Do two points determine a unique line in 4D space? I wish to generalize the notion of two points determining a unique line to four dimensions, but with the additional condition that all points on the line are a unit distance from the origin and the "line" is not straight, but forms a least-distance curve between the two points. This is easy to do in the case of three dimensions: the line is a great circle and is defined by p.x=0, where p is found by the cross product of the two points. But I can't seem to generalize this to four dimensions. Is there not a unique geodesic curve that passes through two points? If there is, how would I express it in terms of the two points?

The origin and the 2 given points determine a plane in 4-space, so now you can use your "line" in that plane.
