How can we prove the curvature of a space curve using regular parameterization?

In summary, the conversation discusses the proof that the curvature of a space curve can be expressed as the magnitude of the cross product of the first and second derivatives of the parameterization, divided by the cube of the magnitude of the first derivative. The conversation also mentions a helpful resource for understanding this concept.
  • #1
kidsmoker
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Homework Statement



Let [tex]\underline{r}[/tex] be a regular parameterisation of a space curve [tex]C \subset R^{3}[/tex]. Prove that

[tex]\kappa=\frac{\left\|\underline{\dot{r}}\times\underline{\ddot{r}}\right\|}{\left\|\underline{\dot{r}}\right\|^{3}}[/tex] .

The Attempt at a Solution



We have

[tex]t(u)=\frac{\frac{dr}{du}}{\left\|\frac{dr}{du}\right\|}[/tex]

so differentiating both sides wrt u we obtain

[tex]\frac{dt}{du}=\frac{\frac{d^{2}r}{du^{2}}}{\left\|\frac{dr}{du}\right\|}+\frac{dr}{du}\frac{d}{du}(\frac{1}{\left\|\frac{dr}{du}\right\|})[/tex].

Since

[tex]\frac{dt}{du}=\kappa\underline{n}[/tex]

this gets me the curavture in terms of the desired bits (with n too) but I can't seem to get it to the desired result :\

Thanks for your help!
 
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  • #2
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1. What is the definition of curvature of a space curve?

The curvature of a space curve is a measure of how much the curve deviates from being a straight line at a given point. It is the reciprocal of the radius of the circle that best approximates the curve at that point.

2. How is the curvature of a space curve calculated?

The curvature of a space curve is calculated using the formula K = |T'(t)| / ||r'(t)||, where K is the curvature, T'(t) is the tangent vector, and ||r'(t)|| is the magnitude of the derivative of the position vector.

3. What does a high curvature value indicate about a space curve?

A high curvature value indicates that the curve is highly curved at a given point, meaning that the curve deviates significantly from being a straight line. This can be seen as a sharp turn or bend in the curve.

4. How does the curvature of a space curve affect its behavior?

The curvature of a space curve affects its behavior by determining how the curve changes direction at a given point. A higher curvature value means a sharper change in direction, while a lower curvature value indicates a smoother change in direction.

5. Can the curvature of a space curve be negative?

Yes, the curvature of a space curve can be negative. This indicates that the curve is bending in the opposite direction from the positive curvature. Negative curvature can be seen in curves such as a saddle shape or a figure eight.

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