Converting Between Polar and Rectangular Coordinates

Definition Put the pole at the origin and the polar axis along the positive $x$-axis. If a point has polar coordinates $(r, \theta)$, its rectangular coordinates are

These hold for every real $r$, including $r = 0$ and $r < 0$. For $r > 0$ the point is $r$ times the point $(\cos\theta, \sin\theta)$ of the unit circle. For $r < 0$ the point is also $(-r, \theta + \pi)$, and $-r\cos(\theta + \pi) = r\cos\theta$ and $-r\sin(\theta + \pi) = r\sin\theta$, so the same formulas come out.

Definition If $(r, \theta)$ is any polar name of the point $(x, y)$, then

So $\lvert r \rvert = \sqrt{x^2 + y^2}$, and when a positive radius is wanted, $r = \sqrt{x^2 + y^2}$ for every point other than the origin.

Definition For a point $(x, y)$ with $x \ne 0$, every polar angle with a positive radius satisfies $\tan\theta = y/x$. That equation alone does not settle the angle. It has two solutions in $[0, 2\pi)$, one pointing at $(x, y)$ and one pointing at the opposite point $(-x, -y)$, and $\arctan(y/x)$ always lies in $(-\frac{\pi}{2}, \frac{\pi}{2})$, so it only ever points into the right half of the plane. The two-input function $\operatorname{atan2}(y, x)$ fixes this. For $(x, y) \ne (0, 0)$ it is the unique angle $\theta$ in $(-\pi, \pi]$ with

and case by case it equals

It is undefined at the origin. Note the order of the inputs, $y$ first. If you want an angle in $[0, 2\pi)$, add $2\pi$ to a negative result.

Definition To convert a polar point $(r, \theta)$ to rectangular form, compute $x = r\cos\theta$ and $y = r\sin\theta$. To convert a rectangular point $(x, y) \ne (0, 0)$ to polar form with a positive radius, take

and add an integer multiple of $2\pi$ to $\theta$ if another range is wanted. The origin converts to $(0, \theta)$ with any $\theta$.

Definition A rectangular equation in $x$ and $y$ becomes a polar equation when you substitute $x = r\cos\theta$ and $y = r\sin\theta$. A polar equation becomes rectangular when you rewrite it in terms of

often after multiplying both sides by $r$. The graph of a polar equation is the set of points that have at least one polar name satisfying it. A conversion is correct when the two equations have exactly the same graph.

Definition Some algebra steps change the set of points an equation describes. Multiplying both sides by $r$ can add the pole. Dividing both sides by an expression can lose the points where that expression is $0$, and squaring both sides can add points where the two sides had opposite signs. The domain check asks where each formula is defined and whether such a step added or lost points. The pole needs its own test, since it lies on the graph of a polar equation exactly when some pair $(0, \theta)$ satisfies the equation. The tracing check follows $r = f(\theta)$ as $\theta$ increases and records which part of the curve is drawn, so you know an interval of $\theta$ that draws the whole curve once.