# Various forms of trigonometry

Plane trigonometry is the measurement of angles…pie slices of a circle.

There is also Hyperbolic trigonometry.

Are there trigonometric forms of other conic sections? Parabolas and elipses?

The obvious – there’s Spherical Trig. Useful for your navgation types.

i

I think the OP is wondering if there are analogous functions to sinh, tanh, etc. for other conic sections. Not that I know of, but that means nothing.

FWIW,
Rob

Just did a little reasearch:

Elliptic trig IS circular trig…a.k.a. high school trig.

Hyperbolic trig is useful in computing formulae in Electromagnetizm and Special Relativity.

Parabolic Trig is apparently useful in computing Gallilean and Lorentz Transformations.
Here’s a question…are the apparently related mathematical properties of the various trig functions merely manifestations of a much deeper trig based on the unit cone itself, instead of merely symmetrical branches of mathematics based on cross-sections of the unit cone?

FWIW there is the fascinating ancient tradition of Japanese temple icons, which feature geometric exercises. Whereas the Greeks worked out a great deal of geometry based on lines and angles, the Japanese did so based on circles and spheres. They would work out something like the ideal sphere diameters for maximum packing fraction, and create an icon representing the process. AFAIK the Greeks and Japanese did these things in mutual isolation (and not necessarily contemporaneously). Of course the education I got here in the USA included math topics that evolved out of the work of the Greeks, but I still wonder at how math might look different today if it evolved differently early on by the merest of accidental choices, and enjoyed these icons as a suggestion of that.

Hyperbolic trigonometry has less to do with the hyperbola as a conic section, and more to do with non-Euclidean geometry. If you start out with the assumption that for any line m and point P, there are infinitely many lines parallel to m passing through P, and work out trigonometry, you’ll find that you get the hyperbolic trigonometric functions. There is a variation of trigonometry for the case where there are no lines through P parallel to m, but it doesn’t seem to get as much press.