The astroid is defined by these equations:
c(t) = RR * [ cube(cos(t)), cube(sin(t)) ]
Of course it is a special case of the
Hypocycloids
with frequency = -3 and stick = 1. It is shown separately because
of its second construction:
The tangent segment between the coordinate axes has constant length;
it is often viewed as a ladder from the ground to a wall. The normals
of the astroid are tangents of twice as large an astroid, rotated by
45 degrees. This pair of orthogonal ladders allows a simple circle & ruler
construction of the astroid - watch the demo.
"Rolling" means that the point of the rolling circle that touches the
"street"-circle has velocity zero.
A motion with a point at rest is a rotation around this rest point.
Applied to the drawing pen on the stick
this means: the curve tangent at the point just drawn is orthogonal to its
connection to its rest point.
Compare Cycloid