The most common parametrization of ellipses is:
c(t) = [ a * cos(t), b * sin(t) ]
Here a is constant and with the ratio parameter
above b = a/ratio.
1. The circular directrix is
the circle of radius 2a
around one (here: the left) focal point. Draw, for any
point P on the directrix, its radius and the segment to
the other focal point F. The symmetry line between P
and F is the the tangent to the ellipse at its
intersection with the radius of P.
2. Clearly, this ellipse is an image of the circle of
radius a under the affine map
(x,y) -> (x, b/a*y).
The demo constructs this map to get the ellipse, point
3. The rolling construction obtains the ellipse
by rolling a circle of radius R inside a circle of radius 2R. The rolling circle
draws the ellipse with a stick of length L > R:
c(t) = R*[cos(t),sin(t)] + L*[cos(t), sin(-t)], hence a = R+L, b = L-R.
These three constructions give ellipses with their tangents. For a joint
construction of all conics see Conic Sections.