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LectureNotes7U

# LectureNotes7U - Oct 1 20041 Math 497C Curves and Surfaces...

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Math 497C Oct 1, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 7 1.17 The Frenet-Serret Frame and Torsion Recall that if α : I R n is a unit speed curve, then the unit tangent vector is defined as T ( t ) := α ( t ) . Further, if κ ( t ) = T ( t ) = 0, we may define the principal normal as N ( t ) := T ( t ) κ ( t ) . As we saw earlier, in R 2 , { T, N } form a moving frame whose derivatives may be expressed in terms of { T, N } itself. In R 3 , however, we need a third vector to form a frame. This is achieved by defining the binormal as B ( t ) := T ( t ) × N ( t ) . Then similar to the computations we did in finding the derivatives of { T, N } , it is easily shown that T ( t ) N ( t ) B ( t ) = 0 κ ( t ) 0 - κ ( t ) 0 τ ( t ) 0 - τ ( t ) 0 T ( t ) N ( t ) B ( t ) , where τ is the torsion which is defined as τ ( t ) := - B , N . Note that torsion is well defined only when κ = 0, so that N is defined. Torsion is a measure of how much a space curve deviates from lying in a plane: 1 Last revised: September 27, 2007 1

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Exercise 1. Show that if the torsion of a curve α : I R 3 is zero everywhere then it lies in a plane. ( Hint : We need to check that there exist a point p and a (fixed) vector v in R 3 such that α ( t ) - p, v = 0. Let v = B , and p be any point of the curve.) Exercise 2. Computer the curvature and torsion of the circular helix ( r cos t, r sin t, ht ) where r and h are constants. How does changing the values of r and h effect the curvature and torsion.
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