre Varignon in 1704. The roulette of the pole of a hyperbolic spiral rolling on a straight line is a tractrix. inverts to the spiral of Archimedes r = a www-history.mcs.st-and.ac.uk /history/Curves/Hyperbolic.html   (144 words)

 SPIRAL - LoveToKnow Article on SPIRAL   (Site not responding. Last check: 2007-10-22) A group of spirals are included in the parabolic spirals given by the equation r=aO~ the more important are the Archimedean spiral, r =aO (fig. Its p r equation is p=rf/~ (a2+rf), and the angle between the radius vector and the tangent equals the vector angle. The second, called hyperbolic on account of the analogy of its equation (polar) to that (Cartesian) of a hyperbola between the asymptotes, is the inverse of the Archimedean. www.1911encyclopedia.org /S/SP/SPIRAL.htm   (491 words)

 Law of spiral symmetry transformation The spiral numbers in the sunflower discs are in direct dependence on their "age", i.e. The hyperbolic geometry of the inanimate nature is based on the classical hyperbolic functions, the essence of which is the e-number being one of the most important numerical constants of mathematics. The hyperbolic geometry of the animate nature is based on the hyperbolic Fibonacci and Lucas functions, which essence is expressed with the Golden Section, the fundamental numerical constant of the animate nature. www.goldenmuseum.com /1607SpiralSymmetry_engl.html   (834 words)

 [No title] It is a special case of the spiral of Archimedes." hypocycloid::usage="t->hypocycloid[a,b][t] is the parametrized curve that is traced out by a point P on the circumference of a circle (of radius b) rolling inside another circle (of radius a). A nephroid is a special case of an epicycloid." nielsenspiral::usage="t->nielsenspiral[a][t] is Nielsen's spiral of radius a." parabola::usage="t->parabola[a][t] is the vertical parabola with vertex at the origin, focus at {0,a} and directrix t->{t,-a}. The case m=0 is the cissoid t->cissoid[a/2][t] and the case m=1 is the strophoid t->strophoid[a][t]." tanhspiral::usage="t->tanhspiral[a][t] is the hyperbolic tangent spiral of radius a." tractrix::usage="t->tractrix[a][t] is a tractrix. xtsunxet.usc.es /cordefg/curves2d.m   (1099 words)

 Archimedean spiral   (Site not responding. Last check: 2007-10-22) This spiral is a generalization of Archimedes' spiral (a=1), named to the Greek Archimedes (225 BC). The inverse of the spiral with a constant a is an Archimedean spiral with a constant -a. An Archimedean spiral with parameter a has as polar inverse an Archimedean spiral with parameter -a: so the lituus and Fermat's spiral are inversely related, and also the hyperbolic and the Archimedes' spiral. www.2dcurves.com /spiral/spirala.html   (101 words)

 [No title]   (Site not responding. Last check: 2007-10-22) Hyperbolic tilings are used in many of Escher's works to create the effect of a figure getting smaller and smaller while preserving angles. Hyperbolic tilings appealed to Escher because he liked the idea of similar (not congruent in Euclidean geometry) figures. The lines are not hyperbolic, but they do bear a striking resemblence to lines in the Poincar\'{e} model. www.yellowpigs.net /classes/escher   (1756 words)

 hyperbolic spiral   (Site not responding. Last check: 2007-10-22) This spiral is a kind of Archimedean spiral. The hyperbolic name reflects the relation between radius and angle being the same as the x,y relation for the hyperbola. When a hyperbolic spiral is rolling over a line, then the path of the pole forms a tractrix, being a roulette of the spiral. www.2dcurves.com /spiral/spiralh.html   (87 words)

 [No title]   (Site not responding. Last check: 2007-10-22) In accordance with another aspect of the present seal assembly, a hyperbolic elastic seal is located in an annular channel within the valve cage. The hyperbolic elastic seal has an internal cavity that is expanded into the valve body or the valve cage by fluid pressure within the valve to create a pressure-assisted seal. A single hyperbolic elastic seal 56 is positioned in an annular channel 55 of the valve cage 50 and is engaged by a second internal flange 57 formed within the valve bonnet 15. www.wipo.int /cgi-pct/guest/getbykey5?KEY=04/106790.041209&ELEMENT_SET=DECL   (3727 words)

 Christian Lange This is the hyperbolic function rotated around the Y-axis rapresenting the three dimensional acustic law. On this hyperbolic cone you can create a hyperbolic spiral when the rate of curvature and climbing are equal. This means that the hyperbolic cone or the three dimensional acustic law contains the laws of planets movements and he represent an universal law od nature. www.sectioaurea.com /sectioaurea/the_golden_angle.htm   (833 words)

