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Unformatted text preview: Curvature Vector: from Newtons Second Law to Beam Deflection and to Relativity Shina Tan 1 Curvature Vector Any point r on a curve can be parameterized according to the length s from some starting point on the curve to point r . We define s = d r ds = r ( s ) . (1) Using the formula ds 2 = d r d r , (2) we can check that s is a unit vector, because d r ds d r ds = dr d r ds 2 = ds 2 ds 2 = 1 . s is the local tangential direction on the curve. If s changes direction as you go along the line, the line is curved . We can define the local curvature vector c : c d s ds = r 00 ( s ) . (3) If the curve changes its local tangential direction by 1 per meter, its curvature is c = 1 1m = ( / 180)radians 1m = 0 . 01745 / m . The local curvature vector c is always perpendicular to the local direction vector s : c s = 0 . (4) Proof: c s = d s ds s = 1 2 d ( s s ) ds = 1 2 d 1 ds = 0 . Example 1 At any point on a circle of radius R , there is a local curvature vector whose magnitude is c = 1 /R , and whose direction is toward the center of the circle. This is obviously perpendicular to the local tangential direction vector. Remark Unlike the local tangential direction vector which reverses direction if you choose a different end of the curve as the starting point of s , the local curvature vector c does not depend on the parameterization. For instance, no matter how you parameterize the length on a circle, the curvature vector always points toward the center of the circle. 1 Figure 1: Curvature vectors at various points on a parabola. Example 2 Find the local curvature of any planar curve y = y ( x ). Solution: the local direction on the curve can be represented by the angle, = arctan y ( x ). So c = d ds = d arctan y p 1 + y 2 dx = y 00 (1 + y 2 ) 3 / 2 . Thus the general formula for the curvature of a planar curve is c ( x ) = y 00 ( x ) 1 + y 2 ( x ) 3 / 2 . (5) Example 3 The local curvature of a parabola y = 1 2 kx 2 is c ( x ) = k (1+ k 2 x 2 ) 3 / 2 . The curvature is largest at the parabolas apex, and diminishes along its wings. Obviously, the wings are straighter than the bottom. See Figure 1. Example 4 Find the curvature of a solenoid. Solution: A solenoid can be described by the parametric equations x = R cos s R , y = R sin s R , z = 1- 2 s, where x 2 ( s ) + y 2 ( s ) + z 2 ( s ) = 1 so that s is the length parameter of the solenoid. Taking the second order derivatives of x ( s ), y ( s ), and z ( s ), we find c ( s ) =- 2 R cos s R ,- 2 R sin s R , , and c ( s ) = 2 /R . The curvature vector points toward the axis of the solenoid, and is perpendicular to the cylindrical surface on which the solenoid lies. If = 1, the solenoid degenerates into a circle with radius R , and c = 1 /R . If < 1, the curvature is smaller than 1 /R ....
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- Spring '11