# [Proj] Intersection test for two geodesics goes wrong for very long distances

Charles Karney charles.karney at sri.com
Thu Feb 7 09:45:01 EST 2013

```On 2013-02-07 08:45, Mikael Rittri wrote:
> I suppose you are familiar with the article
>
>      L. E. Sjöberg,
>      Intersections on the sphere and ellipsoid,
>      Journal of Geodesy (2002) 76: 115-120.
>
> He writes: "each of the problems of intersection ... is solved without any limitation of arc length." I tried to read it, but the math was beyond me.
>

Yes, I'm familiar with this paper.  His basic technique for solving for
the intersection point is sound (see my comments in Sec 11 of
http://arxiv.org/abs/1102.1215 ).  But it requires that you've got a
reasonable first estimate of the intersection latitude and that you know
how many times each geodesic crosses that latitude before the
length" is merely an "aspirational" statement.

If you're going to start worrying about all the weird cases, you will
need to tighten up the definition of the intersection point.  Aside from
some degenerate cases, two 2 geodesics from points 1 and 2 with
prescribed azimuths intersect each other infinitely many times.
Presumably you're interested in a "first" intersection which I would
propose you define by requiring that the the intersection point 3
minimize

abs(s_13) + abs(s_23)

Depending on the application you might want to stipulate that s_13 >= 0
and s_23 >= 0 (i.e., you don't consider going backwards on the
geodesics).  One of the points of my previous E-mail is that there's no
guarantee that either of the geodesics 13 and 23, individually, be a
shortest path.

--Charles
```