
	Where results make sense

Topic: Hyperbolic coordinates

Related Topics

Hyperbola

Geometric mean

In the News (Wed 16 Dec 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	Models of the Hyperbolic Plane
		In the Klein model of the hyperbolic plane, the "plane" is the unit disk; in other words, the interior of the Euclidean unit circle.
		The "distance" between two points with coordinates z and w in the complex plane is: 2 arctanh{(z-w)/(1-wz)}, where z is the complex conjugate of z.
		Thus as in the Klein model, the "distance" to the boundary of the disk is infinite, and postulate 2 holds.
	www.geom.uiuc.edu /docs/forum/hype/model.html (519 words)

	Hyperbolic coordinates - Wikipedia, the free encyclopedia
		affords the hyperbolic geometry structure to Q that is erected on HP by hyperbolic motions.
		The hyperbolic lines in Q are rays from the origin or petal-shaped curves leaving and re-entering the origin.
		Quantifying exchange rate fluctuation through hyperbolic angle provides an objective, symmetric, and consistent measure.The quantity Δu is the length of the left-right shift in the hyperbolic motion view of the currency fluctuation.
	en.wikipedia.org /wiki/Hyperbolic_coordinates (275 words)

	History of the upper-air observation at the World Ocean, VII International Congress
		To neutralize the effects of ship rolling upon the measured angular coordinates of the radiosonde, the radar antenna column was mounted on a platform of stabilized post.
		Accumulating these coordinates step by step one may find the horizontal projections of the radiosonde trajectory to retrieve both wind speed and direction in different layers of the atmosphere.
		No linear coordinates are then needed, and the balloon flight level is found using the barometric formula, substituting into the actually measured pressure.
	vitiaz.ru /congress/en/thesis/140.html (2008 words)

	Hyperbolic Space
		Further explanation of the ramifications of the hyperbolic metric can be found in one of the many mathematical textbooks which cover hyperbolic geometry [Mar75] [Wol45].
		Although we could simply place euclidean objects into hyperbolic 3-space and move them around according to the rules of hyperbolic geometry, we would not be exploiting the exponential amount of room available in hyperbolic space.
		Distinct hyperbolic coordinates which are too far from the origin will be projected so close to the surface of the unit ball that there are not enough bits to distinguish between their euclidean coordinates.
	graphics.stanford.edu /papers/h3/html/node4.htm (968 words)

	Trigonometry Functions (Site not responding. Last check: 2007-11-06)
		Converts polar coordinates to rectangular coordinates, where the input is expressed in terms of angle and radius.
		Converts rectangular coordinates to polar coordinates, where the result is expressed in terms of angle and radius.
		Hyperbolic arc-sine; the reverse of the hyperbolic sine.
	www.dreamcalc.com /dc3help/userguide_trigonometry_functions.htm (533 words)

	NonEuclid: X-Y Coordinate System (Site not responding. Last check: 2007-11-06)
		In the Euclidean Geometry, Cartesian coordinate system, the coordinates of any point in the first quadrant are defined to be the ordered pair, (x,y) where x is the perpendicular distance from the point to the x-axis, and y is the perpendicular distance from the point to the y-axis.
		This might make it seem like the point (1,1) is undefined in Hyperbolic Geometry; however, the point (1,1) does exist, and it is located at point P. The length of the perpendicular from P to the x-axis is 1.0 units.
		This coordinate system sets up a one-to-one correspondence between all of the points in the Hyperbolic plane and all ordered pairs (x,y) where x and y are real numbers.
	cs.unm.edu /~joel/NonEuclid/coordinate.html (438 words)

	CMFT 6 (2006), 223--242 (Site not responding. Last check: 2007-11-06)
		In a simply connected hyperbolic region hyperbolic polar coordinates possess global Euclidean properties similar to those of hyperbolic polar coordinates about the origin in the unit disk if and only if the region is convex.
		For example, the Euclidean distance between travelers moving at unit hyperbolic speed along distinct hyperbolic geodesic rays emanating from a common point is increasing if and only if the region is convex.
		A consequence of this is that the ends of distinct hyperbolic geodesic rays in a convex region cannot be too close.
	www.heldermann.de /CMF/CMF06/CMF061/cmf06013.htm (158 words)

