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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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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 0 , g 0 , h 0 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 0 ( t )] 2 + [ g 0 ( t )] 2 + [ h 0 ( t )] 2 dt and this can be put in more compact form: L = Z b a | r 0 ( t ) | dt r 0 ( t ) = h f 0 ( t ) , g 0 ( t ) , h 0 ( 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 , π/ 2) to the point (1 , 0 , 2 π ). 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

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to an arbitrary value of the parameter t : s ( t ) = Z t a | r 0 ( u ) | du = Z t a q [ f 0 ( u )] 2 + [ g 0 ( u )] 2 + [ h 0 ( u )] 2 du d s d t = | r 0 ( 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 , 0) in the direction of increasing t . Solution. (1 , 0 , 0) corresponds to t = 0. Also, ds dt = | r 0 ( t ) | = 2 s = Z 2 0 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 0 ( t ) is continuous and r 0 ( 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 0 ( t ) turns continuously when t varies. 2
The unit tangent vector. T ( t ) = r 0 ( t ) | r 0 ( t ) | . The curvature of a curve C at a given point is a measure how quickly the curve changes direction at that point. Specifically, we define it to be the magnitude of the rate of change of the unit tangent vector with respect to the arc length. We use the arc length in order the curvature to be independent of the parametrization.

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topics_14_3_14_4_First_Kepler_s_Law - 14.3 Arc Length and...

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