Homing verses Tracking

"John Bell" wrote in message m...
I got interested in the effects of homing on a waypoint as opposed to
tracking. I have seen the illustrations and have understood the concept for
some time, but I have never seen any numbers. If you are interested here is
the results of my playing around with Excel:

http://www.cockpitgps.com/other_arti...rack_error.htm

John Bell
www.cockpitgps.com
www.smallboatgps.com

Curiously enough, my very first posting to Compuserve's AVSIG forum,
almost twenty years ago, was on exactly this subject, in response to a
discussion between Bob Dubner and Barry Schiff.

Suppose an airplane flying at unit speed starts homing on the origin
of the (x,y) plane, starting at (1,0) in a crosswind of u. The
equations of motion are

dx/dt = -x/sqrt(x^2 + y^2)
dy/dt = -y/sqrt(x^2 + y^2) + u

with x-1, y=0 at t=0

so

dy/dx = (y - u sqrt(x^2 - y^2))/x

You can verify the the solution of this ODE is:

y =(x/2) * (x^(-u) - x^u)
and that for the homing to succeed, we must have u 1 (less
crosswind than airspeed!)

From this we can derive a couple of interesting results:

(1) The time to home is 1/(1-u^2), which we can compare to the time
to track, which is 1/sqrt(1-u^2). Reverting to dimensional units, we
can say that it takes longer by 1/sqrt(1 - (xwind/TAS)^2) to home than
to track in a direct crosswind.

(2) The maximum cross-track displacement (where dy/dx=0) is

y_max = (1/2) ( ((1-u)/(1+u))^(1/2u -1/2) - ((1-u)/(1+u))^(1/2u
+1/2) )


A great deal more numerical resolution is required to get accurate
results from your spreadsheet for other than small u, particularly
near the origin (homing point), where the track ends up coming in at
right angles to the course, however small u (but non-zero) may be.

Ed

http://williams.best.vwh.net