this is my first time posting here. I am not sure how to type everything so please forgive my notation.
I have the following question and I have worked throught it most of the way, there is just one step that I am not sure on and cannot proceed until I know I have done it right.
So, I started out with the following parameterization for the line: X(t) = a + t(b-a)
My first step was to follow through with an arc length parameterization. I used the formula
S(t) = The integral from t0 up to t of the magnitude of X'(t) with respect to t.
I found x'(t) to be b-a and therefore the magnitude is just |b-a|. However, my concern is in taking there integral of this.
Is it just t|b-a|? I am confused because this is more or less where I am started with the exception of the magnitude.
I know that after I find S(t), I will invert the function to get something in the form t(s) = something and that will be the unit speed parameterization I am looking for.
Any help would be appreciated! Thanks in advance
You have for the length $S=\int_0^t\|\mathbf b-\mathbf a\|du=\|\mathbf b-\mathbf a\|t$.
So $t=\frac{S}{\|\mathbf b-\mathbf a\|}$ hence your arc-parametrization is $$X(S)=\mathbf a+\frac{S}{\|\mathbf b-\mathbf a\|}(\mathbf b-\mathbf a).$$ The range for $S$ is $0\le S\le \|\mathbf b-\mathbf a\|$, since $X(0)=\mathbf a$ and $X(\|\mathbf b-\mathbf a\|)=\mathbf b$.
It is easy to see that $\|\frac{dX}{dS}\|=1$.