Question by : What is the matrix used to reflect a point across an arbitrary line in the coordinate plane?
In geometry class, I studied transformational geometry. In my textbook, there is a graphing calculator activity where you imput matrices to transform objects. So I was fascinated and began to learn matrices on my own. I have matrices for arbitrary rotation, dilation, and translation, but none for reflection. So I researched it, but found nothing. Is there a matrix for reflection ? If so, what are the entries ? If not, explain why.

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Best answer:

Answer by Indica
First consider the line y=tan(θ)x which goes through O

The reflection of a point P(x,y) with polars (r,α) is the point (r,2θ–α)

∴ reflected point has x'=rcos(2θ–α), y'=rsin(2θ–α)

Expanding and using x=rcos(α) and y=rsin(α) gives reflected coordinates as

x' = cos(2θ)x +sin(2θ)y, y' = sin(2θ)x – cos(2θ)y

For the line y=tan(θ)x+c first shift the origin to (0,c). Apply the reflection with above transformation and then restore original origin.

Then x' = cos(2θ)x +sin(2θ)(y–c), y' = c + sin(2θ)x – cos(2θ)(y-c)

This can be written as [ x', y' ] = A [ x, y ] + b

where (by row) A = { cos(2θ), sin(2θ) }, { sin(2θ), −cos(2θ) }

and b = c[ -sin(2θ), 1+cos(2θ) ]

Note that det(A)=-1 but for rotational matrices det(A)=+1.

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