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Rolling Circles along curves

This section gives a combination of calculus, vectors, and parametric curves in an attempt to describe a circle rolling along a parametric curve. The circles will have either a constant radius R orlp a variable radius. Our circles will be tangent to the base curve along which they are rolling in both versions, since this will make the rolling appear closest to the actual physical process. In order to visualize this process, a vector format for our curves and their tangents is necessary.

Let a parametric curve in the plane be defined by

\[ x = f(t) \qquad\mbox{and}\qquad y = g(t) , \]
with t in the interval [𝑎, 𝑏]. Its vector format is x(t) = (f(t), g(t)). This is called the position vector field of the parametric curve. Its derivative vector field or tangent vector field is
\[ \mathbf{T} (t) = \frac{{\text d}\mathbf{x}(t)}{{\text d}t} = \left( \frac{{\text d}f}{{\text d}t} , \frac{{\text d}g}{{\text d}t} \right) . \]
A vector field perpendicular to T(t) is (not unique)
\[ \mathbf{P} (t) = \left( -\frac{{\text d}g}{{\text d}t} , \frac{{\text d}f}{{\text d}t} \right) . \]
Indeed, its dot product with T(t) is zero:
\[ \mathbf{T} (t) \cdot \mathbf{P} (t) = \left( \frac{{\text d}f}{{\text d}t} , \frac{{\text d}g}{{\text d}t} \right) \cdot \left( -\frac{{\text d}g}{{\text d}t} , \frac{{\text d}f}{{\text d}t} \right) = 0 , \]
Also, P(t) can be made length R, corresponding to the radius of our circles, by first making it a unit vector field and then multiplying by R. Then the center vector field (t) for our circles is given by
\begin{equation} \label{EqCircle.1} \mathbf{C} (t) = \mathbf{x}(t) + \frac{R}{|\mathbf{P}|}\, \mathbf{P}(t) . \end{equation}
If we now write C(t) = (h(t), k(t)), then our rolling circles are given by the parametric curve:
\begin{align*} \left( x(s) , y(s) \right) &= \left( R\,\cos s + h(t) , R\,\sin s + k(t) \right) \\ &= R \left( \cos s , \sin s \right) + \mathbf{C} (t) . \end{align*}
As s varies in the interval [0, 2π], we move around this circle while we get one circle for each fixed value t ∈ [𝑎, 𝑏]. In this situation, s is the variable that moves us around each circle while t is the rolling variable that moves us from one circle to the next circle because it controls the location of the circles’ centers.

The formula for C(t) in \eqref{EqCircle.1} may appear complicated; however, if we take a minute to examine each piece, we should begin to see how simple it really is. Since C(t) is the center of the circle that we wish to roll along the curve, we need to start on the curve, x(t) and move R units perpendicular to x(t). Since P(t) is orthogonal to T(t), which in turn is tangent to x(t) we have that P(t) is orthogonal to x(t). To proceed R units in the P(t) direction, we must first make P(t) unit length by dividing it by its magnitude, then multiplying it by R. This gives the second piece of the sum found in Eq.\eqref{EqCircle.1}.    

Example 1:    ■
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Example 2:    ■
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Example 3:    ■
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Example 13:    ■
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