[Math] polar coordinates vector equation of a rectangle

Question

We can write the equation of the circle in vector form in polar coordinates as:

$$\vec{r}=R\hat{r}$$ ; where 'R' is the radius of the circle.

Similarly, can we write the vector equation for a rectangle in polar coordinates? If so, how?

Answer 1

Let $(\pm a,\pm b)$ be the vertices of the rectangle.

When $x=a$, $r\cos \theta=a$

When $y=b$, $r\sin \theta=b$

Hence, $$r(\theta)= \left \{ \begin{array}{ccc} \displaystyle \frac{a}{|\cos \theta \,|} & , & \displaystyle |\tan \theta \,| \leq \frac{b}{a} \\ \displaystyle \frac{b}{|\sin \theta \,|} & , & \displaystyle |\tan \theta \,| \geq \frac{b}{a} \\ \end{array} \right.$$