The area remaining in a rectangle after cropping by a circle is calculated below.
This information is useful for the optical designer who is designing a lens for a rectangular image format. In the correction of aberrations across the field of view, the designer weights the relative importance of off-axis image points.
Equal weights for evenly spaced points lying on a line from the center to the corner of a rectangular image is not optimum as can be seen from the example illustrated at the right. A rectangular image format of width W and length L is circumscribed by a reference circle of radius Ro.. A circle of lesser radius, R, defines the cropped corner areas. In this example, R/Ro = 0.9. Weighting of the aberration correction for the corner point obviously should be much less than for other field points. For example, the 0.85 field point representing the annrular zone between the 0.8 and 0.9 points, comprises a far greater area than that represented by the 0.95 field point.
A logical weighting for field points can be made by reference to the graph below. This graph plots the remaining area of a rectangle after cropping by circles of varying radii.The difference between the areas defined by two radii define the area of the image represented by the midpoint between them.
The graph plots the remaining area in a rectangle of half-diagonal=1.0, half-width, W/2= 0.6, and half-length, L/2=0.8, after cropping by a circle of varying radii, normalized to the rectangle half-diagonal. Similar curves can be plotted for other rectangle aspect ratios via the equation below using the accompanying spreadsheet.