Ed, I found your ten-year-old message by searching the forum archives on the curve name. I hope you're still active in the community discussions.
This French webpage has an investigation of Viviani's Curve. It shows a variety of intersections of 3D surfaces that can generate the curve. Here's a sample from the webpage:
The page is in French; current MacOS Safari will translate it to English.
I also have generated the curve in the Wolfram Language with a parametric plot. This is part of a project to visualize and understand RMT Ropes (AKA "Flow Rope"): an interesting way to move (but not jump) with a jump rope:
Very neat!! Glad I could help :)