7
$\begingroup$

How do I calculate the distance between the line joining the two points on a spherical surface and another point on same surface? I have illustrated my problem in the image below.

Sphere

In the above illustration, the points A, B and X lies on a spherical surface, I need to find the distance between points (A,B) and X. I am not a mathematics guy. If possible please illustrate me the solution as non-mathematics guys could understand. Thanks.

  • 0
    Does this line lie on the sphere or inside the sphere?2011-02-21
  • 0
    geodesic distance on the sphere?2011-02-21
  • 0
    Have you tried to set up the standard parametrized integral for the distance between X and an arbitrary point on the line, then vary the integral to find where it obtains it's minimum?2011-02-21
  • 0
    How are A, B, and X specified? For example, by latitude and longitude? And is the distance along the sphere or a straight line through the sphere?2011-02-21
  • 0
    Use the [Law of Sines](http://en.wikipedia.org/wiki/Law_of_sines#Spherical_case) to get the altitude XC in terms of the length of XB (use the [Haversine formula](http://en.wikipedia.org/wiki/Haversine_formula)) and the spherical angle ABX (computed as the angle between planes OAB and OXB; O = center of earth). Planar angles are computed by converting A, B, X to Cartesian coordinates, using cross-products to obtain the planar normals, and taking their dot products (to get the cosine). Check that angles ABX and BAX are less than 90 degrees; if not, you need the distance BX or AX instead.2011-02-21
  • 1
    @Ross Millikan: the points A, B and X are in Latitudes and longitudes. the distance is along the sphere.2011-02-23

4 Answers 4