Suppose we wish to animate a bouncing ball. We are given keyframes for the ball's motion. The

key frames are at times $T_f=\{0,1,2,3,4,5,6\},$ with associated height values of

$H_f=\{0,5,8,9,8,5,0\}.$

a$)$ Plot the graph of height versus time using linear interpolation between key frames.

b$)$ Give an equation for the height h$($t$)$ over the interval t$=4..\mathrm{.}5$ using linear interpolation.

c$)$ In general, suppose we are given times $T_f=\{t_0,t_1\},$ and associated heights $H_f=\{h_0,h_1\},$ what is the equation for $h(t)$ in the interval $t_0,t_1$ using linear interpolation?

d$)$ Suppose we wish to achieve a smoother animation by using Catmull$-$Rom interpolation. What would the parametric derivatives of the interpolated curve be at $t=4$ and $t=5?$

e$)$ Give $2$D cubic Bezier control points for a Hermite interpolation given two sequential points $(t_0,h_0)$ and $(t_1,h_1)$ with associated derivatives $(v_0,v_1).$