A = HO = HF + FO = FO − FH = x − f,
A′ = H′O′ = H′F′ + F′O′ = F′O′ − F′H′ = x′ − f′,
whence
x = A + f and x′ = A′ + f′.
Using xx′ = ff′, we have (A + f)(A′ + f′) = ff′, which leads to AA′ + Af′ + A′f = O, or
| 1 + | f′ | + | f | = O; |
| A′ | A |
this becomes in the special case when f = -f′,
| 1 | − | 1 | = | 1 | . |
| A′ | A | f |
To express the linear magnification in terms of the principal point distances, we start with equation (4) (f − x)/(f′ − x′) = −x/f′. From this we obtain A/A′ = -x/f′, or x = −f′A/A′; and by using equation (1) we have β = −fA′/f′A.
In the special case of f = −f′, this becomes β = A′/A = y′/y, from which it follows that the ratio of the dimensions of the object and image is equal to the ratio of the distances of the object and image from the principal points.
The convergence can be determined in terms of A and A′ by substituting x = −f′A/A′ in equation (4), when we obtain γ = A/A′.