Archimedean spiral in cartesian coordinates

Question

I know that the Archimedean spiral can be represented using the polar coordinate system very easily. But I was wondering if it can be represented using the Cartesian coordinate system, and if so what is the function?

Answer 1

Here is a solution for a double Archimedean spiral (see figure below).

Let us consider the simplest Archimedean spiral with polar equation:

$$\tag{1}r=\theta.$$

Using the following formulas:

$$\tag{2}\begin{cases}r^2=x^2+y^2\\\tan{\theta}=\tfrac{y}{x}\\\end{cases},$$

(1) can be transformed into the following implicit cartesian equation:

$$\tag{3}\arctan(\tfrac{y}{x})+k\pi=\sqrt{x^2+y^2} \ \ \ \ \ \ \ \ \ \ (x \neq 0).$$

Taking $\tan$ on both sides gives the solution:

$$\tag{4}y= \pm x \tan\sqrt{x^2+y^2}.$$

Remarks:

1. Note that, by multiplying by $x$, restriction $x \neq 0$ is no longer needed. Indeed, a consequence of condition $x \neq 0$ in (3) was that points $(0,(4k+1)\tfrac{\pi}{2}), \ k \in \mathbb{Z},$ were excluded from the curve, which is not the case with form (4).

2. In fact, equation (4) defines a double Archimedean spiral (changing $(x,y)$ into $(-x,-y)$ doesn't change this equation). See picture below where the red curve is the Archimedean spiral, strictly speaking, and the magenta curve is its copy through a central symmetry.

3. Equation (4) is less manageable than (3). (3) can be given under the form of an infinite number of implicit equations

$$f_k(x,y)=\arctan(y/x)+k\pi-\sqrt{x^2+y^2}=0, \ \ k=0,1,2,...$$

or by replacing $\arctan(y/x)$ by $\operatorname{atan2}(y,x)$.

(see SAGE program below).

4. (4) cannot be transformed into a polynomial implicit equation $P(x,y)=0$. A simple reason : any straight line intersects an Archimedean spiral in an infinite number of points, which is impossible for a polynomial equation.

5. It is easy from there to obtain a cartesian equation analogous to (4) for the general Archimedean spirals $r=a\theta.$

SAGE program :

 var('x y');
 n=10;m=3*n
 g=line([[-m,0],[m,0]]);g+=line([[0,-m],[0,m]]);
 for k in range(n) :
    g+=implicit_plot(atan(y/x)+k*pi-sqrt(x^2+y^2),(x,-m,m),(y,-m,m))
 g