Trig functions based on convex curves

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.

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.

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.