Suppose that x: I R3 is a path of class C3 parametrized by arclength. Then the unit

Chapter 3, Problem 33

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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 3235 concern these notions.Suppose that x is parametrized by arclength. Show that x is a straight-line path if and only if its tangent spherical image is a constant path. (See Example 7 of 3.2 and Exercise 30.)