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 and tangent.

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 and tangent.

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.

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.