This is Differential Geometry explain and solve make sure the answer is correct

10. As shown in Figure 1.13, Freddy Flintstone wishes to drive his car with square wheels along a strange road. How should you design the road so that his ride is perfectly smooth, i.e., so that the center of his wheel travels in a horizontal line? (Hints: Start with a square with vertices at (1, 1), with center FIGURE 1.13 C at the origin. If ee(s) = (x(s). y(s)) is an arclength parametrization of the road, starting at (0. 1), consider the vector OC = OP + PQ + QC, where P = (s) is the point of contact and Q is the midpoint of the edge of the square. Use QP = sa'(s) and the fact that Q C is a unit vector orthogonal to 10 CHAPTER 1. CURVES i 1 QP . Express the fact that C moves horizontally to show that s = - : you will need to differentiate x'(s) unexpectedly. Now use the result of Exercise 4 to find y = f(x). Also see the hint for Exercise 9.)