The oloid is also called a two-circle roller (TCR) because its parametrical description is based on two points
on two unit circles
that are perpendicular to each other and mutually contain the center of the other.
are conjugated by the point
are tangents to the respective circle. The line
is called the generator line.
is symmetrical to
with respect to the
is also a generator, which is chosen here because when TCR is set on a plane,
get in contact with the plane and so are all points on
With the generator line, the oloid can be described parametrically so it is necessary to know the position of
at any time. In the original coordinate system with origin at the center of the oloid, the location of
is determined by the angle
; to describe the moving oloid, a new coordinate system in the plane is needed in which
must also depend on
coincide with the point
in the plane when
as the origin of the new coordinate system. On the way to the position shown, the point
leaves a trace on the plane, the blue arc. It has the same length as the arc on
and because this depends on
, the arc length of the blue trajectory can be expressed as a function of
. The arc length of the trajectory can be calculated by integrating over the curvature
We have that
and there are two more integrals to get the parametrization of the arc-length. In  an exact solution is given and in  the integrals are solved numerically, but this slows the process of calculation dramatically so it is not acceptable for interactivity. The author converted the integrands of some of the integrals into power series to get an approximate solution to get the vector equation for the coordinate transformation, but this was not possible for all of the needed functions, so some of the solutions in  are used also. For a more detailed discussion of the mathematics involved, you may refer to the excellent articles  and .
The staggering motion makes the oloid great for stirring, ideally suited to agitate liquids .