I have a question about the following problem:
Find an appropriate parametrization for the given piecewise-smooth curve in $\mathbb{R}^{2}$, with the implied orientation.
The curve $C$, which goes along the circle of radius 3, from the point $(3, 0)$ to the point $(−3, 0)$, and then in a straight line along the x-axis back to $(3, 0)$.
My question is in regards to the segment connecting the points $(-3, 0)$ and $(3, 0)$.
Using the formula $c(t) = (1 - t)<x_{0}, y_{0}> + \ t<x_{1},y_{1}>$, I arrived at the following solution:
$c(t) \ = \ (1 - t)<-3 , 0> + \ t<3, 0>$
$=\ <3t - 3, 0> + <3t, 0>$
$=\ <6t - 3, 0>, t \in [0, 1]$
However, in the text I was working from, I was given this solution:
$<6t - 9, 0>, t\in [1,2]$
How would I go about obtaining this solution for this particular segment?
You can obtain the parametrization $\langle 6t - 9, 0 \rangle, t \in [1,2]$ from the parametrization $\langle 6t - 3, 0 \rangle, t \in [0,1]$ by the change of variable $t \leadsto t-1$. In the second parametrization, $t$ runs from $0$ to $1$, and so the new $t$ (that in $t-1$) must run from $1$ to $2$ (so that $t-1$ runs from $0$ to $1$). You might do this because you might want the parameter in the parametrization for the line segment to run from $1$ to $2$, so that the parameter in the parametrization for the circular segment can run from $0$ to $1$, giving a combined parametrization for the entire curve $C$ with the parameter $t$ varying from $0$ to $2$.