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!