3.1 · Polar coordinates, conversions, and curves

Coordinates describe a directed radius

Polar coordinates $\displaystyle (r,\theta)$ locate a point by distance and direction. A negative radius plots in the direction opposite the specified angle. Equivalent representations include $\displaystyle (r,\theta+2\pi k)$ and $\displaystyle (-r,\theta+\pi+2\pi k)$.

The origin has many angle representations. When converting a nonzero rectangular point, use its quadrant rather than relying on a one-argument arctangent alone.

Use the coordinate identities

The identities are $\displaystyle x=r\cos\theta$, $\displaystyle y=r\sin\theta$, and $\displaystyle r^2=x^2+y^2$. For a point $\displaystyle (2,\pi/3)$, they give $\displaystyle (1,\sqrt{3})$.

For $\displaystyle r=2\cos\theta$, multiply by $\displaystyle r$ to obtain $\displaystyle r^2=2r\cos\theta$, hence $\displaystyle x^2+y^2=2x$. Completing the square identifies a circle centered at $\displaystyle (1,0)$ with radius $\displaystyle 1$.

$$x=r\cos\theta,\quad y=r\sin\theta$$

Plot samples and symmetry before naming a curve

Equations such as $\displaystyle r=a\cos(n\theta)$ generate roses, but radius signs affect which direction each point lies. Evaluate several angles and inspect symmetry rather than treating $\displaystyle r$ as a rectangular height.

A zero radius marks the pole. A sketch should reflect the actual equation and interval; a curve may be traced multiple times as the angle runs through a full revolution.