OK, let's maybe wrap this up. Spoiler: I developed more cool new stuff, but didn't wrap this up. Recap: I set out to see what it would take to map the skips for a QSO, and then try to extrapolate those back to the launch angle of the dipole that KO6BTY and I use for Project TouCans at our home QTH . Along the way, I learned a lot including another use for the cross product. I also remembered, just now, that dipoles can have more than one launch angle. Taken from Radio Antenna Engineering That'll be important later. Today, as promised, I tried the three skip solution. Here's the angle of radiation from the SF station with three skips. But what's the numeric quantity for the angle in degrees? Well, I worked through yet more math. It's getting easier though. I added the code for the cross product of two vectors (position vectors in our case) as well as the usage I'm making of the cosine and sine laws to to earthmid.py . The numeric result of three skips is 7