Question

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

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?

Answer 1

see below

Explanation:

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))`