How would I go about parameterizing the "unit square"?

Question

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?

Answer 1

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.