Problem $3\mathrm{.}1$

A robot arm with a rapid laser is used to do a quick quality check, such as the radius of hole, on six holes

located on a rectangular plate. Coordinates of the centers of the holes are given below

The path of the robot going from one point to another point needs to be smooth so as to avoid sharp jerks in

the arm that can otherwise create premature wear and tear of the robot arm.

a$)$ One such path can be the interpolating polynomial $P_5(x)$ that passes those points. Find it$.$

b$)$ Other paths can be obtained by cubic spline interpolation. Find several cubic splines interpolating these

data, plot them together with $P_5(x)$ and propose the one with the shortest path.