This construction creates an ellipse shape made from the arcs of 4 circles. It is simply an approximation to an ellipse, sometimes used by craftsmen, because in woodwork, a pair of compasses can be replaced by a router
This diagram shows the semi-ellipse along with the 4 centres and some tips for the construction (no colour this time because it is simply a tip on how to construct!)
Let major axes equal 2a, minor axis = 2b. You can construct the distance a-b by drawing a circle centre the origin of radius a or b. so no need to use a ruler
Draw 4 lines connecting "extremes" of the ellipse
From the minor axes extreme, measure a distance a-b from this point along one of the lines just drawn.
Construct the perpendicular bisector between this point and the major extreme point. Where this perpendicular bisector meets the minor axis will be one of the centres for your circle. You will need symmetry to construct the mirror image of this point
Set the compass to a radius from this point to the opposite extreme on the minor axis and draw an arc.
The arcs that create the part of the ellipse passing though the major axis extremes have a centre a distance a-b from the major axis extreme. These arcs "merge together reasonably smoothly to create the ellipse shape
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