Ski Jump, $3$ Bezier Curvas

Control vertices for the curve that models flight are: $p_0=(220,70),p_1=(170,62),p_2=(150,60),p_3=(100,30).$ Control vertices for the curve that models the jump are: $,80),q_2=(235,72),q_3=(220,70).$ Assume the units are meters.

Answer the following questions:

a$.$ Derive the equation for the cubic Bezier curve that corresponds to the given control vertices. Simplify the equation into the form: $a_3{u}^3+a_2{u}^2+a_{1}u+a_0=k(u)$

b$.$ Compute the point on the curve at $u=0\mathrm{.}7$

c$.$ Derive or state the relationship between Bezier curve control vertices and Hermite curve tangent vectors. Compute the tangent vectors at the end of the jump curve $(q_3)$ and the beginning of the flight curve $(p_0).$

d$.$ What is the continuity condition between the jump and flight curves? Justify your answer quantitatively $($i$.$e$.,$ with calculations$).$

e$.$ Compute the control vertices in the flight curve for $C^1$ continuity between the jump and flight curves.