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For small angles or points near the vertex of a parabola we can approximate a parabolic surface with a circle. The focus of the parabola is a unique point specifically for optics (Parallel light will converge at the focus), and vice versa.

Has anyone come across an derivation that shows in the limit of small angles or points near its vertex, the focus of a parabola is equivalent to R/2 where R is the radius of a circle used to approximate the parabola near its vertex.

I've seen the derivation where we obtain

1/s+1/s'=2/R,

but they we always conclude that 2/R must be 1/F.

I was wondering if there is a derivation starting with the focal point of a parabola and then approximates to a circle.