[Haskell-cafe] Geometry

[Stefan O'Rear]

On Mon, Aug 27, 2007 at 11:04:58AM +1000, Tony Morris wrote:
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> I went camping on the weekend and a friend of mine who is a builder
> asked me many questions on geometry as they apply to his every day work
> - - most of which I could answer.
> 
> However, there was one that I couldn't and I am having trouble googling
> a solution (for lack of keywords?). I'm hoping a fellow Haskeller could
> help me out (in Haskell of course).
> 
> The problem is finding the unknown x from the two knowns a and b in the
> given image below (excuse my Microsoft Paintbrush skills). I may have
> misunderstood his problem (we were drawing in dirt) and actually, it is
> the straight line between the two points on the circumference that are
> known and not the specified 'b', but I figure I could derive one
> solution from another if I have misunderstood him.
> 
> Here is my image:
> http://tinyurl.com/2kgsjy

This is a fairly simple exercise in trigonometry.  Call the angle
subtended by b, θ.  Then:

b     = a sin(θ/2)
a - x = a cos(θ/2)

by the relation between circles and trig functions.  From this we can
(algebraicly) derive:

sin(θ/2) = b / a
x = a - a cos(θ/2)
x = a - a (1 - b² / a²)^½   (nb, I'm assuming θ is less than 180° here)

And as you request:

problem a b = a - a * sqrt (1 - b*b / a*a)

Stefan
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