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LectureNotes7U - Math 497C Oct 1, 2004 1 Curves and...

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Unformatted text preview: 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 : I R n is a unit speed curve, then the unit tangent vector is defined as T ( t ) := ( t ) . Further, if ( t ) = k T ( t ) k 6 = 0, we may define the principal normal as N ( t ) := T ( t ) ( 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 ) = ( t )- ( t ) ( t )- ( t ) T ( t ) N ( t ) B ( t ) , where is the torsion which is defined as ( t ) :=-h B , N i . Note that torsion is well defined only when 6 = 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 Exercise 1. Show that if the torsion of a curve : 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 h ( t )- p, v i = 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...
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LectureNotes7U - Math 497C Oct 1, 2004 1 Curves and...

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