## 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 _{x}_{y},

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 z_{X} be the complex number associated with a point X. Then arg(z_{M}) 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(z_{M}).

If _{A} = ae^{it}_{B} = be^{it}_{M} = (a + b)/2·e^{it}

z_{P} = (a + b)/2·e^{it} + (a - b)/2·e^{-it}.

(Euler's formula shows that this is exactly the same as _{P} = a cos(t) + ib sin(t),

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.

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