Consider the parabola y=0.5*(x^2), and specifically that portion of the x-y plane above the curve. In this area, construct a new curve, defined as being the locus of points, each of which is fixed normal distance, d (d>=0), from each point on the original parabola. For d=0, the new curve is the original parabola. For relatively small d, (e.g. d=0.1), the new curve is similar to but not a parabola.

a) Is there a name for this constructed curve?

Something interesting happens to the constructed curve when d becomes equal to and then exceeds a specific finite value.

b) What is this value for d and what happens?

c) Can you generalize what happens at this transition to any parabola by relating the critical value of d to a mathematical property of said parabola?