How to calculate the mirror point along a line?

Question

In 2D plane, I have a point and a line. How to get the mirror point along this line?

Answer 1

The details depend on how your line is represented. If you represent it as an arbitrary point P on the line together with a unit column vector n along the line, then the mirror point Q' to any point Q is given by:

Q' = Q + 2(I - nn^T)(P - Q)

(Here, I is the 2x2 identity matrix, n^T is the transpose of n (treating n as a 2x1 matrix), and nn^T is the 2x2 matrix formed by standard matrix multiplication of n with n^T.) It's not too hard to show that Q' will not change if you move P anywhere on the line.

It's not hard to convert other line representations into a point/unit vector representation.