	H3: Laying Out Large Directed Graphs in 3D Hyperbolic Space
		The first hyperbolic visualization system described in the information visualization literature was the 2D hyperbolic tree browser from Xerox PARC [LR94].
		Hyperbolic methods are very effective at providing global overviews and displaying many nodes at once.
		In our hyperbolic variation of the cone tree algorithm, we draw a dense neighborhood around the focus of interest by laying out nodes on a spherical cap covering the mouth of the cone rather than its perimeter.
	graphics.stanford.edu /papers/h3/html.nosplit (6925 words)

	Body
		This second paper and tape hyperbolic surface (used in classes and workshops for the next 11 years) was the one that Daina witnessed in use.
		This is the usual upper half plane model of the hyperbolic plane thought of as a map of the hyperbolic plane in the same way that we use planar maps of the spherical surface of the earth.
		Thus, we have established that the annular hyperbolic plane is the same as the usual upper half plane model of the hyperbolic plane.
	www.math.cornell.edu /~dwh/papers/crochet/crochet.html (3801 words)

	MathDL \| Special Relativity and Conic Sections
		, the hyperbolic interval is "time-like", and the common intersection is an ellipse.
		, the hyperbolic interval is "space-like", and the common intersection is a hyperbola.
		The time coordinates that he measures for points in that wavefront are not constant, but, as we saw, the curve forms, in the Euclidean structure that the plane inherits from the hyperbolic geometry, an ellipse.
	mathdl.maa.org /mathDL/4/?pa=content&sa=viewDocument&nodeId=375&pf=1 (7227 words)

	Cartesian coordinate system Summary
		In mathematics, the Cartesian coordinate system is used to uniquely determine each point in the plane through two numbers, usually called the x-coordinate and the y-coordinate of the point.
		To define the coordinates, two perpendicular directed lines (the x-axis or abscissa and the y-axis or ordinate), are specified, as well as the unit length, which is marked off on the two axes (see Figure 1).
		Using the Cartesian coordinate system geometric shapes (such as curves) can be described by algebraic equations, namely equations satisfied by the coordinates of the points lying on the shape.
	www.bookrags.com /Cartesian_coordinate_system (3956 words)

	Another Symmetry
		Under an inversion of the coordinate system about the surface at an absolute interval R from the origin (which may be chosen arbitrarily), each event located on a given ray through the origin is moved to another point on that ray such that its absolute interval from the origin is changed from s to R
		Thus the hyperbolic surfaces outside of R are mapped to surfaces inside R, and vice versa.
		Thus inversion of a system of inertial coordinates yields a system of coordinates in which all the null intervals are preserved.
	www.mathpages.com /rr/s1-08/1-08.htm (2017 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Because of this, drawing a hyperbolic circle through a given point with a given center is different from drawing a Euclidean circle through a given point with a given center, and it is not clear (to me) that the former construction can be reduced to the latter (or to any other Euclidean construction).
		One might also consider scaling down a putative hyperbolic trisection to the infinitesimal scale to obtain a Euclidean trisection (which we know to be impossible), but the hyperbolic plane admits no scaling transformations (unlike the Euclidean plane), and anyway the limit of a valid construction may not be a valid construction (degeneracies may arise).
		Since this interpretation fails for other geometries such as the hyperbolic plane or the sphere, it would be nice to understand the problem in a way that would extend naturally to other geometries.
	www.math.niu.edu /~rusin/known-math/98/hyperbolic_tris (457 words)

	Geometries
		In hyperbolic and elliptic geometry there are two such midpoints (points of equal distance to the defining points).
		In the Euclidean view the horizon of hyperbolic geometry is shown as a thin circle.
		Hyperbolic lines are represented by circular arcs that cross the boundary of the disk at right angles.
	www.unl.edu /tcweb/fowler/Documentation/Texts/Geometries.html (1124 words)

