Need help with this homework problem for my differential geometry class. Please show work, thank you!

10. As shown in Figure 1.13, Freddy F lintstone 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 (i1, i1), with center FIGURE 1.13 C at the origin. If at(s) = (x (s), y(s)) is an arclength parametrization of the road, starting at (0,1), consider the vector 08 = 0P + P + Q8, where P = a(s) is the point of contact and Q is the > midpoint of the edge of the square. Use QP = sa' (5) and the fact that Q8 is a unit vector orthogonal to 10 CHAPTER 1. CURVES I y (s) x'(s) unexpectedly. Now use the result of Exercise 4 to nd y = f (3:). Also see the hint for Exercise 9.) 55. Express the fact that C moves horizontally to show that s = ; you will need to differentiate