simple

Two - CW and CCW.

Can you please explain more about what you want to know? Are you talking about drawing with paper & pencil? Easiest way (and the only way I know, apart from plotting the co-ordinates from the equation & playing join-the-dots) is 2 drawing pins (thumbtacks) and a loop of string.

Horizontal and vertical and anything in between.

A CNC machine and a plotter can do a nice one.

Long ago they used mechanical integrators.

I am not sure but I think a "trammels" can be used as well.

http://www.sciencemuseum.org.uk/images/I032/10302774.aspx

Say it were a stone bridge arch, for example. The tutorial, pencil-and-paper, method of doing it is as follows:

• draw a horizontal construction-line between the upper faces of the two springer stones.
• Mark the centre-point of this horizontal line.
• draw a semicircle in construction-line, touching the two springers of the arch and perpendicular to the upper faces.
• draw another, smaller, semicircle in construction line that just touches the underside of the keystone.
• Draw radii from the centre every 10deg or so in construction-line so that they intersect the two semicircles.
• Where each radius intersects the inner semicircle draw a horizontal construction-line contained within the space between the two semicircles.
• Where each radius intersects the outer circle, draw a vertical construction-line contained within the space between the two semicircles.
• Where the verticals and horizontals intersect is a point on the semi-ellipse. Mark the point in bold-line, and join the dots with either French Curves or Flexi-curves.
• Rub-out the construction-lines.

Job done.

Eat yer heart out, AutoCad!

Hammer in two nails a desk. Not completely.

Get a thread which length is longer than distance between nails (nails represent your ellipse poles).

Tie both ends of thread to nails.

Stretch a thread by means of pencil.

Having thread constantly tight draw an ellipse with care.

Yes, you can do it both ways CW and CCW.

I learnt this 65 years back.

When/who discovered this?

I was taught and even demonstrated this method by my math Prof 25 years ago. And he didn't pretend that he was inventor of this.