Parametric Representation for a Square with Side $1$ Centered at the Origin as a Function of the Angle Measured from the Positive $x$-Axis

Question

While playing with some graphics progamming in OpenGL, I've encounterd this problem:

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Find the Parametric representation for a square with side $1$ centered at the origin as a function of the angle $\theta$ measured from the positive $x$-axis.

Here's an image I created to illustrate the problem:

Here the vector $v$ must satisfy $||v||_2 = 1$ and the vector $u$ must satisfy $||u||_{\infty} = 1$

We can parameterize $v$ by using the angle $\theta$ measured from the positive $x$-axis as follows:

$v(\theta) = (sin(\theta), cos(\theta))$

Now how can we parameterize $u$ by using the angle $\theta$ measured from the positive $x$-axis?

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Thanks for any help.

Answer 1

$$u(\theta)=\frac{v(\theta)}{\max\left(\left|\sin\theta\right|,\left|\cos\theta\right|\right)}=\frac{1}{\max\left(\left|\sin\theta\right|,\left|\cos\theta\right|\right)}\left(\cos\theta,\sin\theta\right).$$

Answer 2

Based on the answer of Jack D'Aurizio:

The parameterization $u(\theta) = \frac{v(\theta)}{||v(\theta)||_{\infty}}$ will work since:

$||u(\theta)||_{\infty} = ||\frac{v(\theta)}{||v(\theta)||_{\infty}}||_{\infty} = \frac{1}{||v(\theta)||_{\infty}} ||v(\theta)||_{\infty} = 1$.

(I.e we normalize $v(\theta)$ with respect to the $|| \bullet ||_{\infty}$ norm)