Phrak
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I couldn't sort-out your math. 
Our terminology is getting in the way of communicating, which is why I could say 90+beta, and it means the same when you say 90-beta. One of the problems in defining angles. This seems to originate where the angle between two lines or vectors could be taken to mean a particular angle, or its complimentry angle.
I've found Microsoft Word is pretty easy at drawing simple diagrams that aren't too crude, once you get started. It relys heavy on right mouse clicks to select operations.
On a second note, all of these models we've come-up with are flawed, as the sail and keel don't stall. Upwind, stalling is called 'going into 'irons', I'm sure you know. I don't know what the downwind problem is called. Geometrically, you can see what happens. For sailing upwind, for instance, the sail is doing nicely in a fast apparent wind, but the keel is moving through the water relatively slowly. To balance the force from the sail, the keel is at a large angle of attack. Beta, as defined in this link
http://www.nalsa.org/Articles/Cetus/Iceboat%20Sailing%20Performance-Cetus.pdf"
grows large, and fairly quickly thereafter the keel stalls. It's the sail that stalls going downwind. This is why beta isn't actually constant, but varies for a sail boat and an ice boat as well. You should run that boat program I sent you, if you haven't. It isn't actually a 'toy' program--all the vector are there, and some work went into accurately modeling the LD curves of keel and sail. So it does model going into irons.
Our terminology is getting in the way of communicating, which is why I could say 90+beta, and it means the same when you say 90-beta. One of the problems in defining angles. This seems to originate where the angle between two lines or vectors could be taken to mean a particular angle, or its complimentry angle.
I've found Microsoft Word is pretty easy at drawing simple diagrams that aren't too crude, once you get started. It relys heavy on right mouse clicks to select operations.
On a second note, all of these models we've come-up with are flawed, as the sail and keel don't stall. Upwind, stalling is called 'going into 'irons', I'm sure you know. I don't know what the downwind problem is called. Geometrically, you can see what happens. For sailing upwind, for instance, the sail is doing nicely in a fast apparent wind, but the keel is moving through the water relatively slowly. To balance the force from the sail, the keel is at a large angle of attack. Beta, as defined in this link
http://www.nalsa.org/Articles/Cetus/Iceboat%20Sailing%20Performance-Cetus.pdf"
grows large, and fairly quickly thereafter the keel stalls. It's the sail that stalls going downwind. This is why beta isn't actually constant, but varies for a sail boat and an ice boat as well. You should run that boat program I sent you, if you haven't. It isn't actually a 'toy' program--all the vector are there, and some work went into accurately modeling the LD curves of keel and sail. So it does model going into irons.
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