Nerd question on ULF coverage

I'm trying to figure out what the widest lenses I can try to use on various ULF cameras. I know the minimums in term of coverage required per format but information regarding image circles is always given at infinity when the lens is closest to the film plane. Obviously if you focus on objects closer than infinity you increase the distance of the lens from the film plane, which also increases the image circle size.

Does anyone know of a mathematical formula (trig maybe?) to calculate the rate that an image circle will enlarge based on the distance of the lens from the film plane if you have a given knowledge of the image circle size at infinity, degree of coverage and of course focal length?

its actually fairly simple in my rudamentary math mind.

lenses create an image circle, that image circle is calculated by the angle listed (usually larger than the manufacturers listed one)
so I would think if you drew a simple cross section of the camera layout
such as the one sketched up below. it might help to calculate it.

figuring out what the key figures are would be the film to lens distance, coverage angle, and the excess coverage distances, then as the distance from lens to film plane increases, the excess coverage will increase at a certain ratio... as that seems like it would be different for different angles of lens coverage.

maybe the illustration will help.

A high school Physics book will give you all distance calculations (formulas) need. This one form the 50's or 60's the new books a just the new math stuff and makes no since.

It's the same angles. The difference is the triangle instead of having one side equal to the focal length is equal to the distance focussed at.

Think of the image leaving the back of the lens being a cone. The further you get the bigger the cone.