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.
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.