Editing Circle of confusion (section)

===Determining a circle of confusion diameter from the object field===
[[File:Circle of confusion calculation diagram.svg|thumb|400px|right|Lens and ray diagram for calculating the circle of confusion diameter {{mvar|c}} for an out-of-focus subject at distance {{math|''S''<sub>2</sub>}} when the camera is focused at {{math|''S''<sub>1</sub>}}. The auxiliary blur circle {{mvar|C}} in the object plane (dashed line) makes the calculation easier.]]
[[File:Long Short Focus 1866.jpg|thumb|400px|An early calculation of CoC diameter ("indistinctness") by "T.H." in 1866.]]

To calculate the diameter of the circle of confusion in the image plane for an out-of-focus subject, one method is to first calculate the diameter of the blur circle in a virtual image in the object plane, which is simply done using similar triangles, and then multiply by the magnification of the system, which is calculated with the help of the lens equation.

The blur circle, of diameter {{mvar|C}}, in the focused object plane at distance {{math|''S''<sub>1</sub>}}, is an unfocused virtual image of the object at distance {{math|''S''<sub>2</sub>}} as shown in the diagram. It depends only on these distances and the aperture diameter {{mvar|A}}, via similar triangles, independent of the lens focal length:

<math display=block> C = A {|S_2 - S_1| \over S_2} \,.</math>

The circle of confusion in the image plane is obtained by multiplying by magnification {{mvar|m}}:

<math display=block> c = C m \,,</math>

where the magnification {{mvar|m}} is given by the ratio of focus distances:

<math display=block> m = {f_1 \over S_1} \,.</math>

Using the lens equation we can solve for the auxiliary variable {{math|''f''<sub>1</sub>}}:

<math display=block> {1 \over f} = {1 \over f_1} + {1 \over S_1} \,,</math>

which yields

<math display=block> f_1 = {f S_1 \over S_1 - f} \,,</math>

and express the magnification in terms of focused distance and focal length:

<math display=block> m = {f \over S_1 - f} \,,</math>

which gives the final result:

<math display=block> c = A {|S_2 - S_1| \over S_2} {f \over S_1 - f} \,.</math>

This can optionally be expressed in terms of the [[f-number]] {{math|1= ''N'' = ''f/A''}} as:

<math display=block> c = {|S_2 - S_1| \over S_2} {f^2 \over N(S_1 - f)} \,.</math>

This formula is exact for a simple [[paraxial]] thin lens or a symmetrical lens, in which the entrance pupil and exit pupil are both of diameter {{mvar|A}}. More complex lens designs with a non-unity pupil magnification will need a more complex analysis, as addressed in [[depth of field]].

More generally, this approach leads to an exact paraxial result for all optical systems if {{mvar|A}} is the [[entrance pupil]] diameter, the subject distances are measured from the entrance pupil, and the magnification is known:

<math display=block> c = A m {|S_2 - S_1| \over S_2} \,.</math>

If either the focus distance or the out-of-focus subject distance is infinite, the equations can be evaluated in the limit. For infinite focus distance:

<math display=block> c = {f A \over S_2} = {f^2 \over N S_2} \,.</math>

And for the blur circle of an object at infinity when the focus distance is finite:

<math display=block> c = {f A \over S_1 - f} = {f^2 \over N(S_1 - f)} \,.</math>

If the {{mvar|c}} value is fixed as a circle of confusion diameter limit, either of these can be solved for subject distance to get the [[hyperfocal distance]], with approximately equivalent results.