Plotting the cardioid

Statement

Plot the cardioid $\rho = 1-\sin\theta$ by tabulating $\rho$ for $\theta = 0, \tfrac{\pi}{4}, \tfrac{\pi}{2}, \tfrac{3\pi}{4}, \pi, \tfrac{5\pi}{4}, \tfrac{3\pi}{2}, \tfrac{7\pi}{4}$.

Solution

Compute $\rho = 1 - \sin\theta$ for each angle (approximate values with $\tfrac{\sqrt2}{2}\approx 0{,}71$):

$\theta$	$0$	$\tfrac{\pi}{4}$	$\tfrac{\pi}{2}$	$\tfrac{3\pi}{4}$	$\pi$	$\tfrac{5\pi}{4}$	$\tfrac{3\pi}{2}$	$\tfrac{7\pi}{4}$
$\sin\theta$	$0$	$0{,}71$	$1$	$0{,}71$	$0$	$-0{,}71$	$-1$	$-0{,}71$
$\rho$	$1$	$0{,}29$	$0$	$0{,}29$	$1$	$1{,}71$	$2$	$1{,}71$

The radius is greatest ($\rho=2$) towards the bottom ($\theta=\tfrac{3\pi}{2}$) and vanishes at the top ($\theta=\tfrac{\pi}{2}$), where the curve forms its characteristic cusp. Plotting the points $(x,y)=(\rho\cos\theta,\ \rho\sin\theta)$ on the plane and joining them yields a heart-shaped curve, with the cusp pointing upwards: hence the name cardioid.

Links

Topics: Goniometry
Concepts: Cardioid · Polar coordinates · Polar equation · Cartesian plane
Skills: Calculating · Plotting a graph
Exercise types: Reading a graph