Consider the set of 6 points $((-12,-3), (3,-6), (3,7), (9,-3), (11,-1), (12,1))$. They make an ellipse. No lines generated by the points are parallel. When these points are used to make a Hexagrammum Mysticum, all of the 60 Kirkman Points, 20 Steiner points and 15 Salmon Points are distinct and finite. The 60 Pascal lines, 20 Cayley lines, and 15 Plücker lines are also distinct. As an aside, I learned from Dick Tahta's wonderful Fifteen Schoolgirls that these guys hated each other.
When the initial six points are all rational or in the same root space, the generated points are in the same space. The task at hand is to generate a set of points with at least the first condition, and hopefully more.
- Generate a set of 6 simple points on a conic section with no parallelism.
- The 95 Kirkman Steiner Salmon points should be finite and distinct.
- The 101 points should be human-eye distinguishable in a single image.
I got these 6 points by looking at subsets from 13 point ellipse and Small Lattice Ellipses.
My solution above fails #3. Maybe different root spaces, different approaches or different conic sections will work better.
