Breaking Kidney Stones with an Ellipse

A lithotripter fires shock waves from one focus of an elliptical reflector. Every wave that bounces off the ellipse heads straight for the other focus, where doctors position the kidney stone. The stone takes the full blow while the tissue along the way doesn't.

How it works

Ellipses and their foci

In one sentence

An ellipse with foci $F_1, F_2$ is the set of points $P$ with $|PF_1| + |PF_2| = 2a$ . Centered at the origin with its major axis on the x-axis it's $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ( $a > b > 0$ ), with foci $(\pm c, 0)$ where $c = \sqrt{a^2 - b^2}$ . A ray from one focus reflects off the ellipse through the other.

General formula

\frac{x^2}{{\htmlData{sym=0}{a}}^2} + \frac{y^2}{{\htmlData{sym=1}{b}}^2} = 1, \qquad {\htmlData{sym=2}{c}}^2 = {\htmlData{sym=0}{a}}^2 - {\htmlData{sym=1}{b}}^2, \qquad \text{foci at } (\pm {\htmlData{sym=2}{c}}, 0)

Tie a string to two pins and trace around them with a pencil: the string's fixed length is why every bounce path between the foci is equally long.

What the symbols mean

$a$
Semi-major axis
Half the ellipse's length; every bounce path is 2a long.
$b$
Semi-minor axis
Half the ellipse's height.
$c$
Focal distance
How far each focus sits from the center.

An ellipse is the set of points whose distances to two fixed points, the foci, add up to the same total $2a$ . In standard form it's $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ .
The foci sit at $(\pm c, 0)$ with $c^2 = a^2 - b^2$ : the flatter the ellipse, the farther apart they are.
Because every route from one focus to the ellipse and on to the other focus has the same length, a wave leaving one focus reflects to the other, and all the reflected waves arrive together.
So the energy concentrates at the second focus, the kidney stone, and stays spread out everywhere else.

Learn the math

EllipsesAlgebra · Lesson 29, Common Graphs