User:Kieff/Square sine and cosine functions
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The square sine and square cosine functions are akin to their trigonometric counterparts, but instead of defining an unit circle, they define a square of "radius" 1 (that is, side 2). I'm not sure if such functions are already properly defined in the mathematical community, but I never heard of them. I doubt I'm the first to toy with this concept, though.
The square sine ("sinsk") can be written as:
Where sgn is the very useful sign function. The signal function works here as a very useful inline hack for a piecewise function construction. There's probably a better way to do all this, but hey, it works so far!
The square cosine ("cosk") is defined as:
- cossq(x) = sinsq(x + π / 2)
[edit] Approximations
An interesting approximation can be done by using iterated trigonometric functions:
Define a function ts such as:
- ts0(x) = x
- ts1(x) = tan(sin(x))
- tsn(x) = tan(sin(tsn − 1(x)))
The square sine can then be approximated by:
Which gives a smooth curve that differs no more than 0.1082300356377... from the square sine. I wonder if there's a better approximation...
