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Pardon my naivety, but I wonder if much use has been found for trigonometric functions defined in terms of a centrally symmetric convex curve $K$ replacing the circle $C$. For example, here is the equivalent of the sine function defined on a diamond, and on a quadratic curve (with the true $\sin \theta$ function superimposed for comparison):
           Trig-like Functions
Perhaps for certain $K$ nice properties are retained for the corresponding trig functions: trig identities, orthogonality, Fourier series, etc.?

In some sense I am seeking to understand why the standard trig functions are so ubiquitous and useful, by imagining a variant. Thanks for any insights!

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Joseph, it looks like you're thinking of $\theta$ as an angle in the ordinary way. An alternative might be to view it as parameterizing arc length along the curve $K$. For the diamond, this would give sawteeth for the sine and cosine equivalents, which at the least, I think, would have nice orthogonality properties. – Barry Cipra May 9 '12 at 15:32
@Barry: Excellent point! Yes, I was thinking in terms of $\theta$. But arc length may be more natural, more fundamental. – Joseph O'Rourke May 9 '12 at 15:45
Another way would be to think of $\theta$ as area. Definition of the hyperbolic functions can be done that way. – Gerald Edgar May 9 '12 at 16:33
Thanks, Gerald! That's another viewpoint that did not occur to me. – Joseph O'Rourke May 9 '12 at 17:25
up vote 6 down vote accepted

In some sense elliptic functions are "the" answer to your question. First I will mention an introductory reference which is written in the spirit of the question and explains the history of how elliptic integrals arose from trying to answer similar questions on curves such as ellipses, lemniscates etc.

The remarkable sine functions, by A. I. Markushevich

The argument that I find most relevant for this answer is that for a generalization of circular trigonometry, one would want a new parametrized curve $P(t)=(C(t),S(t))$ which satisfies some kind of addition formulae, similar to the usual trigonometric functions. A theorem of Weierstrass says that a function of a complex variable with an algebraic addition theorem ($P(t+r)$ can be expressed as an algebraic function of $P(t)$ and $P(r)$) must necessarily be a limit of some elliptic function. I.e. it must be one of the following three cases:

  • A rational function.
  • A rational function of $e^{px}$, $p\in \mathbb C$ (here you have hyperbolic and circular trigonometric functions).
  • A rational function of the Weierstrass elliptic function and its derivative.

This is a very important trichotomy in mathematics and appears in several places. "A Brief History of Elliptic Integral Addition Theorems" by J. Barrios gives a nice short historical account from this point of view. It also illustrates with some examples why one might care about addition theorems even if one is interested in geometric properties of the curve.

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I could not hope for a better answer! That the addition formulae play such a key role is a revelation to me. Thanks, Gjergji! – Joseph O'Rourke May 11 '12 at 11:14

A comment (in answer form) on @Barry's comment: for ellipses, the functions you get (if you parametrize by arc length) are elliptic functions, and there are plenty of identities, by far the best source for which is C. L. Siegel's function theory book.

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The sine function for the first one is $$f(\theta)=\frac{\sin \theta}{\sqrt{2} \sin( \frac{\pi}{4}+\theta)}$$ for $\theta\in [0,\frac{\pi}{2}]$.

The second one is $$g(\theta)=\frac{-\tan \theta+\sqrt{\tan^2\theta+4}}{2}\tan\theta$$ for $\theta\in [0,\frac{\pi}{2}]$.

Then extend this symmetrically for other values of $\theta$. Perhaps, you might be able to obtain desired identities using usual $\cos$ and $\sin$ identities, but judging from above formulas, I don't see the reason that there should be simple identities of this sort.

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@i7077107: Yes, I can confirm those equations. They are the same I used to make the plots. – Joseph O'Rourke May 9 '12 at 15:51

One of the ways to define the usual sine function is to consider the boundary value problem of the second order ODE $y''+ cy =0, y(0)= y(\pi)=0.$ Generalizing this definition, P. Lindqvist studied generalized trigonometric functions [L]. Numerous other authors have worked on this topic thereafter. Some of the latest papers include [BV],[BE],[EGL], [T].

[BV] B. A. Bhayo and M. Vuorinen: On generalized trigonometric functions with two parameters. J. Approx. Theory 164 (2012) 1415--1426, doi: 10.1016/j.jat.2012.06.003

[BE] P. J. Bushell and D. E. Edmunds: Remarks on generalised trigonometric functions. Rocky Mountain J. Math. 42 (2012), Number 1, 25--57.

[EGL] D. E. Edmunds, P. Gurka, and J. Lang: Properties of generalized trigonometric functions. J. Approx. Theory 164 (2012) 47--56, doi:10.1016/j.jat.2011.09.004.

[L] P. Lindqvist: Some remarkable sine and cosine functions. Ricerche di Matematica, Vol. XLIV (1995), 269--290.

[T] S. Takeuchi: Generalized Jacobian elliptic functions and their application to bifurcation problems associated with p-Laplacian. J. Math. Anal. Appl. 385 (2012), 24--35, doi:10.1016/j.jmaa.2011.06.063.

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