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Topic: Hyperbolic angle

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Hyperbolic sector

Hyperbolic function

Hyperbola

Conic section

	Java Gallery: Hyperbolic Triangles (Site not responding. Last check: 2007-10-18)
		One of the most surprising facts in hyperbolic geometry is that there is an upper limit to the possible area a triangle can have, even though there is not an upper limit to the lengths of the sides of the triangle.
		In hyperbolic geometry, the sum of the angles of a triangle is always less than 180 degrees (PI radians).
		Thus the area is proportional to the defect, with the above proportionality constant (k is 1 for the model of the hyperbolic plane we're using).
	www.geom.uiuc.edu /java/triangle-area (275 words)

	Hyperbolic angle - Wikipedia, the free encyclopedia
		A hyperbolic angle in standard position is the angle at (0,0) between the ray to (1,1) and the ray to (x,1/x) where x > 1.
		The magnitude of the hyperbolic angle is the area of the corresponding hyperbolic sector which is log x.
		The hyperbolic functions sinh, cosh, and tanh use the hyperbolic angle as their independent variable since their values may be premised on analogies to circular trigonometric functions when the hyperbolic angle defines a hyperbolic triangle.
	en.wikipedia.org /wiki/Hyperbolic_angle (178 words)

	Hyperbolic triangle - Wikipedia, the free encyclopedia
		In hyperbolic geometry, a hyperbolic triangle is a figure in a hyperbolic plane, analogous to a triangle in Euclidean geometry.
		The vertices are usually considered to be in the hyperbolic plane, but sometimes one considers some of the vertices to be at the circle at infinity.
		A property of hyperbolic triangles in hyperbolic geometry is that the sum of the angles of the triangle is less than 180°.
	en.wikipedia.org /wiki/Hyperbolic_triangle (278 words)

	3 H3: 3D Hyperbolic Quasi-Hierarchical Graphs
		Hyperbolic geometry is one of the non-Euclidean geometries developed at the turn of the century.
		Hyperbolic and spherical geometry are the only two non-Euclidean geometries that are homogeneous and have isotropic distance metrics: in other words, there is a uniform, meaningful, and continuous concept of the distance between two points.
		Rigid hyperbolic objects that translate along the hyperbola will appear in the projection to grow to a maximum when at the pole and then shrink, and after translating infinitely up the hyperbola their projection will be infinitely close to the line segment border.
	graphics.stanford.edu /papers/munzner_thesis/html/node8.html (15446 words)

	Trigonometric Functions
		Returns the hyperbolic angle in radians for the given cosine.
		Returns the hyperbolic angle in radians for the given sine.
		The arctangent is the angle in radians between the lines drawn from {0,0} to {x-value, 0} and from {0, 0} to {x- value, y-value}.
	gobe.com /gp/help/Spreadsheet/SSFtrig.htm (219 words)

	m331test3f03sol.htm (Site not responding. Last check: 2007-10-18)
		For example the angle sum of a triangle is always less than 180 degrees in hyperbolic, but the angle sum in neutral geometry is always less than or equal to 180 degrees.
		In hyperbolic geometry, if two parallel lines are cut by a transversal, the alternate interior angles are congruent and the corresponding angles are congruent.
		In hyperbolic geometry, the altitude to the hypotenuse of a right triangle forms two triangles, each of which is similar to the original triangle and to each other.
	www.elmhurst.edu /~jonj/MTH331/M331T3sol/m331test3f03sol.htm (489 words)

	MAM2003 Essay: Hyperbolic Art and the Poster Pattern
		For example, all the triangles in Figure 1 are the same hyperbolic size, as are all the fl fish (or white fish) of Figure 2.
		In the Poincaré model, equidistant curves are represented by circular arcs that intersect the bounding circle in acute (or obtuse) angles.
		For any acute angle and hyperbolic line, there are two equidistant curves ("branches"), one on each side of the line, making that angle with the bounding circle [Gr1].
	www.mathaware.org /mam/03/essay1.html (2110 words)

	Hyperbolic Geometry - Triangles (Site not responding. Last check: 2007-10-18)
		The Poincare half-plane model is conformal, which means that hyperbolic angles in the Poincare half-plane model are exactly the same as the Euclidean angles (with the angles between two intersecting circles being the angle between their tangent lines at the point of intersection.
		Since the hyperbolic line segments are (usually) curved, the angles of a hyperbolic triangle add up to strictly less than 180 degrees.
		Observe that the smaller the traingle, the closer the sum of the angles is to 180 degrees.
	www.math.ksu.edu /math572/tri.html (665 words)

	Hyperbolic Trigonometric Functions
		The McLaurin's series for the hyperbolic sine and cosine may be obtained from their definitions and the series for the exponential function.
		The McLaurin's series formulae for the hyperbolic arc sine, arc cosine, arc cotangent, arc secent, arc cosecent, arc versed sine, arc coverssed sine, and arc haversed sine are not interesting.
		The values of the inverse hyperbolic trigonometric functions have to be obtained from the foregoing arctangent, by solving the quadratic equations of the
	www.geocities.com /ResearchTriangle/2363/hyperbol.html (1630 words)

