We begin this collection of weird and wonderful mathematical clocks with the Morphing Clock that changes shape as time passes. The Geometry Expressions file is included so you can see how it was made and make new versions for yourself!

Is this clock a circle, or a square, or something in-between, or something beyond? Not only do the hands move in this clock, but its shape changes with time. At the hour it is circular, at the quarter hours it is square and at the half hour, it has gone beyond square to a concave curved square shape. The morphing is achieved by taking a linear combination of the parametric representation of a square and a circle.

Geometrically, the morph curve is constructed by joining two corresponding points on the original curves by a line, and then positioning a third point at a specific proportion along this line. Drag the middle red dot to see the morph curve change.

The form of the morph curve does depend
on how the original curves are parametrized.
Changing the parametrization of the original curves does not change their appearance when displayed, but it does change the appearance of the morph curve.

A morphing operation can be defined between any pair of parametric curves, open or closed. This app shows morphing a circle into a line segment. Try moving the location of the destination line segment relative to the start of the circle’s parametrization by dragging the red dots to get diﬀerent morph curves.