The mechanical linkage below appears in the work of Frans van Schooten, a Dutch mathematician who lived in the 17th century.
The rod through points C and A is attached to a board at point C and pivots around this point. Point F is also a stationary point attached to the board. As you drag point A, notice that rhombus AEFD expands and contracts. Point B (the traced point) lies at the intersection of the rod through CA and the rod passing through rhombus vertices D and E.
Even though the hyperbola has two separate branches, notice how the linkage operates in one smooth, continuous motion. Can you describe the position of the linkage's arms at the two asymptotes of the hyperbola?
Perhaps the best way to understand why this linkage draws hyperbolas is to first study the hyperbola portion of the Folded Circle construction. After you've done so, press the 'Show' button above. You'll see a circle with center at C passing 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 AEFD?