Note that w(s0) is parallel to the rectifying plane to x(s0). The direction of t...

Note that w(s₀) is parallel to the rectifying plane to x(s₀). The direction of the Darboux vector w gives the axis of the “screwlike” motion of the path x and its length gives the angular velocity of the motion. Exercises 43–45 concern the Darboux vector.

(a) Use the Frenet–Serret formulas to establish the Darboux formulas:

(b) Use the Darboux formulas to establish the Frenet– Serret formulas. Hence the two sets of equations are equivalent. (Hint: Use Exercise 42.)