Rolling Circles

There are many explorations of the loci created as a circle rolling along a surface. When a circle rolls along a straight line, the locus of a point on the circle is called a CYCLOID.

If the point is located some distance from the center of the circle other than the radius, then the locus is a TROCHOID. For d < r, the trochoid is a curate trochoid; for d > r the trochoid is a prolate trochoid.

Here is a GSP file to demonstrate these three types of curves as a circle moves along a straight surface.

Problem:Derive parmetric equations for the cycloid with a circle of radius

r.The graph is from Graphing Calculator 3.5 using parametric equations. Circle with r = 1.

Curate Trophoid from parametric equations:

Circle with r = 1, d = .5

Prolate trophoid from parametric equations:

Circle with r = 1, d = 2

For one cycle of the Cycloid:

-- the area under the curve is 3 times the area of the circle.

-- the length of the arc is 8 times the length of the radius of the circle.

These properties usually require the calculus to derive.