Rearranging the Sag formula

[Bezza]

hi there i wonder if any of you talented folks can help me with this little problem, my maths isnt great and i need to rearrange the exact sag formula
s=r-SQRT(r^2 - y^2) to make r the subject. I have spent all afternoon trawling through my ophthalmic lens textbooks and I can see plenty of other derivations of the formula itself but none are quite what im looking for.
I think its something like r=y^2 + s^2/ 2s but i cant seem to figure out how to derive this from the exact sag formula.
If anyone out there can show me how to work this out it'd be a great help as this assignment is really stressing me out right now.:angry:
Thanks in advance for any assistance :)

[Darryl Meister]

It's easier to start with the original form of the sag formula derivation, and then work from there.



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)

There's also a discussion related to this topic in This Thread.

[Bezza]

Excellent, thanks for that, I was struggling to derive it from the exact formula but seeing it explained that way, using the basic principles, makes a lot more sense.
Hehehe, one of these days maybe ABDO will revise their assignment questions to make them a little less misleading, had me barking up the wrong tree for ages on that one and now i feel a little bit silly for not spotting the easy route sooner.
Thanks again Darryl your a star