Polar Coordinates and Curves

A structured guide to representing points and curves in polar coordinates, converting between coordinate systems, analyzing graphs and tangents, and calculating polar areas and arc lengths.

Representing points and converting coordinates

A polar point is written as $(r,\theta)$ . The pole is the origin, $r$ is the from it, and $\theta$ is measured counterclockwise from the positive horizontal axis. Angles are usually expressed in radians.

The same point may have several representations. For any integer $k$ ,

(r,\theta)=(r,\theta+2\pi k)

and

(r,\theta)=(-r,\theta+\pi+2\pi k).

The second identity explains how a reverses the direction of the point.

To convert from polar to rectangular coordinates, use

x=r\cos\theta,\qquad y=r\sin\theta.

To convert in the other direction, use

r^2=x^2+y^2,

and determine the quadrant before finding $\theta$ from $\tan\theta=y/x$ . The signs of $x$ and $y$ , or a two-argument inverse tangent, prevent quadrant errors.

For example, starting with $r=4\cos\theta$ , multiply by $r$ and substitute $r^2=x^2+y^2$ and $r\cos\theta=x$ :

x^2+y^2=4x.

Completing the square gives

(x-2)^2+y^2=4,

which is a circle centered at $(2,0)$ with radius $2$ .

Takeaway: Interpret the sign of $r$ , use the conversion equations systematically, and check the quadrant when determining an angle.

Graphing polar curves and recognizing families

A polar curve is commonly written as $r=f(\theta)$ . To graph it, select values of $\theta$ , compute the corresponding values of $r$ , plot the resulting points, and connect them smoothly. Negative values of $r$ should be plotted in the direction opposite the listed angle.

Symmetry can reduce the amount of computation. For symmetry about the polar axis, replace $\theta$ by $-\theta$ . For symmetry about the vertical axis, replace $\theta$ by $\pi-\theta$ . For symmetry about the pole, replace $r$ by $-r$ , or replace $\theta$ by $\theta+\pi$ . If the equation is unchanged, the corresponding symmetry is present. These tests are sufficient but not necessary, so a failed test does not prove that symmetry is absent.

Several families have recognizable forms:

$r=a$ is a circle centered at the pole with radius $|a|$ .
$r=a\cos\theta$ and $r=a\sin\theta$ are circles passing through the pole.
$r=a\pm b\cos\theta$ and $r=a\pm b\sin\theta$ produce limacons and cardioids.
$r=a\cos(n\theta)$ and $r=a\sin(n\theta)$ produce rose curves.
$r=a\theta$ produces an Archimedean spiral.

Before selecting an interval for a rose curve or another repeating curve, determine where $r=0$ , whether the curve returns to the pole, and whether any portion is retraced.

Takeaway: Combine a table of values with symmetry and the known curve family, then verify that the chosen interval traces the desired curve exactly once.

Derivatives and tangent lines

Treat the polar equation as a parametric curve with parameter $\theta$ :

x(\theta)=r(\theta)\cos\theta, \qquad y(\theta)=r(\theta)\sin\theta.

Differentiation gives

\frac{dx}{d\theta}=\frac{dr}{d\theta}\cos\theta-r\sin\theta,

and

\frac{dy}{d\theta}=\frac{dr}{d\theta}\sin\theta+r\cos\theta.

Therefore, when $dx/d\theta\ne0$ ,

\frac{dy}{dx}=\frac{\dfrac{dy}{d\theta}}{\dfrac{dx}{d\theta}}.

To find the tangent line at $\theta=\theta_0$ , first calculate

r_0=f(\theta_0),\qquad x_0=r_0\cos\theta_0,\qquad y_0=r_0\sin\theta_0.

Then evaluate the slope and use

y-y_0=m(x-x_0).

A horizontal tangent generally occurs when $dy/d\theta=0$ and $dx/d\theta\ne0$ . A vertical tangent generally occurs when $dx/d\theta=0$ and $dy/d\theta\ne0$ . If both derivatives vanish, further analysis is needed because the point may be a cusp, a singular point, or a temporary stop in the parametrization.

For $r=1+\sin\theta$ at $\theta=\pi/2$ , one has $r=2$ and $dr/d\theta=0$ . Thus $dy/d\theta=0$ , $dx/d\theta=-2$ , and the slope is $0$ . The point is $(0,2)$ , so the tangent line is $y=2$ .

Takeaway: Differentiate both coordinate functions, check the denominator before using the slope formula, and calculate the point before writing the tangent equation.

Finding areas with polar integrals

A small sector swept through an angle change $\Delta\theta$ has approximate area

\Delta A\approx\frac12r^2\Delta\theta.

Taking a limit gives the area formula

A=\frac12\int_\alpha^\beta r^2\,d\theta.

To find the area inside one curve, first identify zeros of $r$ , intersections, or symmetry intervals. Then select limits that trace the intended region without retracing it. For two curves, solve their intersection equation and determine which curve is outer on each interval. The correct difference is

A=\frac12\int_\alpha^\beta\left(R(\theta)^2-r(\theta)^2\right)d\theta,

not $(R(\theta)-r(\theta))^2$ . The curves may switch roles, so test a convenient angle whenever necessary. Also check for an intersection at the pole, even if the equations do not yield the same nonzero radius.

For one petal of $r=3\sin(2\theta)$ , consecutive zeros occur at $\theta=0$ and $\theta=\pi/2$ . Hence

A=\frac12\int_0^{\pi/2}9\sin^2(2\theta)\,d\theta =\frac{9\pi}{8}.

Takeaway: The main challenge in polar-area problems is choosing correct limits and identifying the outer curve; the integration itself follows directly from the sector formula.

Arc length and error checks

For $r=f(\theta)$ , write the curve parametrically as

x=f(\theta)\cos\theta, \qquad y=f(\theta)\sin\theta.

The parametric arc-length formula simplifies to

L=\int_\alpha^\beta\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}\,d\theta.

Use limits that trace the desired portion exactly once. Algebraic simplification may produce an absolute value. For example, $\sqrt{\cos^2 u}=|\cos u|$ , not always $\cos u$ .

For the cardioid $r=2+2\cos\theta$ on $0\le\theta\le2\pi$ ,

\frac{dr}{d\theta}=-2\sin\theta,

r^2+\left(\frac{dr}{d\theta}\right)^2=8+8\cos\theta=16\cos^2\left(\frac\theta2\right).

Therefore,

L=\int_0^{2\pi}4\left|\cos\left(\frac\theta2\right)\right|\,d\theta=16.

A final review should check for retracing, negative-radius behavior, incorrect quadrant selection, missed intersections, incorrect squared differences in area, dropped absolute values, and cases where $dx/d\theta=0$ .

Takeaway: is a parametric arc-length calculation in simplified form; careful interval selection and absolute-value handling are essential.