Let x(t) and y(t) be segments of a B-spline as in Exercise 6. Show that the curve has C² continuity (as well as C¹ continuity) at x(1). That is, show that x"(1) = y"(0). This higher-order continuity is desirable in CAD applications such as automotive body design, since the curves and surfaces appear much smoother. However, B-splines require three times the computation of Bézier curves, for curves of comparable length. For surfaces, B-splines require nine times the computation of Bézier surfaces. Programmers often choose Bézier surfaces for applications (such as an airplane cockpit simulator) that require real-time rendering.

Data From Exercise 6

A B-spline is built out of B-spline segments, described in Exercise 2. Let p_{0,................,}p₄ be control points. For 0 ≤ t ≤ 1, let x(t) and y(t) be determined by the geometry matrices [p₀ p₁ p₂ p₃] and [p₁ p₂ p₃ p₄], respectively. Notice how the two segments share three control points. The two segments do not overlap, however—they join at a common endpoint, close to p₂.