Suppose that x: I → R3 is a path of class C3 parametrized by arclength. Then the unit tangent vector T(s) defines a vectorvalued function T: I → R3 that may also be considered to be a path (although not necessarily one parametrized by arclength, nor necessarily one with nonvanishing velocity). Since T is a unit vector, the image of the path T must lie on a sphere of radius 1 centered at the origin. This image curve is called the tangent spherical image of x. Likewise, we may consider the functions defined by the normal and binormal vectors N and B to give paths called, respectively, the normal spherical image and binormal spherical image of x. Exercises 32–35 concern these notions.
Suppose that x is parametrized by arclength. Show that the normal spherical image of x can never be constant.
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