Re: Use of Inverse LogLog scales Part 4: Hyperbolic functions

Andreas Poschinger


indeed I lost at least one use case: logarithms of numbers smaller than
1. This I will put into part 5. Any other use cases lost?

Hyperbolic functions are of some importance in transmission line
problems. This I will not look upon.

Another application of hyperbolic function is to calculate the geometry
of suspension bridges or cables between electric towers. Chains (and
ropes) follow what in direct German translation we would call "chain
line" but in English actually catenary.

cosh(x) = 1/2(e^x + e^(-x)) so that for arbitrary catenaries something
like b/(e^(x/a) + e^(-x/a)) needs to be evaluated, and thus first
e^(x/a) and e^(-x/a). It is a mixture of the table building in part 4
and simple reading the inverse in part 1:

a on C to D1 (or to D10); at C x e^(x/a) on LL3 and e^(-x/a) on LL03
respectively on LL2/LL02 and LL1/LL01 for 1/10 and 1/100 of x between 1
and 10.

For this application inverse LogLog scales considerably speed up
calculation since calculation of the inverse is not needed.

Though personally I've never seen cosh(x) unit paper the unit function
approach should be also possible.

The use for sinh and tanh=sinh/cosh would follow the same approach.

Best regards


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