	NonEuclid: Example Exercise (Adjacent Angles)
		Hyperbolic Angles are formed by the intersection of Hyperbolic rays analogous to the formation of angles in Euclidean Geometry.
		The adjacent angles formed by a pair of intersecting lines are supplementary and together measure 180° (this was done in the example exercise).
		The diagonals of a Rhombus bisect the Rhombus's angles.
	www.cs.unm.edu /~joel/NonEuclid/exercise.html (2077 words)

	Semi-Regular Tilings of the Plane Part 4: Hyperbolic Results
		Recall that because triangles in the hyperbolic plane have angle sum less than PI, it follows that the measure of the angle at a vertex of a regular p-gon in the hyperbolic plane is less than PI-(2*PI/p).
		To prove part (3), use the Hyperbolic Triangle Lemma to obtain a tiling by triangles with angles with angles PI/p, PI/q, and PI/r.
		The segment from A_(k+1)/2 to B creates two congruent (k+3)/2-gons with (k-1)/2 angles of 2PI/p, an angle* of PI/p at A_(k+1)/2, and an angle of PI/q at B. Since p is even, as in the discussion of Remark 2, part (2), this (k+3)/2-gon tiles the plane via reflections in its edges.
	people.hws.edu /mitchell/tilings/Part4.html (1887 words)

	Class Plan
		Angle: The hyperbolic measure of the angle between two intersecting hyperbolic lines is the measure of the angle defined by the tangent lines to the hyperbolic lines at the point of intersection.
		Determine the hyperbolic distance from A=(1, 10) to B=(7, 8) (and to C=(1, 20)).
		Let a, b, c represent the hyperbolic length of the sides opposite to A, B, and C, respectively.
	www.msci.memphis.edu /~botelhof/XIX.html (295 words)

	Hyperbolic Functions -- from Wolfram MathWorld
		hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cosecant,
		hyperbolic cosine function is the shape of a hanging cable (the so-called catenary).
		Anderson, J. "Trigonometry in the Hyperbolic Plane." §5.7 in Hyperbolic Geometry.
	mathworld.wolfram.com /HyperbolicFunctions.html (256 words)

	The Ring of Hyperbolic Quaternions
		The modulus of biquaternions, hyperbolic quaternions, and coquaternions does not stay non-negative and the topology of these other quaternion-type rings is then, non-Euclidean, as the persistent student learns.
		Now in hyperbolic quaternions M developed by MacFarlane in the 1890s, we trade in versors i, j, k for motors i, j, k and in fact take V ⊂ M to have a sphere S of motors.
		The 3D hyperboloid model of hyperbolic space is important as a model of velocity space in special relativity.
	ca.geocities.com /macfarlanebio/hypquat (1136 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		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)

	[No title]
		I don't think it's possible in hyperbolic space that all the dihedral angles are 120 degrees, but what you really need is just that the sum of twice the pentagon-hexagon angle plus the hexagon-hexagon angle is 360 degrees, which should be perfectly possible.
		The hexagon- hexagon dihedral angle is larger than the hexagon-pentagon dihedral angle in the Euclidean limit (of small edge length) and I think that this always remains true as the edge length is increased (this is the shaky part of the argument).
		One thing I'm not sure about: The hexagon- hexagon dihedral angle is larger than the hexagon-pentagon dihedral angle in the Euclidean limit (of small edge length) and I think that this always remains true as the edge length is increased (this is the shaky part of the argument).
	www.ics.uci.edu /~eppstein/junkyard/buckyball.html (3314 words)

	Hyperbolic Tessellations
		The hyperbolic plane can not be metrically represented in the Euclidean plane, but Poincaré described ways that it can be conformally represented in the Euclidean plane.
		For this representation, a straight line in the hyperbolic plane is represented as the part (in the disk) of a circle that meets the boundary of the disk at right angles.
		For instance, here is a representation of the tessellation of the hyperbolic plane by pentagons where four pentagons meet at each vertex, that is, the {5,4}-tessellation.
	aleph0.clarku.edu /~djoyce/poincare/poincare.html (784 words)

	Visual Basic Trigonometric Math
		Angle of the point (X, Y) where X and Y are real numbers.
		"Inverse Hyperbolic Secant" Angle whose hyperbolic secant is X. Inverse of the SecH function.
		"Inverse Hyperbolic Cosecant" Angle whose hyperbolic cosecant is X. Inverse of the CscH function.
	www.entisoft.com /estools/MathTrigComplex.HTML (483 words)

	Hyperbolic Rotations
		In elementary calculus they breezed past the hyperbolic functions with hardly a glance -- they seemed to be just a curiosity used by bridge builders.
		The inverse hyperbolic functions are named with an ar prefix, as arcosh(x), to indicate that they return the area associated with that value of the function: it's short for "area of the cosh".
		A hyperbolic rotation is what we get when we slide all the points on the hyperbola along by some angle.
	www.physicsinsights.org /hyperbolic_rotations.html (630 words)

