Editing Hyperfocal distance (section)

===Derivation using geometric optics===
[[File:hyperfocal distance definitions.svg|thumb|300px|Accompanying figures]]

The following derivations refer to the accompanying figures. For clarity, half the aperture and circle of confusion are indicated.<ref>{{cite book|title=Optics in Photography - Google Books|isbn = 9780819407634|url=https://books.google.com/books?id=hcq_40I_7egC|access-date=24 September 2014|last1 = Kingslake|first1 = Rudolf|year = 1992| publisher=SPIE Press }}</ref>

====Definition 1====
An object at distance {{mvar|H}} forms a sharp image at distance {{mvar|x}} (blue line). Here, objects at infinity have images with a circle of confusion indicated by the brown ellipse where the upper red ray through the focal point intersects the blue line.

First using similar triangles hatched in green,
<math display="block">\begin{array}{crcl}
           & \dfrac{x-f}{c/2} & = & \dfrac{f}{D/2}  \\
\therefore & x-f              & = & \dfrac{cf}{D}   \\
\therefore & x                & = & f+\dfrac{cf}{D}
\end{array}</math>

Then using similar triangles dotted in purple,
<math display="block">\begin{array}{crclcl}
           & \dfrac{H}{D/2} & = & \dfrac{x}{c/2}                                              \\
\therefore & H              & = & \dfrac{Dx}{c}   & = & \dfrac{D}{c}\Big(f+\dfrac{cf}{D}\Big) \\
           &                & = & \dfrac{Df}{c}+f & = & \dfrac{f^2}{Nc}+f
\end{array}</math>

as found above.

====Definition 2====
Objects at infinity form sharp images at the focal length {{mvar|f}} (blue line). Here, an object at {{mvar|H}} forms an image with a circle of confusion indicated by the brown ellipse where the lower red ray converging to its sharp image intersects the blue line.

Using similar triangles shaded in yellow,
<math display="block">\begin{array}{crclcl}
           & \dfrac{H}{D/2} & = & \dfrac{f}{c/2}                      \\
\therefore & H              & = & \dfrac{Df}{c} & = & \dfrac{f^2}{Nc}
\end{array}</math>