The mechanical linkage below appears in the work of Frans van Schooten, a Dutch mathematician who lived in the 17th century. Drag point A and notice that rhombus ADFE expands and contracts. Point B (the traced point) lies at the intersection of CA and the rod passing through rhombus vertices D and E.
After you've traced an ellipse, drag an endpoint of the segment in the upper left corner. This segment controls the side length of rhombus ADFE. Now drag point A again. Now that you have a new rhombus, do you trace a different ellipse?
Drag point C closer to point F. Click on the red "X" in the lower right corner to clear the trace. Now drag point A again to see if you trace the same ellipse as before.
One way to understand why this linkage draws ellipses is to first study the Folded Circle construction. After you've done so, examine the model below. A circle with center at point C passes through point A. Drag point A around the circle. Do you see the similarity between this construction and the Folded Circle method? What's the purpose of rhombus ADFE?