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.

4. The ladder construction obtains the ellipse by moving a stick of length a+b, keeping the end points on the coordinate axes.

The point, which divides the stick a:b, i.e. (a*cos(t), b*sin(t)), draws the ellipse. Here t is the angle between stick and x-axis.

In cases 3 and 4 the tangent construction is less obvious. The magenta point is the momentary center of rotation for the movement of the drawing mechanism.

The tangent is orthogonal to the momentary radius from this center to the curve point.

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.

4. The ladder construction obtains the ellipse by moving a stick of length a+b, keeping the end points on the coordinate axes.

The point, which divides the stick a:b, i.e. (a*cos(t), b*sin(t)), draws the ellipse. Here t is the angle between stick and x-axis.

In cases 3 and 4 the tangent construction is less obvious. The magenta point is the momentary center of rotation for the movement of the drawing mechanism.

The tangent is orthogonal to the momentary radius from this center to the curve point.

For a joint construction of all conics see Conic Sections.