I am foregoing the calculus and looking at it geometrically. The curve is an ellipse in the horizontal plane z = 1.
At t = 0, r(0) is on minor vertex (1, 0, 1), and the normal is in the x direction.
At t = π the trace point has made one and a half turns around the ellipse. That places r(π) on the opposite vertex, (-1, 0, 1), and the normal is again in the x direction. By principal normal vector do you mean for it to point toward the convex side of the curve? That would be the negative x direction:
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I am foregoing the calculus and looking at it geometrically. The curve is an ellipse in the horizontal plane z = 1.
At t = 0, r(0) is on minor vertex (1, 0, 1), and the normal is in the x direction.
At t = π the trace point has made one and a half turns around the ellipse. That places r(π) on the opposite vertex, (-1, 0, 1), and the normal is again in the x direction. By principal normal vector do you mean for it to point toward the convex side of the curve? That would be the negative x direction:
unit normal vector: <-1, 0, 0>