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spacecurve

# spacecurve - Numerical Determination of Unit Vectors The x...

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Numerical Determination of Unit Vectors The x , y , and z coordinates of a space curve can be defined as a function of parameter, ξ , as x = x ( ξ ) , y = y ( ξ ) , z = z ( ξ ) . The tangential unit vector, ˆ e t , can be obtained as ˆ e t = d~ r ds = d~ r ds = ds dx dy dz , in which ds = p ( dx ) 2 + ( dy ) 2 + ( dz ) 2 , and ds = s dx 2 + dy 2 + dz 2 = Δ or ds = 1 Δ The above formulation guarantees that | ˆ e t | = 1 . To find the normal unit vector, perform another derivative with respect to s , 1 ρ ˆ e n = d ˆ e t ds = d ds d~ r ds = d ds d~ r ds = d 2 ~ r 2 ds 2 + d~ r d 2 ξ ds 2 With ds = 1 Δ , it follows that d 2 ξ ds 2 = - 1 2 1 Δ Δ d Δ ds = - 1 2 1 Δ Δ d Δ ds = - 1 2 1 Δ 2 d Δ = - 1 Δ 2 dx d 2 x 2 + dy d 2 y 2 + dz d 2 z 2 Therefore, 1 ρ ˆ e n = d ˆ e t ds = ds 2 d 2 ~ r 2 - ds 4 dx d 2 x 2 + dy d 2 y 2 + dz d 2 z 2 d~ r

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ˆ e t ˆ e t A B C D E F G H e t ) D =
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