Any Math Majors in the House? Help Wanted.

[ender_wiggin]

They aren't parallel, two distinct great circles are never parallel. That's pretty much the defining trait of elliptic geometry.

The specific points of intersection will vary, but for any two that are fixed by points close together on the spiral, the intersection points will be quite a long ways off from the spiral, approximately 90° of central angle.

In a real sphere, yes the great circles can't be parallel, but in 2D projections they can look parallel on paper (ie in Mercator all the longitude lines are parallel). Since great circles never have any curve except for the geodesicurve, and we force the spiral into a straight line, it should stand that all the great circles run parallel to each other off the page. Of course, in order to account for the fact that they aren't actually parallel on the sphere, we have to distort the map. The point of intersection is the maximum point of distortion, which should be the maximum width of the map (like the poles in Mercator).

If you happen to find any links that explain the equations to actually implement this, please do share.

[Hai-Etlik]

In a real sphere, yes the great circles can't be parallel, but in 2D projections they can look parallel on paper (ie in Mercator all the longitude lines are parallel). Since great circles never have any curve except for the geodesicurve, and we force the spiral into a straight line, it should stand that all the great circles run parallel to each other off the page. Of course, in order to account for the fact that they aren't actually parallel on the sphere, we have to distort the map. The point of intersection is the maximum point of distortion, which should be the maximum width of the map (like the poles in Mercator).

Yes, their images under the projection would be parallel, but that is not the same thing as them being parallel themselves. The distortion would actually be more like that in the Equidistant Cylindrical projection than the Mercator projection.