Circle into an Ellipse
Name: David L.
Assume I were looking at a circle. As the angle of my
line of sight with the circle becomes more and more acute, I will begin
to perceive the circle as becoming more and more elliptical. What is the
formula for finding the dimensions of this ellipse based on the angle of
my line of sight?
Let us draw two lines from the observation point, one perpendicular to the
plane of the circle and one connecting to the center of the circle. These two
lines define a plane perpendicular to the plane of the circle.
The segment of the line at the intersection of the above two planes which
falls inside the circle is the major axis of the perceived ellipse, and its
length is given by 2R/sin(theta) where R is the radius of the circle and theta
is the viewing angle with respect to the plane of the circle. The minor axis
of the ellipse is the line in the plane of the circle and perpendicular to the
major axis. Its length is simply 2R. With this information you can find the
equation for the ellipse.
Ali Khounsary, Ph.D.
Advanced Photon Source
Consider not a circle but a horizontal rod. As the rod is rotated in the
horizontal plane, it begins to "look" shorter to you. Draw the rod at a
diagonal with your eye off to the side. You will find trigonometry useful
to determine the component "perpendicular" to your line of sight.
As this "apparent length" of the rod varies, so varies the dimension of the
circle being rotated. Keep "y" as it is, and replace "x" with the modified
Dr. Ken Mellendorf
Let the diameter of the circle be 'a'. You are on the z-axis looking at
the circle in the x/y plane where 'x' is the horizontal axis and 'y' is the
vertical axis. As you "tip" the circle about the 'x' axis by an angle
(theta) the major axis of the ellipse along the 'x' axis does not change. It
remains 'a'. However, the projection of the circle to your line of sight is
a*cos(theta) = 'b' the minor axis of the projected ellipse. You can convince
yourself that this is correct by the limiting cases. If 'theta' = 0 [no
tipping], cos(0) = 1 and you view the circle with diameter 'a' in both the
'x' and 'y' directions. But if 'theta' = 90 degrees [ pi/2 radians ], you
will be viewing the tipped circle "edge-on" and the minor axis 'b' = 0.
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Update: June 2012