Microphone Pickup Patterns

A microphone's sensitivity depends on the direction sound comes from, so it's drawn in polar coordinates: distance from the center is the sensitivity at each angle. Vocal mics use a heart-shaped cardioid, so they hear the singer and ignore the speakers behind them.

How it works

Polar coordinates and tangents

In one sentence

A polar point $(r, \theta)$ is $x = r\cos\theta$ , $y = r\sin\theta$ ; a polar curve $r = f(\theta)$ has slope $\frac{dy}{dx} = \frac{f'(\theta)\sin\theta + f(\theta)\cos\theta}{f'(\theta)\cos\theta - f(\theta)\sin\theta}$ .

General formula

{\htmlData{sym=0}{r(\theta)}} = {\htmlData{sym=2}{a}} + (1 - {\htmlData{sym=2}{a}})\cos{\htmlData{sym=1}{\theta}}, \qquad \frac{dy}{dx} = \frac{r'({\htmlData{sym=1}{\theta}})\sin{\htmlData{sym=1}{\theta}} + {\htmlData{sym=0}{r(\theta)}}\cos{\htmlData{sym=1}{\theta}}}{r'({\htmlData{sym=1}{\theta}})\cos{\htmlData{sym=1}{\theta}} - {\htmlData{sym=0}{r(\theta)}}\sin{\htmlData{sym=1}{\theta}}}

Instead of left-right and up-down, describe each point by which way it lies and how far. For direction-dependent things like microphones, that's the natural grid.

What the symbols mean

$r(\theta)$
Sensitivity
How strongly the mic picks up sound from angle θ.
$\theta$
Direction
Angle of the sound source, 0 = straight in front.
$a$
All-around share
1 hears everything equally; 0 is a figure-8.

In polar coordinates a point is $(r, \theta)$ : distance $r$ from the origin at angle $\theta$ , with $x = r\cos\theta$ and $y = r\sin\theta$ .
The pattern $r = a + (1 - a)\cos\theta$ mixes an all-around part ( $a$ ) with a figure-8 part ( $\cos\theta$ ).
At $a = \frac12$ it's a cardioid: full sensitivity in front ( $\theta = 0$ ), none from behind ( $\theta = \pi$ ).
The slope of the pattern comes from differentiating $x$ and $y$ in $\theta$ , which gives the tangent formula above.

Learn the math

Polar CoordinatesCalc II · Lesson 20, Parametric Equations and Polar Coordinates Tangents with Polar CoordinatesCalc II · Lesson 21, Parametric Equations and Polar Coordinates