[ATM] Right house, wrong barn

["Richard F.L.R. Snashall" <rflrs@rcn.com>]

And the moral of the story is "don't try to do math while composing...
especially trying to convert someone elses notation to your own"!


Richard F.L.R. Snashall wrote:
 >
 >
 > Vladimir Galogaza wrote:
 >
 >  > In this case general (ASCII) expression for the conic is
 >  >
 >  > z(x)=x^2/{R*[1+sqrt(1-(K+1)*(x^2/R^2)]}            (1)
 >  >
 >  > R stands for paraxial radius of curvature
 >  > K is conic constant ( -1 for parabola, 0 for circle)
 >  >
 >  > For given R and K and zone radius xo= yn
 >  > corresponding conic ordinate z(xo)=yc is
 >  > obtained from the conic expression (equation) (1)
 >  >
 >  > tangent to the conic z(x) in the point (xo,yo) is:
 >  >
 >  > t(x,xo,K) = d1z(xo)*(x-xo) + z(xo,K)
 >  >
 >  > normal to this tangent in the point (xo,yo) is:
 >  >
 >  > n(x,xo,K) = (-1/d1z(xo))*(x-xo) + z(xo)   ( this is the line you are
 > looking
 >  > for)
 >

S = (R/(K+1)) * ( 1 - sqrt( 1 - (K+1)*(yc/R)^2 ) )

Q = R * sqrt( 1 - (K+1)*(yc/R)^2 )

S + Q = (R/(K+1)) * ( 1 + K*sqrt( 1 - (K+1)*(yc/R)^2 ) )

yn/yc = (S + Q)/Q = 1/( (K+1)*sqrt( 1 - (K+1)*(yc/R)^2 ) ) + K/(K+1)

or:

yn/yc - K/(K+1) = 1/( (K+1)*sqrt( 1 - (K+1)*(yc/R)^2 ) )

or (squaring):

( yn/yc - K/(K+1) )^2 = 1/( (K+1)^2 * ( 1- (K+1)*(yc/R)^2 ) )

and (multiplying by yc^2 * (K+1)^2 * ( 1 - (K+1)*(yc/R)^2 ) )

( 1 - (K+1)*(yc/R)^2 ) * ( (K+1)*yn - K*yc )^2 = yc^2


Now, hopefully, this iteration is correct...

-- 

Rick S.

http://users.rcn.com/rflrs


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