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I was looking at the equation |x|+|y|=1 and an interesting thought came to my mind. Could you parameterize this equation by angle into something akin to a square sine and square cosine? How exactly would you go about doing this?

O.S.
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  • With a square sine function (actually probably the triangle sine function is what you want) https://en.wikipedia.org/wiki/Square_wave – pancini Aug 18 '20 at 22:59
  • You could always parameterise it piecewise. –  Aug 18 '20 at 23:02

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Any angle in standard position $\theta$ has a unique endpoint on the square $|x| + |y| = 1$ given by $$(x_\theta, y_\theta) = \left( \frac{\cos(\theta)}{|\cos(\theta)| + |\sin(\theta)|}, \frac{\sin(\theta)}{|\cos(\theta)| + |\sin(\theta)|} \right).$$ I suppose you could call those functions the "square cosine" and "square sine", respectively.

Rivers McForge
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