Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

enter image description here

Given the parametric equation of a unit circle $$ \vec r(\theta) = \begin{bmatrix} \cos\theta \\ \sin\theta \end{bmatrix}, \quad 0 \leq \theta \leq 2\pi $$

It seems that there is some function $$ f : \mathbb{R} \rightarrow \mathbb{R} $$

such that $$ \vec s(\theta) = f(\theta)\vec r(\theta), \quad 0 \leq \theta \leq 2\pi $$ where $\vec s(\theta)$ is the parametric equation of a square with side length $2$.

Can this function $f$ be found, and if so, what is it?

share|cite|improve this question

3 Answers 3

up vote 2 down vote accepted

Such function is just:

$$ f(\theta) = \frac{1}{\max(|\sin\theta|,|\cos\theta|)}.$$

share|cite|improve this answer
I don't doubt that this is correct, but could you provide at least an informal proof as to why? –  Ryan Aug 11 '14 at 2:32
Such a choice can be understood in terms of the $L^\infty$ norm or supremum norm $$||\boldsymbol x||_\infty = \max \{|x_1|, |x_2|, \ldots, |x_n|\}.$$ Then we have the natural parametrization for $n = 2$ $$\boldsymbol x (\theta) = \frac{\boldsymbol r(\theta)}{||\boldsymbol r(\theta)||_\infty}$$ where $\boldsymbol r(\theta) = (\cos \theta, \sin \theta)$ is the unit circle. –  heropup Aug 11 '14 at 2:35
Exacly, it is just the standard way to map the unit ball with respect to the euclidean norm (the circle) into the unit ball with respect to the supremum norm (the square). –  Jack D'Aurizio Aug 11 '14 at 2:36
@Clockwise It is not difficult to prove. Try looking at just the first quadrant. The key is to observe that for $0 \le \theta < \pi/4$, $\cos \theta > \sin \theta$, and for $\pi/4 < \theta \le \pi/2$, $\sin \theta > \cos \theta$. –  heropup Aug 11 '14 at 2:38

For the parameterization of the square, We can define such a function piecewise. For the first(and last) octant, consider that we have a right triangle, with one leg 1, the adjacent angle $\theta$. Therefore $x=1$ and $y = \tan(\theta)$.

This gives $f = \frac 1 {\cos \theta}$ on this region.

You can construct similar parameterizations for the other 4 pieces with rotations about the origin; yielding $f$ as described by Jack, $\frac{1}{\max(|\sin(\theta)|,|\cos(\theta)|)}$

share|cite|improve this answer

Found this post and thought I would share an equation I derived for "converting" a circle to a square using a ratio method that is also a parametric equation, I am sure that someone else has probably done something similar.

Start by plotting a circle of radius R: Parametric Equation of Circle

I have just given the equation, but this is represented as: Parametric Curve Representation of a Square from a Circle

You rotate the square just by offsetting theta by 45°, what also comes out of this equation is some nice plots, eg a heart: Parametric Equation of a Heart others: enter image description here

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.