 FanFiction.Net : Dictionary & Thesaurus   (Site not responding. Last check: 2007-10-22) (Rhet.) Relating to, containing, or of the nature of, hyperbole; exaggerating or diminishing beyond the fact; exceeding the truth; as, an hyperbolical expression. [1913 Webster] Hyperbolic functions (Math.), certain functions which have relations to the hyperbola corresponding to those which sines, cosines, tangents, etc., have to the circle; and hence, called hyperbolic sines, hyperbolic cosines, etc. Hyperbolic logarithm. Hyperbolic spiral (Math.), a spiral curve, the law of which is, that the distance from the pole to the generating point varies inversely as the angle swept over by the radius vector. www.fanfiction.net /dictionary.php?word=hyperbolic   (167 words)

 Spiral This spiral was studied by Archimedes in about 225 BC in a work On Spirals. Archimedes was able to work out the lengths of various tangents to the spiral. The cam consists of one arch of the spiral above the x-axis together with its reflection in the x-axis. www-groups.dcs.st-and.ac.uk /~history/Curves/Spiral.html   (160 words)

 Essay2 In general, a spiral is a curve on a plane which winds about a fixed point. In the Spiral of Archimedes, each turn of the spiral is the same distance from the previous turn. The Logarithmic Spiral is also called the Equiangular Spiral because for any point on the spiral, the angle between the tangent line and the line to the origin is always the same. jwilson.coe.uga.edu /EMT668/EMAT6680.2000/Umberger/EMAT6690smu/Essay2smu/Essay2smu.html   (536 words)

 [No title] It is used in the definitions of s->alysoid[a,b,c][s] and s->alysoidnds[smin,smax][a,b,c][s]." archimedesspiral::usage="t->archimedesspiral[n,a][t] is the spiral of Archimedes of radius a and degree n. The hyperbolic spiral was first studied by Varigon in 1704 and Cotes in 1722. It is a trumpet-shaped curve that is the locus of points P such that the square of distance of P from the origin is inversely proportional to the angle theta that p makes with the horizontal axis. www.ma.umist.ac.uk /kd/mmaprogs/CURVES.m   (5260 words)

 [No title] As the cluster spirals in, successive outer shells of the mass distribution are peeled off by the tidal field. After the cluster core begins to disintegrate, it continues to spiral inwards until torques from the polarization cloud and the material lost previously become ineffective, that is, until the debris has spread in angle by $\Delta\phi\sim\pi$. Independent of the rate of cluster accretion, the recent arrival in the GC of a cluster with a large number of massive stars would have profoundly changed the physical conditions in the central parsec and may in fact be responsible for the present lack of activity of Sgr A$^\ast$. www.aoc.nrao.edu /~gcnews/gcnews/Vol.12/Ortwin.Gerhard@unibas.ch_Hestars.txt   (1971 words)

 SPIRAL - Online Information article about SPIRAL   (Site not responding. Last check: 2007-10-22) positive pedals, inverse, polar reciprocal and evolutes are all equal equiangular spirals. group of spirals are included in the " parabolic spirals " given by the equation r=aO'; the more important are the Archimedean spiral, r =aO (fig. If B =o, we have p = r-%i A, and the locus is the equiangular spiral. encyclopedia.jrank.org /SOU_STE/SPIRAL.html   (585 words)

 Article - Viktor Schauberger - Water Ways of Life Nature frequently uses the hyperbolic spiral which is externally centripetal and internally moves towards the centre, such spirally movements are found in the spiral nebula of galaxies in space, in the natural flow of water, blood and sap. He concluded that the water passing through the trouts' gills created a hyperbolic centripetal spiral movement, this combined with the trace elements within the gills, and changed the passing water into 'juvenile' water which by its new characteristic reacted with the surrounding stream water creating a secondary system of water circulation around the trouts' bodies. Viktor also observed that like the trout, birds move through air using hyperbolic, centripetal spiral movements, when air flows through their feathers during flight, a strong counter circulation of updraft is created carrying the birds forward and upwards. www.lightnet.co.uk /frontier/viktor.htm   (5275 words)

 Hyperbolic oxygen chambers Information   (Site not responding. Last check: 2007-10-22) Hyperbolic Oxygen Chambers are great for when you're looking to get better at hyperbolic oxygen chambers for selfish purposes. If you need help locating hyperbolic oxygen chambers then you've come to the right place because we have all the hyperbolic oxygen chambers you could want. portable hyperbaric, hyperbolic, hyperbolic therapy, hyperbolic treatment, oxygen therapy, sechrist, mild hyperbaric, mild... oxygen.11intershare7.info /oxygen-bar-accessory/hyperbolic-oxygen-chambers.html   (301 words)

 writeup 11-Polar Equations The spirals described on shells, and called concho-spirals, are such as would result from winding plane logarithmic spirals on cones. As a grows larger, the spiral becomes larger and r is greater. The distance between successive coils of a logarithmic spiral is not constant as with the spirals of Archimedes. jwilson.coe.uga.edu /EMT668/EMT668.Folders.F97/Anderson/writeup11/writeup11.html   (1020 words)

