
Learn how to draw a torus.

A common shape studied in mathematics is a torus or donut. To draw a torus by-hand like a pro is easy. Start by drawing an ellipse:

Now make that ellipse smile!

Finally, add in an upside down arc:

And like magic, we have drawn a torus! On the other hand, if you want to use a computer to draw a torus, perhaps you should use the parametric formula:

where $$ is the radius from the center of the torus to the center of the “tube,” $$ is the radius of the “tube,” $$, and $$. However, listen, you could have derived a parametric formula using the techniques you’ve learned. Let’s do it.

If you’re paying attention, you may notice that we now have a very similar formula to the one given above, except that we have some minus signs where before we had plus signs. What happened here?
We made a mistake in our work in the example above. We lied to you when we give the initial parametric formula for the torus. We just broke math. Everybody wins. Both formulas draw a torus.