How to find transformation matrix when the centre of rotation of a diagram is not the origin?

Question

The transformation that maps triangle T onto triangle W is a rotation of 270 clockwise about point (2, -2). How can I find the matrix that represents this?

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Answer 1 · 2018-07-19T14:53:06

see below

if it were the origin then

#((x'), (y')) = ((cos t, sin t),(-sin t, cos t)) * ((x), (y)) = R(x,y)#

but we need translation #T(x,y) = (x + 2, y - 2)#

take the point #(2, -2) -> (0, 0) -> text{rotate} -> text{Apply }T#

#M(x,y) = T circ R circ T^-1(x,y)#

#M(x,y) = T circ R(x - 2, y + 2)#

#M(x,y) = T(((cos t, sin t),(-sin t, cos t)) * ((x-2), (y+2)))#

#M(x,y) = ((0, -1),(1, 0)) * ((x-2), (y+2)) + ((2), (- 2))#

#M(x,y) = ((- y - 2 + 2), (x - 2 - 2))#

#M(x,y) = ((- y), (x - 4))#