Elliptic/circular Trigonometry = High School trig...measuring triangles an' stuff.
Hyperbolic Trig = useful in programming formulae in electromagnetizm and special relativity.
Parabolic trig = useful in computing gallilean and lorentz transformations.
Circle, ellipse, parabola, hyperbola...all cross-sections of a unit cone...thus are called conic sections.
(trivial/degenerate sections like the point and intersections of lines are probably unimportant to this discussion)
Do the similar trig functions amongst the conic sections imply an underlying symmetry?
A much deeper trig based on the unit cone itself?
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I'm not completely sure what it is you mean to ask, but you could formally do something analogous to trigonometry along any kind of family of curves. It becomes fundamental only when the structure of your space is connected to it.
Take a look at the Euclidean plane, which has the metric ds²=dx²+dy². The curves of r²=x²+y² are circles, and one can define a parameter θ such that rθ is the length along the circle counterclockwise from (r,0). Thus, every point has some value of θ associated with it (in fact, here more than one). A point (x,y) on the r-circle with some parameter θ can thus be used to define the standard trigonometric functions: (r.cos θ, r.sin θ) = (x,y).
We can consider the Minkowski plane, which has the metric ds²=dt²-dx². The curves of r²=t²-x² are hyperbolas, and you can define a parameter α such thar rα is the length along the hyperbola in the t-direction from (r,0), and likewise take (r.cosh α, r.sinh α) = (t,x) for a point along the r-hyperbola as definitions of the hyperbolic trigonometric functions, satisfying cosh²α - sinh²α = 1.
Back in Euclidean space, we can still do hyperbolic trigonometry, at the const of α losing its meaning as a measure of arc length. Moreover, we can formally do this along any kind of family of curves. Having a parabolic spacetime if the metric is degenerate, in particularly completely independent of time intervals--e.g., ds² = dx² even though there is a time coordinate t. Since Newtonian mechanics (Galilean relativity) is like this, that's the connection you're referring to in the OP.
Lorentz transformations are hyperbolic. Well, if by transformations one means boosts (usual nomenclature doesn't distinguish them). For the full Lorentz group SO(1,3), there is actually a parabolic subgroup of transformations with exactly one eigenvector that happens to be lightlike. This sort of degeneracy is there because Minkowski spacetime is only a semi-inner product space, i.e., contains vectors that are null--have length zero but are not the zero vector. So representative generators can be taken to be:
hyperbolic: rotation about the time axis
elliptic: rotation about a space axis
parabolic: rotation about a null vector
The parabolic transformations can actually be seen as a limit of the first two kinds of transformations involving an infinite boost. There is another class sometimes identified as distinct that can also be built from first two classes.
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