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There is an interesting forum thread at
http://www.virtualregatta.com/forums/viewtopic.php?f=35&t=11678
The guy studied how VR algorithm moves boats in regattas.

(zedstar http://www.virtualregatta.com/forums/memberlist.php?mode=viewprofile&u=1560205)

To me there are 3 possibilities:
1- Move boats using flat trigonometry.
2- Move boats using spherical trigonometry (Great Circle formulas).
3- Move boats using an ellipsoid model (such as WGS84).

Number 3 probably gives the results closest to the real thing
(since gps devices give wgs84 coords)
This calculation can be done using Vincenty formulas. Check this site:
http://www.movable-type.co.uk/scripts/latlong-vincenty-direct.html
Needless to say, the formula is big and slow to calculate.

I tried to check the differences between the results of the 3.
Let's say our boat is sailing at 20 knts. In 10 minutes it sails a distance=6.17 km.

for lat1=0, lon1=0
Course=0 --> GC= 0°03.33'N / 0°-0.00'W | flat= 0°03.33'N / 0°-0.00'W | vincenty dest= 0°03.35'N / 0°00.00'W - flat-vinc=34.63 m | flat-sphe=0.002 m
Course=45 --> GC= 0°02.35'N / 0°02.35'E | flat= 0°02.35'N / 0°02.35'E | vincenty dest= 0°02.37'N / 0°02.35'E - flat-vinc=24.96 m | flat-sphe=0 m
Course=90 --> GC= 0°00.00'N / 0°03.33'E | flat= 0°00.00'N / 0°03.33'E | vincenty dest= 0°00.00'N / 0°03.33'E - flat-vinc=6.90 m | flat-sphe=0 m

for lat1=45N, lon1=0
Course=0 --> GC=45°03.33'N / 0°-0.00'W | flat=45°03.33'N / 0°-0.00'W | vincenty dest=45°03.33'N / 0°00.00'W - flat-vinc=3.47 m | flat-sphe=0.002 m
Course=45 --> GC=45°02.35'N / 0°03.33'E | flat=45°02.35'N / 0°03.33'E | vincenty dest=45°02.35'N / 0°03.32'E - flat-vinc=9.25 m | flat-sphe=3.34 m
Course=90 --> GC=44°60.00'N / 0°04.71'E | flat=45°00.00'N / 0°04.71'E | vincenty dest=44°60.00'N / 0°04.70'E - flat-vinc=17.48 m | flat-sphe=2.98 m

for lat1=80N , lon1=0
Course=0 --> GC=80°03.33'N / 0°-0.00'W | flat=80°03.33'N / 0°-0.00'W | vincenty dest=80°03.32'N / 0°00.00'W - flat-vinc=25.70 m | flat-sphe=0 m
Course=45 --> GC=80°02.35'N / 0°13.61'E | flat=80°02.35'N / 0°13.56'E | vincenty dest=80°02.34'N / 0°13.55'E - flat-vinc=26.69 m | flat-sphe=18.93 m
Course=90 --> GC=79°59.99'N / 0°19.17'E | flat=80°00.00'N / 0°19.17'E | vincenty dest=79°59.99'N / 0°19.09'E - flat-vinc=31.80 m | flat-sphe=16.94 m

The larger difference between the 3 formulas seems to be around 34 m (0.55% in 6.17 km)
Pretty small. Note that flat to spherical differences are even smaller.
So it doesn't make much difference which formulas you use..

Thanks Omar, for having made this study and publishing your result.
If you allow me, I'd contest a little your conclusion.
According to me, 0.55% in 10 minutes is a tremendous shifting.
What would it make after 5 hours, which is not that much ?
About 1 km ?

Ops.. removed code for correction... be back soon..

After 5 hours it could be 100 seconds. This may be large when
comparing boat times (=competition).
But if every one is subject to the same formula,
its really not much.. In geological time

Ricoo wrote:Thanks Omar, for having made this study and publishing your result.
If you allow me, I'd contest a little your conclusion.
According to me, 0.55% in 10 minutes is a tremendous shifting.
What would it make after 5 hours, which is not that much ?
About 1 km ?

It may even been more than that, considering that a little shift in lat or lon involves sometimes a change of wind square.

Note: with VR, the travel of the boat does not follow a great circle course but small 10-minutes-long great circle segments.
Each of these segments start the same bearing (boat bearing).

I run an experiment with 3 boats, starting from the
same point ( 36S 56W near Punta) and time.
Velux boat, nice 20 knts SW wind.

Each boat used a different model (formula) for movements.
- Spherical formula - boat passoca (blue)
- Flat Mercator Trigonometry - boat pangrilo (orange)
- Vincenty WGS84 - boat jacare (green)

In the figure below we see the result.

After 7 hours (122.1 NM) the difference between
green and orange is about 0.25 NM.
Flat and spherical are pretty close.
WGS84 (green) is a little out.
Maybe this is because this is geodetic coordinates ?

You say for this to be right, one should integrate a rhumb line?





http://en.wikipedia.org/wiki/Geodetic_system

omar wrote:I run an experiment with 3 boats, starting from the
same point ( 36S 56W near Punta) and time.
Velux boat, nice 20 knts SW wind.

Each boat used a different model (formula) for movements.
- Spherical formula - boat passoca (blue)
- Flat Mercator Trigonometry - boat pangrilo (orange)
- Vincenty WGS84 - boat jacare (green)

In the figure below we see the result.

After 7 hours (122.1 NM) the difference between
green and orange is about 0.25 NM.

Nice test.
Have you used the same value as Zedstar for the Earth radius in the great-circle computation?

One note about the 0.25 nm: at most 1" (angle) = 60 nm/3600 = 0.017 nm
=> Estimation for 0.25nm : about 15"

The Radius of the earth was assumed to be 6371 km
for spherical and flat formulas.
Used the formulas from the excellent website:
http://www.movable-type.co.uk/scripts/latlong.html

For WGS84 ellipsoid it goes:
a = 6378137 m //a & b semi-axis
b = 6356752.3142 m
f = 1/298.257223563

To be really realistic, one should
use a rhumb line ( a segment sailed with fixed course ),
integrated over the 10 min period, using WGS84 ellipsoid.
This is what a real sailor would read in the GPS device.

That would be some formula...

Thanks for your study, Omar.
Practicaly, the roads are so close that it's almost invisible.
Except in one case : if you are very close to the wind limit, and with a 90° bearing, you can pass the limit unwilingly. And the result can be quite bad (I know it by experience ).
Hervé