	Hyperbolic Trigonometric Functions
		Measure the angle theta counter-clockwise from the vertex -- the perihelion.
		This is the angle (in radians) between the rotation-axis of the planet and the orbital revolution-axis.
		B4 is the angle (in radians) between the vernal equinox and the perihelion.
	www.rism.com /Trig/hyperbol.htm (9850 words)

	Math Horizons Article
		For example, all the triangles in Figure 1 are the same hyperbolic size, as are all the fl fish (or white fish) of Figure 2, and the kites of Figure 3.
		This is as it should be, since the backbone arcs are not hyperbolic lines, but equidistant curves, each point of which is an equal hyperbolic distance from a hyperbolic line.
		For any acute angle and hyperbolic line, there are two equidistant curves ("branches"), one on each side of the line, making that angle with the bounding circle.
	www.d.umn.edu /~ddunham/mathhoriz/paper.html (2128 words)

	Is Universe expanding
		This is how author concludes that the redshift of Galaxies are cause by the nature of Hyperbolic space, in which the space stretches the light spherical front.
		However, if we assume the Universe is in Hyperbolic space, very logically, we must derive its rules and formulas from Hyperbolic rules and Hyperbolic formulas.
		If he believed that the Universe was in Hyperbolic space, then there is a space curvature which is the cosmological constant.
	www.angelfire.com /ca6/aliou/expanduniverse.htm (1518 words)

	InfoVis CyberInfrastructure- Hyperbolic Trees
		Hyperbolic graph layout uses a context + focus technique to represent and manipulate large tree hierarchies on limited screen size.
		Hyperbolic trees are based on Poincare's model of the (hyperbolic) non-Euclidean plane.
		The hyperbolic tree implementation is based in part on the hyperbolic tree implementation by Andreas Hadjiprocopis (1999).
	iv.slis.indiana.edu /sw/hyptree.html (650 words)

	Euclid's Elements, Book I, Proposition 29
		Therefore the angle AGH is not unequal to the angle GHD, and therefore equals it.
		Poincaré, however, described a useful model of hyperbolic geometry where the "points" in a hyperbolic plane are taken to be points inside a fixed circle (but not the points on the circumference).
		In the diagram, AB is a "line" in the hyperbolic plane, that is, a circle orthogonal to the circumference of the shaded disk which represents the hyperbolic plane.
	aleph0.clarku.edu /~djoyce/java/elements/bookI/propI29.html (793 words)

	Optional Problems Fall 2003
		Explain why an angle inscribed in a circle is half the central angle subtended by the same arc that subtends the inscribed angle.
		Construct a (hyperbolic) line that passes through a given point and is perpendicular to a given line.
		Construct a (hyperbolic) line that is perpendicular to a given line through a given point that is not on that line.
	www.mtholyoke.edu /courses/jmorrow/problem_options_03.html (942 words)

	About "Hyperbolic Angle" (Site not responding. Last check: 2007-10-18)
		Exposition of a new presentation of the hyperbolic angle, together with hyperbolic functions sinh x and cosh x.
		The author writes: "The argument of hyperbolic functions sinh x and cosh x is given a clear development on this website.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /library/view/41509.html (102 words)

	Lab 18: Hyperbolic Ruler & Compass Constructions
		To make a given angle, start with one of the three random points to serve as the angle vertex, and the other two as arbitrary points on the angle sides.
		Construct a (hyperbolic) angle bisector for a given angle.
		Construct a (hyperbolic) line that passes through P and is perpendicular to the given line segment.
	www.mtholyoke.edu /courses/jmorrow/lab_18.html (769 words)

	3rd Year Honours Mathematical Physics Special Relativity Hyperbolic Trigonometric Functions (Site not responding. Last check: 2007-10-18)
		The behaviour of sin and cos under differentiation can also be determined easily from the exponential representation.
		Hyperbolic trigonometric functions are related to hyperbolae in a manner similar to that in which trigonometric functions are related to circles.
		They are actually much simpler, because they do not require complex exponentials.
	www.thphys.may.ie /Notes/SR/hyperbolic/hyperbolic.html (269 words)

	MATH 141: Euclidean Geometry (Site not responding. Last check: 2007-10-18)
		Visualize the distortion of length and the preservation of angles in the hyperbolic plane.
		This means that the hyperbolic angle measure is the same as the Euclidean angle measure.
		Try to construct triangles whose sum of angles is as small as possible and triangles whose sum of angles is as large as possible.
	www.math.ucdavis.edu /~cgee/math141/computerlab.html (1437 words)

	MB-Cyclopædia for MultiValue Systems
		Hyperbolic cosine A = Cosh A = -i cosine iA = e
		Hyperbolic tangent A = Tanh A = Sinh A / Cosh A
		Hyperbolic secant = the reciprocal of the hyperbolic cosine Hyperbolic cosecant = the reciprocal of the hyperbolic sine Hyperbolic cotangent = the reciprocal of the hyperbolic tangent
	hometown.aol.com /mbpublish/mmt57.html (487 words)

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