A Trisectrix from a Carpenter’s Square

UPDATE: The article is now published. Read it in Mathematics Magazine.

Yesterday I posted an article to the arXiv, “A Trisectrix from a Carpenter’s Square.

Abstract: In 1928 Henry Scudder described how to use a carpenter’s square to trisect an angle. We use the ideas behind Scudder’s technique to define a trisectrix—a curve that can be used to trisect an angle. We also describe a compass that could be used to draw the curve.

I also made a GeoGebra applet to accompany the article. Give it a try.

$(t - 4 t/(1+t^2), 3 - 4 t/(1+t^2))$ for $-\infty < t < \infty$ and
$((cos(2 a) - cos(a))/sin(a), 1 - 2 cos(a))$ for $-\pi < a < \pi$.
The "drop" part of the curve is given by $|t| < \sqrt{3}$ or $|a| < 2 \pi / 3$.
The drop has area $3\sqrt{3} \approx 5.20$. The perimeter can be written using Elliptic functions (approximately 4.12).
If the drop of the curve is considered to be the cross-section of a solid of revolution, its volume is $(8 \ln(2) - 3) \pi \approx 8.00$ and its surface are is $3 (9 - 4 \sqrt{3}) \pi \approx 19.53$.