 Spiral Wishing Well Fund Raising Machines, Over \$200 million worth of coins tossed in for charities.   (Site not responding. Last check: 2007-10-22) The coins make about 40 revolutions around the vortex funnel, each one spiraling closer to the center, hugging the near-vertical side-walls at eye-blurring speeds before finally spinning out of sight into a locked receptacle 20 to 30 seconds later. Spiral Wishing Wells do not require valuable employee time like most corporate fund raising programs. Some stores use a Spiral Wishing Well as their "official" fund-raising method and avoid the pressure from the dozens of charities who want to solicit their customers. www.divnick.com /wishing_well.htm   (711 words)

 [No title] The bending measure for the logarithmic spiral \ induces a multiple of Lebesgue measure on \[Gamma]. In the \ hyperbolic plane these become lines of constant curvature, at a fixed \ distance from the geodesic \[Gamma].\n\nFinding the bending measure (some \ constant multiple of the Lebesgue measure) along \[Gamma] will enable us to \ compute any type of measurement of bending which interests us. One \ endpoint of \[Alpha] lies in \[CapitalOmega] on the spiral which is a \ geodesic in the hyperbolic metric on \[CapitalOmega]. www.maths.warwick.ac.uk /~dbae/papers/EMM/spiral5.nb   (1467 words)

 Drift Bifurcations of Relative Equilibria and Transitions of Spiral Waves - Ashwin, Melbourne (ResearchIndex) In particular, we investigate how the drift for a parametrized family of normally hyperbolic relative equilibria can change character at what we call a `drift bifurcation'. To do this, we use results of Arnold to analyze parametrized families of elements in the Lie algebra of the symmetry group. Drift bifurcations of relative equilibria and transitions of spiral waves. citeseer.ist.psu.edu /ashwin98drift.html   (615 words)

 The Geometry Junkyard: All Topics This shape, constructed by inscribing circular arcs in a spiral tiling of squares, resembles but is not quite the same as a logarithmic spiral. Ok, renaming a hyperbolic paraboloid a "helical right triangle" and saying that it's "a revolutionary foundation for new knowledge" seems a little cranky but there are some interesting pictures of shapes formed by compounds of these saddles. Packing circles in the hyperbolic plane, Java animation by Kevin Pilgrim illustrating the effects of changing radii in the hyperbolic plane. www.ics.uci.edu /~eppstein/junkyard/all.html   (9742 words)

 Neilos - Centre for Implosion Research Vortex Energiser Frequently Asked Questions Earlier this century the Austrian physicist Walter Schauberger, the son of Viktor Schauberger, recognised that the harmonic spiral space curve, which is based on the harmonic series, is the perfect model of our Universe. Projecting this spiral onto a hyperbolic cone produces the 3D harmonic spiral space curve of the Vortex Energiser. It is a set of three copper tubes, wound in a hyperbolic spiral shape, filled with your imploded water. www.neilos.org /vortexfaq.htm   (2266 words)

 Prime Number Spiral - Free Download   (Site not responding. Last check: 2007-10-22) We start with the central point and arrange the positive integers in a spiral fashion (anticlockwise). There is a tendency for the prime numbers to occur on diagonal lines, however far out into the spiral one goes. It also (a) allows coloring of the prime numbers in various ways, (b) displays a random arrangement to compare with the primes and (c) shows how square and triangular numbers are located on the spiral. windows-education-mathematics.funhosts.com /prime-number-spiral.html   (399 words)

 hypertek, hyperbolic funnel - funnel, hyperbolic funnel, coin funnel, school, vortex, fundraising, school funnel, ... Each coin behaves differently because of the velocity by which it is spun onto the funnels surface; the direction and the weight cause it to travel different paths. When owning a Hyperbolic Funnel you will enjoy the income that is generated. The Hyperbolic Funnel is a perfect income generator and will continue to make money for many years after it is placed on site. www.funnelworks.com   (647 words)

 [No title]   (Site not responding. Last check: 2007-10-22) ] A function whose value is equal to the reciprocal of the value of the hyperbolic sine. ] A function whose value is equal to the value of the hyperbolic cosine divided by the value of the hyperbolic sine. ] A function of pairs of points within a unit circle, where the interior of this circle is a conformal or projective representation of a hyperbolic space used in transmission line theory and waveguide analysis. www.accessscience.com /Dictionary/H/H26/DictH26.html   (1855 words)

 Math This hyperboloid and “hyperbolic beach ball” were ray traced and textured using inverse transformations (the “pull back” method). This can be accomplished by applying a special homography to a single spiral. A pseudosphere is a hyperbolic surface with constant negative curvature. www.bugman123.com /Math/Math.html   (2082 words)

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