Between Major and Minor Circles
In a polar system of coordinates based at the center of ellipse, an ellipse is presented as a curve
P(t) = (a cos(t), b sin(t)),
where a and b are the lengths of its half-axes. Consider two circles (one major, the other minor) associated with the ellipse:
A(t) = (a cos(t), a sin(t)),
B(t) = (b cos(t), b sin(t)).
Both touch the ellipse at its vertices. If components of a generic point
P(t) = (A(t)x, B(t)y).
This explains the configuration illustrated by the applet below.
Since ΔABP is right, the midpoint M of its hypotenuse AB is also the circumcenter:
OM = (a + b) / 2.
MP = |a - b| / 2.
Let zX be the complex number associated with a point X. Then arg(zM) is the angle formed by the radius-vector OM with the (positive) x-axis. For a > b, the angle formed by MP with the (positive) x-axis is -arg(zM).
zP = (a + b)/2·eit + (a - b)/2·e-it.
(Euler's formula shows that this is exactly the same as
In an ellipse, conjugate diameters correspond to complementary values of parameter t. The applet illustrates this point too: OA is perpendicular to OA', although PP' and QQ' are not perpendicular, except when they coincide with the axes of the ellipse.
Copyright © 1996-2018 Alexander Bogomolny