The scale of the image can be expressed as a ratio. We are talking about the actual size (height) of an object vs. the height of its formed image. This ratio intertwines object distance (symbol u) with image distance (symbol v) and focal length (symbol F). These are the traditional symbols used in optical formula. I would like to know the origins of these symbols so if anyone knows, please enlighten me.
The focal length of a lens is the measured distance from rear nodal of the lens array to the focused image when the object being imaged is at infinity. This lens-to-image distance elongates if the object is closer than infinity. While the focal length of the lens does not change, this elongated back focus is substituted for the focal length. However, the following equations provide suitable results using F as engraved on the lens barrel for all subject distances except for close-up work (less than 1 yard /meter).
r = v ÷ u (strictly a linear ratio)
Note: use the same unit of measure when solving the equation (I choose millimeters)
Suppose a 1 meter height object is imaged from 10 meters distance using a 50mm lens
F = 50mm (focal length)
v = object distance = 10m X 1000 = 10,000mm
u = image distance (for 50mm lens) 50mm
o = object size = 1000mm
r = 10000 ÷ 50 = 200
For an object 1000mm height, image height will be 1000 ÷ 200 = 5mm
Suppose a 1 meter height object is imaged from 5 meters distance using a 50mm lens
F = 50mm (focal length )
v = object distance = 5m X 1000 = 5,000mm
u = image distance (for 50mm lens) 50mm
o = object size = 1000mm
r = 5000 ÷ 50 = 100
For an object 1000mm height, image height will be 1000 ÷ 100 = 10mm
When doing close-up work F must be elongated to equal the back focus distance
To accomplish we convert focal length and object distance to diopter units.
Suppose a 4mm height object is imaged from 200mm using a 50mm lens
Find the revised F or back focus distance
50mm lens express in diopter units 1/50 X 1000 = 20d
200mm object distance expressed in diopter units 1/200 X 1000 = 5d
Subtract to find back focus 20 – 5 = 15d
Convert back to millimeters to find back focus
1/15 X 1000 = 66.7mm Back Focus
F = 66.7mm (focal length )
v = object distance = 200mm
u = image distance (for 50mm lens back focus distance ) 66.7
o = object size = 1000mm
r = 200 ÷ 66.7 = 3
For an object 4mm height, image height will be 4 ÷ 3 = 1.3mm