Equations in polar coordinates

Many curves have a polar equation far simpler than their Cartesian one. It is enough to express $\rho$ as a function of $\theta$ (or to impose a condition relating them).

Property — Elementary curves in polar coordinates

• Circle centred at the pole, radius $a$: $\rho = a$.
• Line through the pole, inclination $\theta_0$: $\theta = \theta_0$.
• Line perpendicular to the polar axis at distance $d$: $\rho\cos\theta = d$ (that is, $x = d$).
• Circle through the pole, of radius $a$ and centre on the polar axis: $\rho = 2a\cos\theta$.
• Cardioid: $\rho = a(1 + \cos\theta)$.
• $4$-petal rose: $\rho = a\cos(2\theta)$ ($a>0$).
• Archimedean spiral: $\rho = a\theta$, $\theta\ge 0$.
• Logarithmic spiral: $\rho = a\,e^{b\theta}$.

Cardioid $\rho = 1+\cos\theta$.

$4$-petal rose $\rho = \cos(2\theta)$.

Archimedean spiral $\rho = \theta$.

Example — From polar to Cartesian: the circle $\rho = 2a\cos\theta$

Multiply both sides by $\rho$: $\rho^2 = 2a\rho\cos\theta$, that is $x^2+y^2 = 2ax$. Rewriting: $(x-a)^2 + y^2 = a^2$: it is the circle of centre $(a,0)$ and radius $a$, passing through the origin.

Example — From Cartesian to polar: the line $y = x$

Substituting $x = \rho\cos\theta$, $y = \rho\sin\theta$: $\rho\sin\theta = \rho\cos\theta$, that is $\tan\theta = 1$, whence $\theta = \pi/4$ (one ray) or $\theta = -3\pi/4$ (the other ray).

Links

Topics: Goniometry
Concepts: Cardioid · Polar coordinates · Polar equation · Cartesian plane · Spiral
Methods: Polar equation
Skills: Plotting a graph · Using formulae