Circle-Cropped Rectangle

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 Circle-Rectangle Common Area  (Rev A) 06 Aug 15of 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.Slide1

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.