I am working on something and I have an equation that I am having a hard time solving. I think I might be makign a stupid mistake in the math so I am hopeing someone might catch it as I am having a tough time catching it.
The variables:
D = Back Vertex Power
F = Front Curve
n = Index of Lens
q = Back Vertex to Center of Rotation
F2(n + 2) + F[2/q(n2 - 1) + D(n + 2)] + n[D + (n - 1)/q]2 = 0
It looks daunting but heres how far I've gotten:
F2(n + 2) + F[2/q(n2 - 1) + D(n + 2)] + n[D + (n - 1)/q]2 = 0
A = (n+2)
B = [2/q(n2 - 1) + D(n + 2)]
C = n[D + (n - 1)/q]2
So now I can rewrite the equation to read:
AF2 + BF + C = 0
now it's in a simple quadratic form, no I would assume to find the + and - root I would use the quadratci equation:
F- = (-B - [B2 - 4AC]1/2)/2A
F+ = (-B + [B2 - 4AC]1/2)/2A
So now I will get the positive and negative root, which BTW will be the Ostwalt and Wollastan branches of the Tscherning Ellipse. Now here's my delema I have been plugging in numbers left and right and I can't get them to pan out. I will try with an example:
D = 1.00
F = ?
n = 1.5
q = 25mm = 0.025m
A = (1.5 + 2) = 3.5
B = 2/0.025(2.25 - 1) + 1(1.5 + 2)
B = 80(1.25) + 3.5
B = 103.5
C = 1.5(1 + 2(1(1.5 - 1)/0.025) + ((1.5 - 1)2/0.0252)
C = 1.5(1 + 40 + (0.25/0.000625)
C = 1.5(1 + 40 + 400) = 1.5(441) = 661.5
A = 3.5
B = 103.5
C = 661.5
F- = (-103.5 - [103.52 - 4*3.5*661.5]1/2)/2*3.5
F- = (-103.5 - [10712.25 - 9261]1/2)/7
F- = (-103.5 - [1451.25]1/2)/7
F- = (-103.5 - 38.1)/7 = -141.6/7 = -20.2
F+ = (-103.5 + 38.1)/7 = -65.4/7 = -9.3
For some reason those numbers don't look right to me, any suggestions?
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