# Optical Forums > Ophthalmic Optics >  Converting SAG formula to a base curve

## lensgrinder

Greetings All and thanks for any answers in advance.
I was digging through the lab the other day a discoverd an old SAG gauge.  I know the formula for finding the SAG  of a lens, but how can you minipulate the formula  so you can find the front curve or back curve?  
I would also like to say that I have been reading posts here for a couple of months and I think this site is fantastic.

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## QDO1

> Greetings All and thanks for any answers in advance.
> I was digging through the lab the other day a discoverd an old SAG gauge. I know the formula for finding the SAG of a lens, but how can you minipulate the formula so you can find the front curve or back curve? 
> I would also like to say that I have been reading posts here for a couple of months and I think this site is fantastic.


Not quite that easy - you would have to know the index of the lens too.  Substitute  it in the formula and rearange for F.  using the rough formula (which is good enough)

sag  = Y squared F / 2000(N-1)

thus sag/f = Y squared / 2000(N-1)

thus F = (Ysquared /2000(n-1))/sag

(y = radius in MM)

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## lensgrinder

Thanks QDO1.  I am trying to do this using the original SAG formula, not the approximation formula.  My problem is seperating r^2 from the square root part of the equation.  I remember years ago we used to enter the SAG of the lens and the computer would convert this to a base curve.

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## JRS

You did not mention if the gauge is a "ball tip" or a "bell" type or even the size (diameter). The formula is different.

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## mullo

Is this what you are looking for?   

s1 = sag of front surface s2 = sag of back surface
n=index of refraction 
r1= radius of curvature of front surface in metres 
r2= radius of curvature of back surface in metres 
F1 = base curve/front curve of lens
F2 = back curve/ocular curve of lens
h = semi-chord or half-diameter of lens
Ö = square root



To solve for F1 (base curve):

s1 = r1 - Ö (r1² - h²) where r1 = (n-1)/F1 

therefore 

s1 = [(n-1)/F1] - Ö (r1² - h²) 



and finally to find F1, isolate F1



*F1  =  n-1    /    [ s1 + Ö (r1² - h²) ]*








or for F2 (back curve)


s2 = r2 - Ö (r2² - h²) where r2 = (1-n)/F2

therefore 
_______
s2 = [(1-n)/F2] - Ö (r2² - h²) 



and finally to find F2, isolate F2



*F2  =   1-n  /    [* *s2 + Ö (r2² - h²) ]*

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## mullo

Sorry for the form, when I type it out it doesn't save as typed so I had to improvise. :shiner:

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## Darryl Meister

The formula for surface power (F) from sag (s) would be:

*F = [s * 2000 * (n - 1)] / (s^2 + y^2)*

where n is the refractive index and y is the semi-diamater of the gauge. For a typical sag gauge, n would generally be 1.530 and y would be 25. This simplifies the equation to:

*F = (s * 1060) / (s^2 + 625)*

As JR pointed out, if this is a ball-tipped gauge, the radius of the tip would need to be factored in; the equation is a little more complicated, since the radius of the lens surface needs to be adjusted by an amount equal to the radius of the ball tip.

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## Darryl Meister

> *F1 = n-1 / [ s1 + Ö (r1² - h²) ]*


I think your equation still has an unknown term in it (r1) that would prevent you from solving for F1 directly, at least without some algebraic manipulation.

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## lensgrinder

Thanks everyone for your input.  Darryl, this is exactly what I was looking for and sorry for the lack of info.  It is a bell SAG gauge with a 50mm diameter.

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## lensgrinder

Darryl would it be possible for you to walk me through how you derived that formula?

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## Darryl Meister

> Darryl would it be possible for you to walk me through how you derived that formula?




Using the Pythagorean theorem on the illustration above, we can see that r^2 = y^2 + (r - s)^2.

