Conic Sections: Converting 'Rho' to 'Sagitta' ?

[cmoher3]

I'm trying to make a polishing lap, for optical work and starting with a 3D printed mold. Since my optical lap is parabolic, I reached for the Create >> Conic Curve tool.

On drawing the two end points, I was prompted to quantify the bulging (or sinking) in the middle as a "Rho value". I have not encountered that metric before. It appears to be somewhat related to the term Sagitta in optics., if it isn't a synonym. Would anyone here know or suggest a conversion formula or value between the two ?

[tommVS2MB]

 How have you tried to answer that; other than posting here.

http://www.viewmold.com/ug_html_files/modeling/crv_2pt_anc.html

[cmoher3]

I already saw that Mr. Snarly Pants, hence my question in the first place. Where do you see Sagitta mentioned in the link you posted ? Or, any mention of the relationship between the two terms ?

[wmhazzard]

https://en.wikipedia.org/wiki/Sagitta_(optics)

https://www.quora.com/What-does-it-mean-by-rho-value-for-conic-section

[laughingcreek · Accepted solution]

If your talking about a true parabolic curve, then Rho has to be .5.  Is Sagitta going to be a known parameter?  Then the length of your anchor point line will be 2 x sagitta.

[cmoher3]

I appreciate the efforts expended here , but none of them produce the mathematical relationship between Rho and Sagitta. Sagitta is used a lot in optics and would be consistent in at least spherical concave lenses. People grinding optical lenses and mirrors use a spherometer to measure the sagitta throughout the grinding process. I've seen these diagrams and formulas for Rho. None of them include any mention of the sagitta. The difference in converting a spherical lens to a parabolic one is usually measured in light waves, subtle as it is, but critical to removing optical aberrations. 

I'm going to check elsewhere, but perhaps the reason sagitta and rho don't show up in the same formula is because sagitta starts to be at least theoretically irrelevant or error prone once a lens moves from spherical or parabolic. I've come to that conclusion in my imaginings of using a spherometer on a parabola. It simply would start reporting erroneously, albeit in errors of light waves....... so I guess I've answered my own question.

[cmoher3 · Accepted solution]

laughingcreek.... It's OK... I got it now. Sagitta is not , with precision, a useful measure in non-linear curves like parabolas, only in spheres. The only time a formula could be devised relating Sagitta to Rho is on a perfect sphere. The only way to accurately describe a parabolas depth (or height in the case of convex) is with Rho.

Having said that, since the difference between sagitta and whatever Rho indicates as depth or crest of curves, comes out to be so small that optical tests that measure to within wavelengths are required. Unfortunately, some have not only rounded off the measurements but actual underlying definitions.

I understand now why Fusion uses Rho and not Sagitta...