Polar Coordinates

Cartesian coordinates measure a point by two perpendicular distances. Many curves that are awkward in that grid become simple when a point is instead located by how far it is from a center and in what direction. That is the polar system.

Non-uniqueness. Unlike Cartesian coordinates, a polar point has many names. A full turn adds nothing, and a negative radius points the opposite way:

The convention for $r<0$ is that $(-r,\theta )$ lies at distance $|r|$ from $O$ on the ray opposite to $\theta$. So $(2,\pi /3)$, $(2,7\pi /3)$, and $(-2,4\pi /3)$ all name the same point.

Converting between systems

Placing the pole at the origin and the polar axis along the positive $x$-axis lines up the two systems. Right-triangle trigonometry reads off the relations in both directions.

• Polar to Cartesian: $x=r\cos \theta$, $y=r\sin \theta$.
• Cartesian to polar: $r^2=x^2+y^2$, $\tan \theta =\frac{y}{x}$.

The first pair is unambiguous. The second is not: $\tan \theta$ repeats every $\pi$, so a value of $\tan \theta$ leaves two candidate angles, and the quadrant of $(x,y)$ decides which one.

Polar curves

The graph of $r=f(\theta )$ is the set of points with at least one polar representation satisfying the equation. A few equations that would be clumsy in Cartesian form are immediate here.

• $r=a$ is the circle of radius $|a|$ centered at $O$.
• $\theta ={\theta }_0$ is the line through $O$ at angle ${\theta }_0$.
• $r=2a\cos \theta$ is a circle of radius $a$ through the pole; substituting $r\cos \theta =x$ and $r^2=x^2+y^2$ gives $(x-a{)}^2+y^2=a^2$.

More elaborate curves are read off a Cartesian plot of $r$ against $\theta$: as $\theta$ advances, watch $r$ grow and shrink, and swing a radial arm to that length. Three families recur often enough to name.

Three more curves round out the standard catalog: the odd rose keeps its petal count, the lemniscate is a figure-eight through the pole, and the Archimedean spiral winds outward at a rate set by the angle.

Sketching the cardioid $r=1+\sin \theta$. As $\theta$ goes $0\to \pi /2$, $r$ rises $1\to 2$; as $\theta$ goes $\pi /2\to \pi$, $r$ falls $2\to 1$; as $\theta$ goes $\pi \to 3\pi /2$, $r$ falls $1\to 0$, pinning the cusp at the pole; and as $\theta$ goes $3\pi /2\to 2\pi$, $r$ rises $0\to 1$. The trace closes into a heart.

Symmetry

Recognizing symmetry halves the plotting work. Each rule is an invariance of the equation under a substitution.

Since $\cos (-\theta )=\cos \theta$, the circle $r=2\cos \theta$ and the rose $r=\cos 2\theta$ are symmetric about the polar axis; the cardioid $r=1+\sin \theta$, unchanged under $\theta \to \pi -\theta$, is symmetric about $\theta =\pi /2$.

Tangents to polar curves

Treat $\theta$ as a parameter. Writing $x=r\cos \theta =f(\theta )\cos \theta$ and $y=f(\theta )\sin \theta$ and applying the parametric slope rule with the product rule gives the slope in terms of $r$ and $\mathbf{d}r/\mathbf{d}\theta$.¹

Horizontal tangents occur where the numerator vanishes (and the denominator does not); vertical tangents where the denominator vanishes. At the pole, where $r=0$, the formula collapses to $\mathbf{d}y/\mathbf{d}x=\tan \theta$, so the tangent line at the pole is $\theta ={\theta }_0$ for each ${\theta }_0$ with $f({\theta }_0)=0$. For the rose $r=\cos 2\theta$, which passes through the pole at $\theta =\pi /4$ and $3\pi /4$, those two angles are the tangent lines there.

Area in polar coordinates

Rectangular area sums thin rectangles. Polar area sums thin circular sectors. The area of a sector of radius $r$ and central angle $\theta$ is $\frac{1}{2}{r}^2\theta$, so a wedge of angular width $\mathbf{d}\theta$ at radius $r=f(\theta )$ contributes $\frac{1}{2}[f(\theta ){]}^2\mathbf{d}\theta$.

Summing over sectors and passing to the limit is a Riemann sum for $\frac{1}{2}[f(\theta ){]}^2$.

The right petal of the rose $r=\cos 2\theta$ is swept as $\theta$ runs from $-\pi /4$ to $\pi /4$, the angular window where $\cos 2\theta \ge 0$.

Region between two polar curves. For $f(\theta )\ge g(\theta )\ge 0$, the area inside $r=f$ and outside $r=g$ subtracts one sector integral from the other:

A caution on intersections. Solving two polar equations can miss crossing points, because a single geometric point has many $(r,\theta )$ names and the pole has no unique name at all. The circle $r=3\sin \theta$ and cardioid $r=1+\sin \theta$ meet at the pole, yet the pole solves the first at $\theta =0$ and the second at $\theta =3\pi /2$, so no common $\theta$ reveals it. Always sketch both curves to find every intersection.

Arc length in polar coordinates

The same parametric substitution converts the arc-length integral. With $x=r\cos \theta$, $y=r\sin \theta$, the sum of squared derivatives simplifies, using ${\cos }^2\theta +{\sin }^2\theta =1$, to a clean expression in $r$ and $\mathbf{d}r/\mathbf{d}\theta$:

Length of the cardioid $r=1+\sin \theta$. Here $\mathbf{d}r/\mathbf{d}\theta =\cos \theta$, so the integrand is $\sqrt{(1+\sin \theta {)}^2+{\cos }^2\theta }=\sqrt{2+2\sin \theta }$, and over the full turn $0\le \theta \le 2\pi$ the integral evaluates to

Polar formulas from parametric ones

Every polar formula above is a polar curve read as a parametric curve in $\theta$, with $x=r\cos \theta$ and $y=r\sin \theta$ substituted in. The slope, area, and length integrals are the parametric ones after that substitution.