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# dghw3 - Math 454 Spring'08 homework assignment#3 due Feb 28...

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Math 454, Spring ’08, homework assignment #3, due Feb 28 Problem 1 Let C be a regular parametrized simple space curve of class C 3 . Prove that for any point P on C , there exists a plane π passing through P and such that all nearby points on C lie on the same side of π . ( Hint: it is one of the “coordinate planes” determined by the orthonormal frame { t , n , b } . Use the local canonical form.) Problem 2 Use vector analysis and the Frenet–Serret formulas for a regular space curve γ ( s ) to compute the following quantities in terms of the vectors t ( s ) , n ( s ) , b ( s ) and the curvature and torsion, κ ( s ) , τ ( s ): 1. b ( s ) × t ( s ); 2. n ( s ); 3. b ( s ); 4. γ ( s ). Prime denotes differentiation with respect to s and you may assume that all the nec- essary derivatives exist. Problem 3 Let γ ( t ) be a circular helix as in Pressley, p. 26. Find the equations of the osculating plane, rectifying plane, and the normal plane at the point P = ( x 0 , y 0 , z 0 ) of the helix. Your answer must be expressed in terms of the Cartesian coordinates of P

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dghw3 - Math 454 Spring'08 homework assignment#3 due Feb 28...

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