Hyperbolic geometry

 
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nextstep
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Joined: 06 Jan 2007
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PostPosted: Sun Nov 09, 2014 1:35 am    Post subject: Hyperbolic geometry Reply with quote

Hi all,
Hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced.
The parallel postulate in Euclidean geometry is equivalent to the statement that, in two-dimensional space, for any given line R and point P not on R, there is exactly one line through P that does not intersect R; i.e., that is parallel to R. In hyperbolic geometry there are at least two distinct lines through P which do not intersect R, so the parallel postulate is false. Models have been
constructed within Euclidean geometry that obey the axioms of hyperbolic geometry, thus proving that the parallel postulate is independent of the other postulates of Euclid.
All six of the standard trigonometric functions have hyperbolic analogs.
The attached image is for Hyperbolic Helicoid



HyperbolicHelicoid by taha_ab, on Flickr
Quote:
{
"Param3D": {
"Name": [
"HyperbolicHelicoidal"
],
"Component": [
"HyperbolicHelicoidal"
],
"Fx": [
"(sinh(v)*cos(3*u))/(1+cosh(u)*cosh(v))"
],
"Fy": [
"(sinh(v)*sin(3*u))/(1+cosh(u)*cosh(v))"
],
"Fz": [
"(cosh(v)*sinh(u))/(1+cosh(u)*cosh(v))"
],
"Umax": [
"pi"
],
"Umin": [
"-pi"
],
"Vmax": [
"pi"
],
"Vmin": [
"-pi"
]
}
}



More at:
https://www.facebook.com/pages/MathMod/529510253833102
http://mathworld.wolfram.com/HyperbolicHelicoid.html
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Abderrahman
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