In the first model of 2D elliptic geometry, "points" and "lines" are the lines and planes through the origin in 3D, respectively. These axioms are satisfied: two "points" determine a "line" (because the two ordinary lines determine an ordinary plane), and two "lines" determine a "point" (intersect the two ordinary planes to get an ordinary line). There are no parallel "lines", because all ordinary planes intersect.
Another definition uses the points and great circles on a sphere with opposite points identified. This definition seems more natural than the first because the points are more point-like and the lines are one-dimensional, but the identification of opposite points is somewhat disorienting, especially for the great circles.
The two models are clearly related: a line through the center of a sphere intersects the sphere in two opposite points and a plane through the center intersects the sphere in a great circle.
For the choice of a pair of planes, the sliders govern the planes' normals.