	NonEuclid: 9: X-Y Coordinate System
		In the Euclidean geometry, Cartesian coordinate system, the coordinates of any point in the first quadrant are defined to be the ordered pair, (x,y) where x is the perpendicular distance from the point to the x-axis, and y is the perpendicular distance from the point to the y-axis.
		This might make it seem like the point (1,1) is undefined in hyperbolic geometry; however, the point (1,1) does exist, and it is located at point P. The length of the perpendicular from P to the x-axis is 1.0 units.
		This coordinate system sets up a one-to-one correspondence between all of the points in the Hyperbolic plane and all ordered pairs (x,y) where x and y are real numbers.
	www.cs.unm.edu /~joel/NonEuclid/coordinate.html (439 words)

	MAC 2312, Calculus with Analytic Geometry II
		Topics include inverse trigonometric functions, hyperbolic and inverse hyperbolic functions, areas, volumes, centroids, work, fluid pressure, length of arc, trigonometric integrals, integration techniques, polar coordinates, indeterminate forms, improper integrals, infinite series, plane curves, parametric equations, conic sections, and computer work.
		The student will be able to apply knowledge of the inverse trigonometric functions and of the hyperbolic and inverse hyperbolic functions to evaluate integrals.
		The student will be able to define basic terms related to polar coordinates and be able to plot points in a polar coordinate system.
	www.spcollege.edu /clw/math_science/courses/MAC2312.htm (818 words)

	Coordinate Geometry in the Hyperbolic Plane
		There are other coordinate systems that can be used, but this is standard.
		Every point has a unique ordered pair of Lobachevsky coordinates , and, conversely, every ordered pair of real numbers is tha pair of Lobachevsky coordinates for some unique point.
		Thus, a line does not have a linear equation in Lobachevsky coordinates, and a linear equation does not necessarily describe a line.
	www.math.uncc.edu /~droyster/math3181/notes/hyprgeom/node74.html (619 words)

	A Unified Algebraic Framework for Classical Geometry (Site not responding. Last check: 2007-11-06)
		The strongest evidence given by the founders for its consistency is the duality between hyperbolic and spherical trigonometries.
		Because stereographic projections are conformal maps, the conformal groups of n-dimensional Euclidean, spherical, and hyperbolic spaces are isometric to each other, and are all isometric to the group of isometries of hyperbolic (n+1)-space, according to observations of Klein [K1872], [K1872].
		Geometric Algebra was applied to hyperbolic geometry by H. Li in [L97], stimulated by Iversen's book [I92] on the algebraic treatment of hyperbolic geometry and by the paper of Hestenes and Ziegler [HZ91] on projective geometry with Geometric Algebra.
	modelingnts.la.asu.edu /html/UAFCG.html (2056 words)

	Elliptical Coordinates
		These so-called elliptical coordinates are based upon the four trigonometric functions: the hyperbolic or circular cosine or sine.
		The polar coordinates (r, phi) may be defined in terms of the Cartesian coordinates (x, y) by the set of two equations
		Substitute for a and b and employ the identity for the hyperbolic cosine, to obtain
	www.rism.com /Trig/elliptical_coordinates.htm (3051 words)

	[No title]
		If the hyperbolic interval is positive, it means that each event lies in the exterior of the other's light cone, and the events are not causally connected.
		This means that it must be possible to synchronize their clocks when they are pairwise stationary with respect to one another, and that each measures the speed of light to be one unit distance per unit time.
		Of course, the hyperbolic orthogonal bisector of a segment does not appear perpendicular to the segment, as it would be in the Euclidean metric.
	www.mathwright.com /library6/focas4.htm (887 words)

	Chapter 5
		A paper model of the hyperbolic plane may be constructed as follows: Cut out many identical annular ("annulus" is the region between two concentric circles) strips as in Figure 5.2.
		The hyperbolic plane can be approximately constructed by using heptagons (7-sided) surrounded by seven hexagons and two hexagons and one heptagon together around each vertex.
		The hyperbolic soccer ball construction is related to the {3,7} construction in the sense that if a neighborhood of each vertex in the {3,7} construction is replaced by a heptagon then the remaining portion of each triangle is a hexagon
	www.math.cornell.edu /~dwh/books/eg00/supplements/AHPmodel/index.html (1566 words)

