9.7: Geodesic Curvature in Orthogonal Coordinates
For a curve X(t) in the horizontal plane in 3-space, we have defined the curvature
T'(t) = -sinθ(t)θ'(t)E1(t) + cosθ(t)θ'(t)E2(t) + cosθ(t)E1'(t) + sinθ(t)E2'(t) Note that= θ'(t) + cosθ(t)E1(t) · (-sinθ(t)E1(t) + cosθ(t)E2(t)) + sinθ(t)E2'(t)(-sinθ(t)E1(t) + cosθ(t)E2(t)) = θ'(t) + cos2θ(t)E1'(t) · E2(t) - sin2θ(T)E2'(t) · E1(t) = θ'(t) + E1'(t) · E2(t) In the case of a plane curve, E1(t) = E1 and E2(t) = E2 are constant so this result reduces to the previous calculation. Furthermore, for plane curves we have: abκg(t)s'(t)dt = abθ'(t)dt = θ(b) - θ(a). If the curve is closed, continuous, and non-self-intersecting, then θ(b) - θ(a) = 2π. For closed curves that contain vertices, we have to add in the external angles, so that θ(b) - θ(a) + ∑ext.angles = 2π. Therefore, abκgds + ∑ext.angles = 2π In the case of a general non-self-intersecting curve bounding a domain D in a coordinate patch of surface in 3-space, we have∂Dκgds + ∑ext.angles = ∂D(θ'(t) + E1'(t) · E2(t))dt = 2π + ∂DE1'(t) · E2(t))dt We will show that ∂DE1'(t) · E2(t))dt = -ΚdA, and that will complete the intrinsic form of the Gauss-Bonnet Theorem. In particular, we will show that E1'(t) · E2(t) is an intrinsic quantity. |