> I have a bow ( u know - like in a bow and arrow ).
>
> The length of the piece of "wood" and the length of the piece
> of string is known.  Assume that the string is attached to the
> ends of the bow and that its curve is contstant ( always
> circular - never eliptical.

> How would i calculate the radius of the bow?

You've got a circular arc with a chord.  Length of the circular
arc, i.e. the bow length, we'll call bowlen.  Length of the bow
string we'll call slen (i.e. the chord).  [I'd use strlen as the
variable name, but it has too may C language overtones. :-) ]

The math changes depending if the arc formed by the bow is less
than or greater than half a circle.  I'll do the less than form
which requires:

    bowlen < (slen * pi) / 2

I'm using  r  for the unknown radius...

[use monospaced font for formulas]

Imagine two radius lines from the intersection points of the
bow and bow string to the center of the circle.  These radius
lines and the chord (slen) form an isoceles triangle.  Split
that triangle in half by adding a perpendicular line from the
center of the chord to the center of the circle.  This forms a
right triangle with known side opposite (slen/2) & hypotenus (r).

The angle in the right triangle is

           ( (slen / 2) )
    arcsin ( ---------- )
           (     r      )

Therefore, the angle formed in the isoceles triangle (i.e. two
radius lines capped by the bow) is

               ( (slen / 2) )
    2 * arcsin ( ---------- )
               (     r      )

This angle allows us to know the portion of a full circumference
which the bow represents.

We know that the circumference of the circle is 2 * pi * r, so
the length of the arc is

             (             ( (slen / 2) )  )
             (  2 * arcsin ( ---------- )  )
             (             (     r      )  )
    bowlen = ( --------------------------- ) * 2 * pi * r
             (           2 * pi            )

Since the wood bow length is a given, we can solve for r as it
is the only unknown.

I don't remember all of my trig formulas any more and don't have
a table of them handy, so I won't attempt to reduce the formula.


> If i shorten the string by 10cm, how much will the radius
> change ?  Is the change linear ? I dont think so.

Given the arcsin fuction, I doubt that it's linear.

                                                Lee Jones

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