|Re: Hyperbolics with the Advantage|
Message #5 Posted by Valentin Albillo on 1 Dec 2003, 4:34 a.m.,
in response to message #4 by Gene
"give me an example of something that uses complex hyperbolics."
They're used all the time in structural engineering, Gene,
since the formula for the shape (called "catenary") of a weighty chain freely hanging from two separate points happens to be an hyperbolic cosine, and most suspension bridges and other such superstructures feature catenaries all the time. See first link below.
This also applies not only to macro-scale, plain-vanilla bridges, but also to nano-scale bridges in superconducting materials, which also happen to be hyperbolic cosines, see second link below.
Also, some cultures use shapes defined by hyperbolic functions in their civil engineering not only for practical reasons, but for spiritual and artistic reasons as well, specially for shrines and temples. The exponential growth of an hyperbolic cosine is felt as much more uprising and tending towards "higher spirituality" than a simple, quadratic parabola, so Japanese temples feature spikes and elevations with hyperbolic shapes. Have a look at the very revealing third link, which makes it clear that people can easily notice that something's not right with a mere parabola, despite initial appearances.
Finally, the fourth link will tell you why you need to apply an "hyperbolic cosine" coating to critical parts of supersonic fighter aircrafts at Boeing.
Superconducting fluctuations ...
Curves in traditional Japanese architecture and Civil Engineering
Surface coating for supersonic fighter aircraft
In case you're wondering about the use of hyperbolic functions with complex (i.e. "imaginary") arguments, the applications are the same, see these references for instance (unfortunately they don't seem to be available in the web, only as printed publications):
Nsugbe EAO & Williams CJK (1999). The Use of the Inverse Hyperbolic Cosine Function of a Complex Variable in Defining a Bridge Geometry. In Astudillo R & Madrid AJ (eds.) Proc. IASS 40th Aniversary Congress, 2: H21-28. CEDEX Sección de Edición, Madrid.
The generation of bone-like forms using analytic functions of a complex variable. E.A.O. Nsugbe, C.J.K. Williams, Journal of Computers and Structures, Jan 1998
[they demonstrated their ideas by actually building a bridge in bone-like shape, the shape being generated by using inverse hyperbolic cosine functions with complex arguments]
There are many more applications, but I hope the above ones will serve you well.
Best regards from V.
Edited: 1 Dec 2003, 6:06 a.m.