TITLE: Trig functions based on convex curves
QUESTION [7 upvotes]: 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):

          


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!

REPLY [2 votes]: 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.