This is not a true Euclidean construction as defined in
Constructions - Tools and Rules but a practical way to draw
an ellipse given its width and height and when mathematical precision is not so important.
It is sometimes called the "Gardener's Ellipse", because
it works well on a large scale, using rope and stakes, to lay out elliptical flower beds in formal gardens.

You can also calculate the positions of the focus points.
See Foci of an Ellipse.

a + b, the length of the string, is equal to the major axis length PQ of the ellipse.

The string length was set from P and Q in the construction.

3

The figure is an ellipse

From the definition of an ellipse:
From any point C on the ellipse, the sum of the distances from C to each focus is equal to the major axis length.
The string is kept taut to ensure this condition is met.

Click here for a printable worksheet containing an ellipse drawing problem.
When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.