what would my path look like using the sun for navigation

A theoretical question which thought about instead of writing my C++ code. I know it is poorly defined however I am now intrigued as to the answer!

If you got into a boat in the middle of the sea before sunrise, and as the sun cam up sailed directly towards it, and continued to do so throughout the day until you were heading west in the evening at sunset, and then stopped at sun down. Repeat this for say a week/month, then look on a GPS device as to the path you have traveled, what shape would your path/footprint be?

Everyone I have asked has come up with a different answer, and short of buying a boat not sure how to work it out.

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You do realise this will depend a lot on your latitude, time of year, boat speed (and other factors such as tides but I'm guessing you want to rule out external factors for this example)

If you are on the latitude that places the sun directly overhead, you will start sailing east until midday, then sail west: a straight line

If you are north of the sun's latitude, your path will be more of an arc, with the amount of southing dependent on your latitude (similarly for being south of the sun's latitude)

So, over a few months you'd end up moving towards the sun's latitude, then when you reach it you will have a daily oscillation east and west and a north-south movement following the sun's latitude.

Update This question is important enough for what-if xkcd to ask a variant of - but for other celestial bodies. This diagram is for Sirius, but follow the link for some interesting ones:

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I am pretty sure that, in general, the sun does not come out exactly at E, or go down exactly at W, even if it is directly above at noon. – Jaime Oct 12 '12 at 17:33
Correct, in fact it only will twice a year. – Rory Alsop Oct 12 '12 at 17:48