This has precious* little to do with Dan's method, but he spurred me locate the Grapher app on my computer (Utilities folder??), take a demo that comes with it (Torus-knot), turn it into a trefoil knot, and then fix the graphics that got broke. This page shows the settings as well as the picture: http://www.mac-guyver.com/switham/2020/05/Torus_knot <http://www.mac-guyver.com/switham/2020/05/Torus_knot/index.html> (As far as I know, the only way to get equations into Grapher is with its own editor. Their file format looks opaque. It can apparently import point sets, though...) --Steve *my precious. Fit for an elven king. Btw, summary of _The Hobbit_: Marty Feldman gets hold of an all-access key card to Cheyenne Mountain.
Dan Asimov <dasimov@earthlink.net> Date: 5/25/20, 5:45 PM
I tried to find equations in R^3 for the trefoil defined initially in R^4 = {(x,y,z,w)}:
(x+iy)^2 + (z+iw)^3 = 0
and
x^2 + y^2 + z^2 + w^2 = 1
but which is then stereographically projected into R^3 = {(X,Y,Z)} via
X = x/(1-w), Y = y/(1-w), Z = z/(1-w).
If my algebra is right, the final (two) equations are
4(X^2 - Y^2)(X^2 + Y^2 + Z^2 + 1) + 8 Z^3 - 6 Z W^2 = 0 & 8 X Y (X^2 + Y^2 + Z^2 + 1) + 12 Z^2 W - W^3 = 0
where W is short for (X^2 + Y^2 + Z^2 - 1).
I can't tell for sure if it looks right when I try to plot it in 3D using Mac "Grapher".
—Dan