After expanding,

r^2 = y^2 + r^2 - 2 * r * s + s^2

And canceling the r^2 terms,

0 = y^2 - 2 * r * s + s^2

And rearranging to solve for r,

2 * r * s = y^2 + s^2

r = (y^2 + s^2) /  (2 * s)

Now, r = 1000 * (n - 1) / F, where F is the surface power. So, substituting for r gives us,

1000 * (n - 1) / F = (y^2 + s^2) / (2 * s)

And, finally, rearranging to solve for F,

F = [s * 2000 * (n - 1)] / (y^2 + s^2)

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## lensgrinder

Thank you very much.  This really helps out.

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## optispares

> The formula for surface power (F) from sag (s) would be:
> 
> *F = [s * 2000 * (n - 1)] / (s^2 + y^2)*
> 
> where n is the refractive index and y is the semi-diamater of the gauge. For a typical sag gauge, n would generally be 1.530 and y would be 25. This simplifies the equation to:
> 
> *F = (s * 1060) / (s^2 + 625)*
> 
> As JR pointed out, if this is a ball-tipped gauge, the radius of the tip would need to be factored in; the equation is a little more complicated, since the radius of the lens surface needs to be adjusted by an amount equal to the radius of the ball tip.


what would the formula be, changed to take into account the ball tip

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## QDO1

> JRS
> 04-20-2002 09:12 PM
> 
> When using a BALL-TIP gauge, the formula must take into consideration where the curve touches the ball. In the case of laps (convex curves), the higher the curve, the more "inside" the ball is touched. Concave just the opposite.
> 
> For your formula to work, you must know the diameter of the ball tip and the distance between. Since you said a 40mm gauge, my guess it is either a LOH gauge or an Essilor gauge. The ball tip size is different even though the distances are the same. Your Coburn chart is probably based on a 50mm distance with ball tip size of 3.9688 mm.
> 40mm Essilor ball tip size = 2.00mm
> 40mm LOH ball tip size = 3.00mm
> 
> ...


 
found in the archives

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## optispares

> JRS04-20-2002 09:12 PMWhen using a BALL-TIP gauge, the formula must take into consideration where the curve touches the ball. In the case of laps (convex curves), the higher the curve, the more "inside" the ball is touched. Concave just the opposite.
> 
> For your formula to work, you must know the diameter of the ball tip and the distance between. Since you said a 40mm gauge, my guess it is either a LOH gauge or an Essilor gauge. The ball tip size is different even though the distances are the same. Your Coburn chart is probably based on a 50mm distance with ball tip size of 3.9688 mm.
> 40mm Essilor ball tip size = 2.00mm
> 40mm LOH ball tip size = 3.00mm
> 
> BELL gauges - ones with the solid brass (usually) ring always use the same contact point regardless of curve.
> 
> 
> ...


 i used the above for a spread sheet but i want to write a script in vb so that when i input the sag it gives the curve and my brain won't function in how to transpose the formula

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## Darryl Meister

It would probably be something like:

*F = [s * 2000 * (n - 1)] / (y^2 + s^2 + 2 * b * s)*

where b is the radius of the ball tip. But I would have to double-check the results (which I won't have time to do for a while).

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## optispares

> It would probably be something like:
> 
> *F = [s * 2000 * (n - 1)] / (y^2 + s^2 + 2 * b * s)*
> 
> where b is the radius of the ball tip. But I would have to double-check the results (which I won't have time to do for a while).


*F = [s * 2000 * (n - 1)] / (y^2 + s^2- (2*b*s)) * 
*for convex lenses seems accurate to the first decimal place when i used the figures for a 10d lens*
*the answer was 10.01 close enough for what i need as it is only a 24mm*
*sag gauge for checking the front curve of lenticular bowls.*

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## HarryChiling

Check out the file directory I just uploaded a calculator for this problem.  Both ball tip and bell gauge are in the calculator. Enjoy

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## optispares

> Check out the file directory I just uploaded a calculator for this problem. Both ball tip and bell gauge are in the calculator. Enjoy


i just finished a calculator myself:hammer: guess there is no point uploading another:cheers: although this one does convex as well

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## Darryl Meister

> for convex lenses seems accurate to the first decimal place


Yes, you would indeed use -2 for convex surfaces, and +2 for concave surfaces, as you point out. I also double-checked the results against a surfacing chart and they are dead-on, so my derivation is apparently correct.

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