Calculating Ellipses

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If you need to find the change in area between a circle and when it is tranformed into an ellipse by tilting it so that its major axis is increased.

When the circular region in 3 space is rotated about one of its diameters, turning it into the new minor axis, the orthogonal radius becomes the major axis, and will be increased by the inverse of either the sine or cosine of the angle, depending on how you measure it.

If the circle is just sitting there minding its own business basking in the light waves arriving along its normal, then we say it's moved 0 degrees, so we use cosine, (since cos 0 degrees = 1, otherwise we run into a problem with dividing by sin 0 = 0) because we will be dividing the other diameter's measurement by the cosine of the angle that we have rotated it.

Tilting 30 degrees from the normal, the major axis grows by 1/cos 30 degrees or 1/.8660 = 1.155. Then this could be plugged into the area of an ellipse equation. Tilting more to 45 degrees gives a lengthening of the diameter by 1/.7071 or 1.414 times.

Notice the areas of the new ellipses compared to the original circle are off by a factor of, you guessed, the cosine of the angle of displacement from the normal! This is why the light meter's reading is attenuated, the area of radiant flux at right angles to the beam is now spread out over a larger area of the transformed ellipse! So there are less photons per unit area! Now does everyone see the wisdom of the detector parallel to the recording material?

This can also be used to calculate the area of coverage of your collimation mirror.