Math32A sample problems

Math32A sample problems - Curves and curvature 1Find the...

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Math32A sample problems Geometry items in two dimensions(from class and homework) Prove: 1 Base angles of an isosceles triangle are equal 2 Altitudes of a triangle meet in a point 3 Medians of a triangle meet in a point 4 Angle bisector of a triangle divides the opposite side into segments with lengths proportional to the adjacent sides 5 Product of lengths of segments of a chord through a point is the same for all chords through the point 6 Product of the length of a whole secant of a circle and its external segment is independent of the secant. Three dimensional geometry of regular tetrahedron: 1Find fourth vertex if three vertices are on the unit circle((1,0,0) etc.) 2Find the angle from the lines from the center to each of two vertices(answer cos angle = -1/3) 3Find the “dihedral angle”= angle between the planes of two faces (answer cos angle = 1/3)
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Unformatted text preview: Curves and curvature 1Find the arc length parameterization of a curve r(t) if the length of r’(t) is equal to a constant S>0. (Answer R(s)= r(s/S) ) Use the previous to find the curvature of a circle of radius A. Find the curvature of the helix r(t)= (A cos t, A sin t, Bt) , A>0 , B>0 (or B<0) 2Suppose r(s) is an arclength parameter curve so that r’(s) has length 1 for all s. (a) Show that r’’(s) is perpendicular to r’(s). (b) Suppose r’’(s) is not 0 and then set N= r’’(s)/ || r’’(s)|| and κ= ||r”(s)||. Then show that dN/ds = -κ T + τ (TxN) for some (numerical) τ (c) Then show that with B= TxN, dB/ds = -τN (cf book problems 59,60, 61 in section 13.3) 3 Find the torsion τ for the curve r(t) = (A cos t, A sin t, Bt ) where A>0 but B can be positive , negative, or 0....
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