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When I was a student learning the difference, I found the unit-speed interpretation to be more helpful. Try using an animation in Matlab, Python, or Mathematica to illustrate this for simple curves by simultaneously tracing out $r(t) = (\cos{2t}, \sin{2t})$ and $r(s)$ or $\tilde{r}(t) = (t, t^2)$ and $\tilde{r}(s)$. Seeing the curve being traced out in its "normal" parametrization and its arclength parametrization will help show that the parameters are describing the same curve but at different speeds (the two parameters for the curve corresponding to $\tilde{r}$ show this especially well since the standard parametrization is not constant speed).

I think students will understand that implicitly, the arclength parametrization is just hiding the composition with a function involving the inverse of the arclength integral and doing so allows you to travel at unit speed. In other words, $r(s)$ really just means $r(s(t))$$r(t(s))$. The technicalities may be better suited for a differential geometry course (see the first chapter of Elementary Differential Geometry by Andrew Pressley), as concepts like regularity of a curve and the Inverse Function Theorem are required.

When I was a student learning the difference, I found the unit-speed interpretation to be more helpful. Try using an animation in Matlab, Python, or Mathematica to illustrate this for simple curves by simultaneously tracing out $r(t) = (\cos{2t}, \sin{2t})$ and $r(s)$ or $\tilde{r}(t) = (t, t^2)$ and $\tilde{r}(s)$. Seeing the curve being traced out in its "normal" parametrization and its arclength parametrization will help show that the parameters are describing the same curve but at different speeds (the two parameters for the curve corresponding to $\tilde{r}$ show this especially well since the standard parametrization is not constant speed).

I think students will understand that implicitly, the arclength parametrization is just hiding the composition with a function involving the inverse of the arclength integral and doing so allows you to travel at unit speed. In other words, $r(s)$ really just means $r(s(t))$. The technicalities may be better suited for a differential geometry course (see the first chapter of Elementary Differential Geometry by Andrew Pressley), as concepts like regularity of a curve and the Inverse Function Theorem are required.

When I was a student learning the difference, I found the unit-speed interpretation to be more helpful. Try using an animation in Matlab, Python, or Mathematica to illustrate this for simple curves by simultaneously tracing out $r(t) = (\cos{2t}, \sin{2t})$ and $r(s)$ or $\tilde{r}(t) = (t, t^2)$ and $\tilde{r}(s)$. Seeing the curve being traced out in its "normal" parametrization and its arclength parametrization will help show that the parameters are describing the same curve but at different speeds (the two parameters for the curve corresponding to $\tilde{r}$ show this especially well since the standard parametrization is not constant speed).

I think students will understand that implicitly, the arclength parametrization is just hiding the composition with a function involving the inverse of the arclength integral and doing so allows you to travel at unit speed. In other words, $r(s)$ really just means $r(t(s))$. The technicalities may be better suited for a differential geometry course (see the first chapter of Elementary Differential Geometry by Andrew Pressley), as concepts like regularity of a curve and the Inverse Function Theorem are required.

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source | link

When I was a student learning the difference, I found the unit-speed interpretation to be more helpful. Try using an animation in Matlab, Python, or Mathematica to illustrate this for simple curves by simultaneously tracing out $r(t) = (\cos{2t}, \sin{2t})$ and $r(s)$ or $\tilde{r}(t) = (t, t^2)$ and $\tilde{r}(s)$. Seeing the curve being traced out in its "normal" parametrization and its arclength parametrization will help show that the parameters are describing the same curve but at different speeds (the two parameters for the curve corresponding to $\tilde{r}$ show this especially well since the standard parametrization is not constant speed).

I think students will understand that implicitly, the arclength parametrization is just hiding the composition with a function involving the inverse of the arclength integral and doing so allows you to travel at unit speed. In other words, $r(s)$ really just means $r(s(t))$. The technicalities may be better suited for a differential geometry course (see the first chapter of Elementary Differential Geometry by Andrew Pressley), as concepts like regularity of a curve and the Inverse Function Theorem are required.