# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Re: Closest point of approach.
From: Peter Hakel
Date: 2012 Jan 3, 07:50 -0800
Hi Antoine,

Using the "brute force" method of:

*) parametrizing both trajectories as linear functions of time (i.e. constant velocities), and,
*) using calculus to minimize the distance between aircraft,

leads to a linear equation for the time of CPA, with a single and unique solution.  You can plug this solution back into the parametrization to get everything else.  This can admittedly be a bit tedious, if a calculator is used.  If the procedure is coded once into a computer, then it's not so bad for the user.  The steps get a bit easier if you place the coordinate system origin on your aircraft and work with relative trajectory and velocity of the second aircraft (which is quite useful).  Then it's about the distance between a line and a point, which can be solved using effective 2-D, as Gary has shown.

One good thing about this is that a purely geometric distance (i.e. with time parametrization removed) between two lines in space is really the worst case scenario.  Two aircraft can be (and often are) in the same place on the planet, just not at the same time!  :-))

Peter Hakel

From: Antoine Couëtte <antoine.m.couette@club-internet.fr>
To: NavList@fer3.com
Sent: Tuesday, January 3, 2012 2:08 AM
Subject: [NavList] Re: Closest point of approach.

... However, the least distance between 2 lines in space - for example the track of 2 A/C at different levels - is not a trivial computation, as I can remember. It showed in the HP41 Calculator Mathematics and Geometry Solutions Booklet as their very last example if I can recall correctly.
And to you, Gary, if you did solve it just lately (sorry, no time to look into it, I will be fly back to Europe shortly) in 3 dimensions, then ... Attaboy.
Happy New Year to all
Kermit
Antoine M. Cou�tte
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