Waves with a compass

You construct a figure called 'Wavy Wave'. It sits on a straight central line AB. It is made of two half circles of the same size: the first bulges up above the line, and the second, right after it, bulges down below the line. Together they make one smooth wave from A to B.

The book takes AB = 8 cm. The two half circles share the line equally, so the first one goes from A to a point X in the middle of AB: AX = 8 ÷ 2 = 4 cm, and XB = 4 cm too.

What radius should the compass have? A half circle stretches across twice its radius, from one side of its centre to the other. The first half circle stretches across AX = 4 cm, so its radius is half of that, 4 ÷ 2 = 2 cm. Its centre, where the compass point goes, is the middle of AX, 2 cm from A. The second half circle has the same radius, 2 cm, with its centre in the middle of XB, and it is drawn on the other side of the line.

The same idea works on a line of any length: split the line into two equal parts, and take half of one part as the radius. For a central line of 12 cm, each part is 6 cm and the radius is 3 cm.

The book also asks for waves smaller than a half circle, like the neck of 'A Person' and the curves of the 'Eyes'. For a flatter curve through the same two points, the compass point can no longer sit on the line. It must go off the line, on the other side from the curve, and the radius becomes longer. The book says to estimate where to keep the tip and try again until both waves match.