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### Distance from a Point to a Great Circle

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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/
```
Associated Topics:
College Linear Algebra
College Non-Euclidean Geometry

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