Homing verses Tracking

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

Slightly off this thread topic, but what formula do you use to calc the XTE?

I've used a variation of Ed Williams' formula "XTD
=asin(sin(dist_AD)*sin(crs_AD-crs_AB))" but if I run my calcs in parallel
with a GPS they are consistently different with the GPS numbers varying
considerably more than mine.

"John Bell" wrote in message
...
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

Slightly off this thread topic, but what formula do you use to calc the
XTE?

I've used a variation of Ed Williams' formula "XTD
=asin(sin(dist_AD)*sin(crs_AD-crs_AB))" but if I run my calcs in parallel
with a GPS they are consistently different with the GPS numbers varying
considerably more than mine.

For anybody on the cross post response list, this refers to Ed William's
Aviation Formulary page. There are some things directly related such as
airspeed and altimetry issues. However, there are also some useful general
navigation formulas with more general applicability:
http://williams.best.vwh.net/avform.htm

Ron,

The way that I did the spreadsheet, I never had to use this formula.
Without checking, here is a guess: I think crs_AB would refer to the GPS
value of course and crs_AD would refer to the GPS value of BRG.

If you go to the very top of the text, Ed talks about using radians to
measure distance:

Great circle distance can be likewise be expressed in radians by defining
the distance to be the angle subtended by the arc at the center of the
earth. Since by definition, one nautical mile subtends one minute (=1/60
degree) of arc, we have:

distance_radians=(pi/(180*60))*distance_nm
distance_nm=((180*60)/pi)*distance_radians

Note: the nautical mile is currently defined to be 1852 meters - which to be
consistent with its historical definition implies the earth's radius to be
1.852 * (180*60/pi) = 6366.71 km, which indeed lies between the currently
accepted ( WGS84) equatorial and polar radii of 6378.137 and 6356.752 km,
respectively. Other choices of the earth's radius in this range are
consistent with the spherical approximation and may for some specialized
purposes be preferred.

Since 1 radian = 180/pi degrees, you can use distance_degrees=
distance_nm/60.

I keep some of Ed's formulas on my Palm PDA. For more info:
http://www.cockpitgps.com/palm/index.htm

John

"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

(Ed Williams) wrote in
om:

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.

It's good to see you here, Ed. Your Aviation Formulary is one of the
treasures of the internet, IMO. I've used it many times. Thank you for
providing it.

Followups set to rec.aviation.ifr.

--
Regards,

Stan

Stan Gosnell wrote:
(Ed Williams) wrote in
om:


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.


It's good to see you here, Ed. Your Aviation Formulary is one of the
treasures of the internet, IMO. I've used it many times. Thank you for
providing it.

Followups set to rec.aviation.ifr.

I'll second that. Glad to know you lurk here, Ed.

Dave