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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.
Use unit tangent vectors and unit normal vectors to derive a parametric
formula for a torus.
Imagine a circular curve in that runs through the donut,
shown in the diagram from the previous problem by the two “dots.”
Give a formula for a vector-valued function that will draw a circle in the -plane,
centered at the origin, of radius , as runs from to . Compute , the function that will
give the unit tangent vector for any value of . Simplify your answer.
Compute , the function that will give the principal unit normal vector for
any value of . Simplify your answer. Compute , the function that will
give the unit binormal vector for any value of . Simplify your answer.
Now if we put these together we can write our torus as
Simplifying and writing the components of this formula out we see:
And done. We’ve given a parametric formula for a torus.
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.
Both formulas are correct, the first one we gave was derived using “outward”
pointing normal vectors. The second one we gave was derived using “inward”
pointing normal vectors.