R Series: R01-07. Spherical Cycloids and Involutes

By David W. Henderson and Daina Taimina

Involutes, Cycloids, Epicycloids, Hypocycloids, and Trochoids on Plane

When a straight line rolls along a stationary circle a point on the line traces a curve called an involute (of the circle). See I04, for a mechanism that uses involutes and see Spiral Pump –Mathematical Tutorial for explanation of how it works.

When a circle rolls along a stationary straight line a point on the circumference of the circle traces a curve called a cycloid.

When a circle rolls along another circle then a point on the circumference of the rolling circle traces out a curve called an epicycloid (if the rolling circle rolls on the outside of the stationary circle) or a hypocycloid (if the rolling circle rolls on the inside of the stationary circle).

In all these cases of rolling circles points not on the circumference trace curves called trochoids.

See http://xahlee.org/SpecialPlaneCurves_dir/specialPlaneCurves.htmlfor detailed descriptions, equations, and pictures of the various forms of these planar curves.

Involutes, Cycloids, Epicycloids, Hypocycloids, and Trochoids on Sphere

All of the curves described above involve straight lines and circles in the plane. However, we can do the same things on a sphere. The curves on a sphere that correspond to straight lines are the great circles(circles that divide the sphere into two equal halves) because great circles have the same symmetries on the spherical surface as do straight lines on the plane. (For further discussions of great circles as straight lines on spheres, see Henderson/Taimina, Experiencing Geometry, Chapter 2. Note that on the sphere the “straight” lines are also circles and thus a spherical involute can also be considered an epicycloid. A circle on a spherical surface forms a cone from the center of the sphere; in the case of a great circle this cone is actually a planar disk. Reuleaux uses these cones and discs to produce on a sphere the rolling of circles on circles.

Several of Reuleaux’s mechanisms involve and illustrate spherical involutes, cycloids, epicycloids, hypocycloids, and trochoids:

R01 is a great circle rolling on a very small circle with the resulting spherical involute/epicycloid marked with wire.

R02 is a small circle rolling inside a larger circle and the resulting spherical hypocycloid and several spherical trochoids are marked with wire. Note that the trochoid traced by the center of the rolling circle is another circle.

R03 is a small circle rolling on the outside of a larger circle and the resulting spherical epicycloid are marked with wire.

R04 is a small circle rolling inside a larger circle with the resulting spherical hypocycloid marked by wire.

R05 has the same circles as in R4, but here the wire marks the spherical epicycloid that results from the larger circle rolling along the smaller circle (which is contained in the larger circle).

R06 is a great circle rolling on a small circle with the resulting spherical involute/epicycloid marked by brass wire and a resulting spherical trochoid marked with copper wire.

R07 is a small circle rolling on a great circle with the resulting spherical cycloid marked.

For equations and other analytic detail of the spherical cycloids, epicycloids, hypocycloids, and trochoids see Section 7.6 of Kinematics of spherical mechanisms (1988) by C.H. Chiang.