R/C Naval Combat

Alright folks, here is the challenge of the evening...  assuming that it even has a discrete solution...  pardon any mistypeing...  I keep coming up with one too few equations to solve (one too many unknowns)....  which either means I have missed something or there is no unique solution...

All angles are in radians... 

The goal (pump related, of course)...   imagine two circles, an inner circle and an outer circle, centered on location (0,0) in the Primary coordinate system.

The parametric equations for the circle are: 

x=r*cos(theta)

y=r*sin(theta)

r=radius of the circle

theta is a angle measured from the X axis, in the standard Right hand rule manner

x & y are the coordinates of the circle for a given theta....  

the tangent angle of the circle will be alpha=theta+Pi/2

Now imagine an involute, who's characteristic circle is centered a distance XX,YY from (0,0).

The parametric equation for the involute is a function of the radius of its characteristic circle, and the tangent angle of the curve....  with the origin at the center of the characteristic circle....

x=a*(cos(t)+t*sin(t))= F(a,t)

y=a*(sin(t)-t*cos(t))= G(a,t)

where a= the radius of the characteristic circle of the involute curve

t is the tangent angle of said curve....

First, one must translate the involute equation into the same coordinate system as the two circles...

X=x + XX

Y=y + YY

Now here is the fun stuff....   

Imagine you know the intersection angle of the circles and the involute...  E.G.  the outer circle and the involute intersect producing an intersection angle of A1.  the inner circle and the involute intersect producing an intersection angle of A2.

in that case.

A1=theta+Pi/2 - t     at the location of the intersection at the outer circle

A2 = theta + Pi/2 -t    at the location of the intersection of the inner circle.

If you match the parametric equations at both intersecting points, this leaves you 5 equations that fully define point 1, and 5 more that define the other intersection point...   all 10 equations are independent...

There is a problem however...  there are 11 unknowns...  which means I need another equation (or there is no single solution)   So I figured I would give you folks a chance to mull it over...  

definition.... will have to wait on an image till tomorrow....

something like this although I can't draw an involute to save my life..   the two spots marked with angles are what I want to define the involute by, assuming that is doable...   I do not know their X,Y positions....   but I do know the intersection angle I want...   I can not simply fix one of them at a point on the circle without allowing the involute's coordinate system to rotate with respect to the circles coordinate system.   Luckily, if I don't fix that point, I don't have to deal with rotating coordinate system transformations..   and all this is a skosh fuzzy already.

Like I said before, there may not be a unique solution, there may even be a set of unique solutions, but operating under the "I need as many equations as I got variables" assumption....    tells me I need one more thing to make this solvable...

I'm not up to the math but I am assuming that what you are trying to do is to create a shape that will tend to a constant increase in the acceleration of the water from when it first encounters the vane until it reaches the tip of the vane.  If this is so, then would it make sense to look at the distance from the inter side of the vane to the out side of the circle and how it changes as you move over the arc?  I don't quite see whay the angle on the outside needs to be 30 degrees...  Isn't the important thing what is happening on the other side?

phill the semi-clueless