	DeLatte, David A. (1991-01-01) Nonstationary normal forms for anosov diffeomorphisms and hyperbolic skew products. ... (Site not responding. Last check: 2007-11-06)
		Birkhoff considered the formal power series of f at the origin and showed that there is a formal change of coordinates, h, which satisfies [...] where g is of the form [...] and where [...] is the (real) eigenvalue of the linear part [...] and [...] is a formal series with [...].
		One may introduce hyperbolic coordinates [...] where [...] the hyperbolic angle, describes the position on the hyperbola xy = c.
		These coordinates give a clear understanding of the local behavior of f; specifically f shifts points along these local hyperbolae.
	etd.caltech.edu /etd/available/etd-03292004-155247 (603 words)

	Behind the scenes
		In general, a point with spatial coordinates (x,y,z) is identified with the point (x/z,y/z,1) of the original plane.
		Finding the coordinates for the points of intersection is nothing else but solving a quadratic equation.
		Between 1815 and 1824, independently from each other, these three people - who are today considered as the discoverers of hyperbolic geometry - came to the point at which they declared their system as free from contradictions, just because they could not find any.
	www.unl.edu /tcweb/fowler/Documentation/Texts/Mathematics.html (5045 words)

	Nonlinear hyperbolic equations
		The key ingredient, though, is a sharp new bilinear estimate of Strichartz type for solutions to the wave equation; this is presented in the Appendix.
		Strichartz estimates for the wave equation in the hyperbolic space and global existence for the semilinear wave equation, Trans.
		It is proved here that the full Strichartz estimates hold for second order hyperbolic operators whose coefficients have two derivatives in $L^1(L^\infty)$.
	math.berkeley.edu /~tataru/nlw.html (1286 words)

	[No title]
		In fact, there is "exponential room" in 3-D hyperbolic space, in the sense that a circle's circumference is an exponential growth function of its radius, and similarly a sphere's area is an exponential growth function of its radius.
		I follow exactly Munzner's two-pass layout algorithm in 3-D hyperbolic space to derive 3-D hyperbolic coordinates for all nodes in an input graph, and then I use a well known conversion calculation to convert from 3-D hyperbolic coordinates to 3-D Euclidean coordinates.
		Then, use the rotation and translation operations specified by the (r, \phi, \theta) coordinate of the child node to obtain the transformation matrix for moving the child node from the origin in the parent's local coordinate system to the final position in the parent's local coordinate system.
	www.cs.utexas.edu /~kliu/graphics/project4 (3982 words)

	libnova: Hyperbolic Motion
		Calculate the time of rise, set and transit for a body with a hyperbolic orbit.
		Functions relating to the Hyperbolic motion of bodies.
		Calculate a bodies equatorial coordinates for the given julian day.
	libnova.sourceforge.net /group__hyperbolic.html (261 words)

	Hyperbolic Equations
		These coordinates are important for at least two reasons.
		The idiosyncratic aspect of the simple string is that these two parts do not change their shape as they propagate along the string.
		To justify these claims it is neccessary to construct this unique solution to a general second order linear hyperbolic differential equation.
	www.math.ohio-state.edu /~gerlach/math/BVtypset/node144.html (488 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		] Coordinates for two particles used in studying the helium atom; they comprise the distance between the two particles, the sum of the distances of the particles from the origin, and the difference of the distances of the particles from the origin.
		] 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 (1230 words)

	Coordinate Systems for Conformal Cerebellar Flat Maps
		a canonical surface-based coordinate system can be imposed by specifying 2 points (such as anatomical or functional landmarks) on the hyperbolic map or 3 points on the spherical map [2, 11]
		Figure 8: The coordinate lines from the hyperbolic map in Figure 7 as they appear on the Euclidean (left) and spherical maps (right).
		three landmarks required for a canonical coordinate system: 1) midline base of precentral fissure was used as the north pole; 2) midline base of horizontal fissure was used as the south pole; 3) midline base of primary fissure was used as an equatorial point
	www.math.fsu.edu /~mhurdal/posters/hbm00coord.html (1224 words)

Try your search on: Qwika (all wikis)