Distance from a Point to a Great Circle

Date: 05/24/2000 at 23:36:17
From: Dean Bellamy
Subject: 'Perpendicular' distance from point to great circle

Here's a variation on the "distance between two points specified by 
latitude and longitude" problem that's frequently asked.

I have a great circle connecting two points A and B (specified by 
their latitude and longitude). I want the shortest great-circle 
distance to that great circle from a third point C (also specified by 
latitude and longitude). For example, given the shortest possible road 
connecting two cities, what is the shortest distance I'd have to walk 
from my position to that road.

I can work out the perpendicular distance in a 2-D situation by 
finding the projection of AC on AB using a dot-product, but I'm not 
sure how to apply this to the surface of a sphere.

Date: 05/25/2000 at 08:33:11
From: Doctor Rick
Subject: Re: 'Perpendicular' distance from point to great circle

Hi, Dean.

Since you sound as if you're vector-savvy, I won't try to work this 
out in detail, but just give you an approach that should do the trick.

Let vectors A and B be unit vectors pointing from the center of the 
earth toward points A and B. Take the cross-product of A and B and 
normalize the result to get a vector N:

     N = (A x B)/|A x B|

N is normal to the plane of the great circle joining A to B.

Now take the dot product of N with C, the unit vector corresponding to 
point C.

     N . C = cos(<NOC)

where <NOC is the angle between N and C. The difference between this 
angle and a right angle is the arc corresponding to the distance from 
C to the great circle. Express it in radians and multiply by the 
radius of the earth, and you're done.

If you need more explanation of any of these steps or concepts, just 
ask. There is some explanation of the use of spherical coordinates 
relative to longitude and latitude, and the use of cross products and 
dot products in this context, in various Dr. Math Archive items that 
you can find by searching for   latitude longitude  , for example. It 
sounds as if you may have seen these.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/