Cardioids in Coffee Cups

Cardioids in Coffee Cups

Numberphile have just done a nice video on how a cardioid shape is formed when a light is shone against the side of a mug. You can see this effect above (from the Numberphile video here). So, I decided to recreate this using Geogebra to get to understand some of the maths behind this shape.

What is a cardioid?

Cardioids can be formed by tracing a fixed point on a circle when that circle rotates around another circle of the same radius (You can see the gif this image was taken from here).

If we start with a cross sectional circle on Geogebra and imagine a light source shining horizontally onto this we generate the following:

We have marked the intersection points of the light with the mug cross section. Then we can consider what will happen to light reflected from this surface. Let’s take the point O on the cross sectional surface of the mug, and draw the tangent to the circle at this point. Then we can use Geogebra to reflect the light line in this tangent. This gives the following:

All we have to do now is repeat the process for the other points on circle. This gives:

Tidying this up we then have:

We can then see the start of a cardioid shape appearing from the envelope of reflected lines.

I then used the basic equation of a cardioid generated by a circle of radius a centred at the origin:

To fit an equation of the cardioid which represented this shape. The equation that I found that fit the Geogebra image was given by:

And so I added this as a dotted blue line and removed the initial horizontal lines. This gave the following image:

So we can see that the envelope of these reflected lines does indeed generate a cardioid. Next time you’re drinking your cup of coffee, have look and see if you can see one!