Trochoids, Roses, and Thorns—Beyond the Spirograph

Abstract

This paper began with experiments using the computer algebra system Mathe? matica to draw trochoids, the kinds of curves produced by the Spirograph? drawing sets. If two tangent circles have their centers on the same side of the common tangent line, and one circle remains fixed while the other is rolled around it without slipping, a hypotrochoid is traced by any point on a diameter or extended diameter of the rolling circle. If two tangent circles have their centers on opposite sides of the common tangent line, and one circle remains fixed while the other is rolled around it without slipping, an epitrochoid is traced by any point on a diameter or extended diameter of the rolling circle. A hypocycloid is a hypotro? choid for which the tracing point is on the circumference of the rolling circle, and an epicycloid is an epitrochoid for which the tracing point is on the circumference of the rolling circle. The term trochoid is used to refer to either a hypotrochoid or an epitrochoid. Either radius, but not both, can be infinite, so that cycloids and trochoids obtained by rolling a circle along a straight line, and also certain spirals and involutes are covered by the nomenclature, but in this paper we shall assume both radii are finite. The Spirograph? produces graphs of trochoids using toothed disks and rings to prevent slipping, but because none of the holes for the pen reach the circumference of the disks, the Spirograph? cannot be used to draw true hypocycloids or epicycloids. Since the graph of a trochoid depends on four parameters that are fixed and one that is variable, all these quantities will be part of the notation. The hypotrochoid denoted by hy[t;n,m,r,a] is generated by a rolling (moving) circle of radius m and a fixed (rconmoving) circle of radius n, with rm the distance from the center of the rolling circle to the tracing point. Assume the center of the fixed circle is at the origin and denote the initial position of the tracing point (and the center of the