This lab is about the vector valued function r(t) = (a cos(t), 2 sin(2t), 4bt), where a, b ? 0. Calculate the torsion at t = ?, as a function of a and b. Suggestion: First calculate v, a, da/dt at t = ?, and then use those to find ?.
Q2 (99 points)
Refer to Question 1, and assume a = 4 and b = 0, with result that the z-component of r(t) is always zero ie r(t) is a curve in the x, y-plane.
A. I have calculated that a global minimum of the radius of curvature occurs at t = 0.6708. Calculate the radius of curvature and the center of the best fitting circle to the curve, at that value of t.
B. Using a computer, plot the curve in the x, y-plane, from t = 0 to t = 2?, and on the same graph, plot the best fitting circle that you calculated in (A).
Suggestion: The radius of curvature is R = \frac{|v|^3}{|v \times a|} and the center is at r + RN. To avoid complicated expressions, first substitute t = 0.6708 into r, v, and a, and then calculate ? and N from those.
