(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Proof that, if a particle moves along a space curve with curvature 0, then its motion is a along a line.

2. Relevant equations

[tex]K=\frac{||r'(t)\times r''(t)||}{(||r'(t)||)^3}[/tex]

(curvature of a space curve)

3. The attempt at a solution

Assume the curve is smooth, so r'(t) cannot be the zero vector. The numerator must be 0. I evaluate the cross product (set it to 0), and get the following equations.

[tex]g'(t)h''(t) = h'(t)g''(t)[/tex]

[tex]f'(t)h''(t) = h'(t)f''(t)[/tex]

[tex]f'(t)g''(t) = g'(t)f''(t)[/tex]

Here I don't know what to do to get to the equation of a line.

Thank you in advance.

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# Proof: Curvature Zero -> Motion along a line

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