hyperbolic trig

[ydoaPs]

what is it? i know hyperbolic geometry is a type of non-euclidean geometry; does it have anythign to do with that? if so, is there a trig for sphereical geometry?

[Crash]

Um hyperbolic geometrey as opposed to "normal" geometry is based on hyperbolas instead of circles, say if trig is a sphere then hyperbolic trig is and inverse sphere sort of like a saddle

[ydoaPs]

how would you define the trig funtions, then?

[Crash]

with a circle, a line starting at the origin and heading out to the right side where the circle has a radius of 1 and sin being the lateral distance and cos being the vertical distance and tan being sin/cos and so on and so fourth for csch, sch, cot etc.

[ydoaPs]

how do you do that with hyperbolic geometry?

[Crash]

you dont, you use a hyperbola as i said in my first post, same principal but use an inverse sphere as i said

[ydoaPs]

that's what i was asking. how are trig functions defined with a hyperbola?

[NeonBlack]

Actually, hyperbolic trig functions are defined by exponentials.

I forget the actual definitions but I can look them up.

[NeonBlack]

[math] sinh (x) = \frac{e^x-e^{-x}}{2}[/math]

 

[math] cosh (x) = \frac{e^x+e^{-x}}{2}[/math]

 

[math] tanh (x) = \frac{sinh(x)}{cos(x)}[/math]

 

[math] sech (x) = \frac{1}{cos(x)}[/math]

 

etc...

[matt grime]

firstly hyperbolic trig and hyperbolic geometry are not necessarily related, indeed they aren't really.

 

secondly neonblack is missing some h's in his post (sech is 1/cosh) but that;s just a typo.

 

hyperbolic trig fucntions are as defined, they also satisfy integrals akin to theordinary trig ones, and can be related to ordinary trig by complex numbers.

 

hyperbolic geometry: a geometry is a set of points with a rule for making geodesics. a geodesic is a "shortest" path. but perhaps you might wnat to think of it is a "path of least resistance" since you may well think that straight lines are always the "shortest" path. geometries satisfy some axioms, things like give two points there is a geodesic passing through them and so on. eulicdean geometry, the one we trhink in of straight lines and planes and such satisfies the parallel poostulate: given a geodesic and a poitn not on that geodesic there is a unique parallel geodesic through that point.

 

for many years, centuries, it was beleived that the parallel postulate was deducible fromthe other axioms (of euclid) but it isn't. it turns out you can define other geometries satisfying all of euclid's rules except the parallel postulaate.

 

indeed we live on one: spherical geometry. we know that planes fly not on straight lines but on great circles as teh shortest path. two great circles always inteserct. so the parallel postulate fails as there is no second geodesic.

 

hyperbolic is the other extreme.

 

there are several models of hyperbolic geometry. the easiest to me is the poincare disc:

 

it is the inside of unit disc in the complex plane (all complex numbers of modulus less than one) and the geodesics are semicircles that intersect the boundary at right angles (note this includes diameters of the disc - cirlces of infinte radius). in this space, given a gedoesic and a point not on the geodesic there are an infinte number of "parallel" geodesics through that point. the easiest example is if the frst geodesic is not a diamter, and the point is the centre of the disc and then there are an infinite number of diameters through the centre not intersecting the geodesic.