# Reflection Matrix in 2D

Reflection Matrix in 2D

Here is a simple setup of a manipulation and reflection matrix in 2D space.

By using a reflection matrix, we can determine the coordinates of the point , the reflected image of the point in the line defined by the vector from the origin.

P

1

P

L=(u,v)

The projection of onto the line is . The point is then determined by extending the segment by . As vectors, =2-P.

P

P

0

P

1

PP

0

||

PP

0

P

1

P

0

If is normalized (so that +=1), the reflection matrix is . Then , that is, the reflection of a reflection is the identity. Also, .

L

2

u

2

v

R=

1-2 2 v | 2uv |

2uv | -1+2 2 v |

R·R=

I

2

det(R)=-1