This question addresses a problem from this question. Sketching a possible approach to attack it. If we have the differentiable function $t(x)$ which we want to create a sequence of circles as described in the previous question. If we define two functions ($c_x(x)$, $c_y(x)$) to be the centerpoint coordinates of a circle laying tangent to the function $t(x)$ at $x$.
That the circle radius vector should be orthogonal to the curve gives us the following dot product equation (which should hold true because tangent of a circle is orthogonal to it's radius):
$$ \left(1,\frac{\partial t}{\partial x} (x)\right) \left(\begin{array}{l}c_x(x) - x\\c_y(x)-t(x)\end{array}\right) = 0$$
This is a linear equation for $c_x$,$c_y$ where everything else involved is assumed to be known, but we should also require these functions be smooth.
The next requirement may prove to be more tricky we want to make sure that we only have exactly one real solution : $x = x_0$ to the intersection between the function and the circle : $$(t(x)-c_y(x_0))^2 + (x-c_x(x_0))^2 = {\epsilon_R}(x_0)^2$$
Can someone help me from here? Do we have enough constraints to determine our center point and radius functions? Or at least show that there exist such functions which have properties we desire?