## Ellipse from cosine theorem

Ellipse with equation
x^2 + y^2 - 2cos(alpha)xy - d^2 = 0
in its principal axes represented:
(x/a)^2 + (y/b)^2 = 1.
With a = d/sqrt(1-cos(alpha)) and
b = d/sqrt(1+cos(alpha))
c = d*sqrt(2*cos(alpha))/sin(alpha)
equivalently
(1+cos(alpha))*x^2 + (1-cos(alpha))*y^2 = d^2 .
For fixed (d) and variable (alpha), all the resulting ellipses pass through the four points W, X, Y, Z, with coordinates (d/sqrt(2))*(±1, ±1).

The ellipse's shape is controlled a) through segment AB (equal to A*B*), defining length (d), and b) through the polar angle (alpha). Switch to the selection-tool ( CTRL+1 ) to modify AB. Switch to the selection-on-contour tool ( CTRL+2 ) to modify (alpha), by moving G on the circle. The blue ellipse is a 45-degrees rotation of the red ellipse. For D moving on the elliptic arc FE, the corresponding coordinates (Dx, Dy) measured on the sides of (alpha) define points A*, B* respectively, such that |A*B*| = |AB|. Switch to the selection-on-contour tool ( CTRL+2 ) to modify C and its 45-degrees rotated D and see the corresponding location of A*B*.