Converting specific equations from Polar to Cartesian

Question

These different equations are given in Polar and my goal is to plot them in Cartesian coordinate system:

1. $r = \cos(4φ)$
2. $ φ = \dfrac r {r-1}$, $r > 1$

I am aware of:

• $x = r \cos( φ )$
• $y = r \sin( φ )$
• $r^2 = x^2 + y^2\\$

However I cannot figure out how to use these traits properly, and I am not able to find any relatable examples.

Any hints how should I approach this?

Answer 1

• Use $x=r\cos\varphi=\cos 4\varphi\cos\varphi$ and similar for $y$
• Use $x=r\cos\varphi=r\cos\dfrac{r}{r-1}$ and similar for $y$.

Answer 2

You are missing the relation $\phi=\arctan\left(\dfrac yx\right)$. Simply substitute

1. $\sqrt{x^2+y^2}=\cos\left(4\arctan\left(\dfrac yx\right)\right)$,

2. $\arctan\left(\dfrac yx\right)=\dfrac{\sqrt{x^2+y^2}}{\sqrt{x^2+y^2-1}}.$

By trigonometric transformations, the first equation can be rewritten as a polynomial of a high degree in $x,y$, while the second has no simple form.

$$64x^{12}-144x^{10}+81x^8-x^6-32x^4y^5+54x^4y^4-12x^2y^6-3x^2y^4+y^8-y^6-16y^4x^6+192y^3x^6+352y^2x^8-108y^2x^6-3y^2x^4+256yx^{10}-288yx^8=0.$$

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As plotting directly in Cartesian coordinates requires numerical root finding methods, it is much more advisable to use paramtric forms,

1. $x=r\cos(\phi)=\cos(4\phi)\cos(\phi),\\y=r\sin(\phi)=\cos(4\phi)\sin(\phi),$
2. $y=r\cos\left(\dfrac r{r-1}\right),\\x=r\sin\left(\dfrac r{r-1}\right).$