# Helmholtz-Lagrange Invariant

is a mapping invariant of paraxial optics, given by the product

where n is the refraction index, is the aperture angle, and y is the object height.

This value doesn’t change, if the object side values are replaced by the corresponding image side values:

From this we get the paraxial Magnification:

# hyperfocal distance

When a lens is focussed to the hyperfocal distance H, the DOF of the lens is maximized: The range of acceptable sharpness then extends from to infinity

There are two Formulas in use:

For f=50mm, F2, and CoC = 0.03mm we get

This is the formula we use here. The results just differ in the focal length of the lens.

and

For f=50mm, F2, and CoC = 0.03mm we get

Where CoC is the circle of confusion, F is the F-number and f is the focal length of the lens.

The hyperfocal distance has curious mathematical properties:

The hyperfocal distance H is the distance at which you have to focus an object to receive the largest depth of field. If, namely, a lens is focussed to H, it is focused from to infinity.

When focussed to , so everything from  to  focused.
When focussed to , so everything from  to  focused.
When focussed to , so everything from  to  focused.

When focussed to , so everything from  to  focused.

The distance

is the Depth of field.

Notice:

The depth of field is getting smaller, the larger is n, say the shorter the working distance is!