[RyanCahoon]

Originally Posted by faust1706

This requires an understanding of calculus 1, the normal line.

This is definitely the simpler/faster way to derive the equations, but requires a separate proof that the shortest distance between the original curve and the boundary curve can be found along the normal line of the original curve. For bonus points:

Let y(x) be the original function [ y(x) = 0.0433x² ]
Suppose f(x) is the equation describing boundary curve that we're trying to find.
Let D(x, x₀) = distance between [x, y(x)] and [x₀, f(x₀)]
D(x, x₀) = √((x - x₀)² + (y(x) - f(x₀))²)

Do the derivation as the intersection between the hyperplane where the distance between the curves is 1 [ D(x, x₀) = 1 ] and the hyperplane where the distance between the curves at the solution points is minimal [ d D(x, x₀) / d x₀ = 0 ]