How can I reflect a vector in a line without using the origin?

[Gregg]

Homework Statement

To find a transformation that reflects a vector in a line that does not pass through the origin.

The Attempt at a Solution

I've been trying to work this out for a little while and I found that I can do it if I use the vector for the 2d transformation with a 3x3 matrix for x = a (below) and y=a similarly. But y=ax + b I cannot do. I cannot see a way to do this just as I can't see a way to rotate around a point other than the origin.

$\left[ \begin{array}{ccc} -1 &0 &2a \\0 &1 &0 \\0 &0 &1 \\ \end{array}\right] \left[ \begin{array}{c} x \\ y \\ 1 \end{array} \right] = \left[ \begin{array}{c} -x+2a \\ y \\ 1 \end{array} \right]$


Where do I start?

 

[Wretchosoft]

I'll give you a hint for rotating around a point other than the origin: The origin is just that, a point. You can perform a coordinate transformation to reduce the case of a point not at the origin to the case at the origin. If you understand that methodology, you should also be able to reflect about lines that don't pass through the origin.

 

[lanedance]

ie as wretchosoft is implying first apply a translation to make your line pass through the origin
$$\textbf{x'} = T_a(\textbf{x}) = \textbf{x} + \textbf{a}$$
then apply your reflection matrix
$$R_f(\textbf{x'}) = R.\textbf{x'}$$

and translate back, then your whole tranformation should be something like
$$T_{-a} \circ R_f \circ T_a(\textbf{x})$$