topics_14_3_14_4_First_Kepler_s_Law

topics_14_3_14_4_First_Kepler_s_Law - 14.3. Arc Length and...

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Unformatted text preview: 14.3. Arc Length and Curvature of Space Curves The length of a space curve is defined in the same way as a length of a planar curve. Given a space curve r ( t ) = h f ( t ) , g ( t ) , h ( t ) i , a t b or equivalently with parametric equations x = f ( t ) , y = g ( t ) , z = g ( t ) , a t b. If f , g , h are continuous on [ a, b ] and the curve is traversed exactly once if t varies from a to b , then its length L is L = Z b a q [ f ( t )] 2 + [ g ( t )] 2 + [ h ( t )] 2 dt and this can be put in more compact form: L = Z b a | r ( t ) | dt r ( t ) = h f ( t ) , g ( t ) , h ( t ) i . Example 1. Find the length of the arc (the part) of the circular he- lix with vector equation r ( t ) = cos( t ) i + sin( t ) j + t k from the point (0 , 1 , / 2) to the point (1 , , 2 ). Different parametrization of a curve. Note that one curve can have many parametric representations. However, the formula for the length of a curve does not depend on the parametric rep- resentation of the curve. Example 1. The vector functions r 1 ( t ) = h t, t 2 , t 3 i , t [1 , 2]; r 2 ( t ) = h e v , t, e 2 v , e 3 v i , v [0 , ln(2)] are two parametric representations of the twisted cubic t = e v , v = ln( t ). Parametrization of a curve with respect to its arc length. Define the arc length function s ( t ) as the length of a curve r ( t ) = h f ( t ) , g ( t ) , h ( t ) i , a t b from the initial value of the parameter a 1 to an arbitrary value of the parameter t : s ( t ) = Z t a | r ( u ) | du = Z t a q [ f ( u )] 2 + [ g ( u )] 2 + [ h ( u )] 2 du d s d t = | r ( t ) | FTC . In general, the function s = s ( t ) is increasing (hence, one-to-one) and from here it has an inverse t = t ( s ). Then the curve can be re-parametrized in term of s by r = r ( t ( s )) . The arc length parametrization is useful because it does not depend on a particular coordinate system at it is coming in a natural way from the shape of the curve. Thus for s = 2, r ( t (2)) is the position vector of the point on the curve that is 2 units of length along the curve from the starting point of the curve. Example 2. Re-parametrize the helix r ( t ) = cos( t ) i + sin( t ) j + t k with respect to the arc length measured from (1 , , 0) in the direction of increasing t . Solution. (1 , , 0) corresponds to t = 0. Also, ds dt = | r ( t ) | = 2 s = Z 2 2 dt = 2 t and from here t = s/ 2 r ( t ( s )) = cos( s/ 2) i + sin( s/ 2) j + ( s/ 2) k . Curvature of a given curve.What is a smooth curve? A parametriza- tion r ( t ) is called smooth on some interval if r ( t ) is continuous and r ( t ) 6 = 0. A curve is smooth is it has at least one smooth parametriza- tion. A smooth curve has no sharp corners and cusps: the tangent vector r ( t ) turns continuously when t varies....
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This note was uploaded on 02/05/2010 for the course MATH 264 taught by Professor Dr.d.dryanov during the Fall '09 term at Concordia Canada.

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topics_14_3_14_4_First_Kepler_s_Law - 14.3. Arc Length and...

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