Chapter 8.6, Problem 21E

TrueType^® fonts, created by Apple Computer and Adobe Systems, use quadratic Bézier curves, while PostScript^® fonts, created by Microsoft, use cubic Bézier curves. The cubic curves provide more flexibility for typeface design, but it is important to Microsoft that every typeface using quadratic curves can be transformed into one that uses cubic curves. Suppose that w(t) is a quadratic curve, with control points p₀, p₁, and p₂.

1. a. Find control points r₀, r₁, r₂, and r₃ such that the cubic Bézier curve x(t) with these control points has the property that x(t) and w(t) have the same initial and terminal points and the same tangent vectors at t = 0 and t = 1. (See Exercise 16.)
2. b. Show that if x(t) is constructed as in part (a), then x(t) = w(t) for 0 ≤ t ≤ 1.