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?
 A: It sounds like you're just trying to carry out the picture of spherical geometry in 3-D to 4-D. 
In the 3-D picture, the surface of the unit sphere is taken to be the set of points, and the "lines" are the great circles. Any two points which aren't antipodal determine a unique great circle. 
The way to look at the great circle in that case is as the intersection of a plane through the origin with the sphere. This is suggestive then that you are looking for the intersection of the plane through your two points and the origin in 4-D, and the line should be the intersection of this plane with the sphere $S_3$.
Looking at it from this perspective, you can believe that two nonantipodal points with the origin of 4-space form a noncollinear set of three points, and hence a unique plane. 
A: The origin and the 2 given points determine a plane in 4-space, so now you can use your "line" in that plane.
A: For any given surface with a metric, there are some differential equations you can solve in order to determine what a geodesic looks like. In your case the surface is the three-sphere, $S^3$, and a metric can be found on the wiki page http://en.wikipedia.org/wiki/3-sphere under hyper-spherical coordinates. The differential equations for a geodesic can then be found by some calculus of variations. I suggest you first see how this works out for $S^2$.
