Let x(t) and y(t) be Bzier curves from Exercise 5, and suppose the combined curve has C

Let x(t) and y(t) be Bézier curves from Exercise 5, and suppose the combined curve has C² continuity (which includes C¹ continuity) at p₃. Set x"(1) = y"(0) and show that p₅ is completely determined by p₁, p₂, and p₃. Thus, the points p_0,.......,p₃ and the C² condition determine all but one of the control points for y(t).

Data From Exercise 5

Let x(t) and y(t) be cubic Bézier curves with control points (p_0, p_1,p_2,p₃) and (p_3, p₄, p₅, p₆), respectively, so that x(t) and y(t) are joined at p³. The following questions refer to the curve consisting of x(t) followed by y(t). For simplicity, assume that the curve is in R².
a. What condition on the control points will guarantee that the curve has C₁ continuity at p₃? Justify your answer.
b. What happens when x'(1) and y'(0) are both the zero vector?