For a university assignment I have a question about rotating a picture in PowerPoint. In PowerPoint a picture can have four transformations. Rotate right (90°), rotate left (90°), flip horizontally and flip vertically. Lets call them R(r), R(l), F(v) and F(h) for short.
We are then asked to compute the matrix multiplication for every pair of possible transformations.
E.g. R(r)*(R(r) or R(r)*F(h)
All of that seems fairly simple to me but they specified something in the question I dont understand.
R(r) X R(l) is read from right to left as “rotate left” then “rotate the result right” because Powerpoint transformations are performed in world coordinates
Can someone explain to me why I would read that left to right? And does that mean when I multiply the matrices together I would in fact do R(l) X (R(r) when asked to do R(r) X r(l)
Why does world co-ordinates effect this?
Would I read all transformations as left to right or just some?
Hope somebody comes with a better answer than mine. 2x2 transformation matrices T1 and T2 are transforming a 2d vector v to a new 2d vector v_t, such that
v_t = T2 * T1 * v
The transformation from left to right is mathematically not possible
v_t = v * T1 * T2
since a 2d vector can not be multiplied with a 2x2 matrix. Let's look at the v * T1 in component form:
_ _ _ _
| vx | | t00 t01 |
v * T1 = | vy | * | t10 t11 | // invalid matrix multiplication
- - - -
Therefore the vector v has to be right of the transformations. If one accepts that, it is kind of understandable that in case of T2 * T1 * v, the transformation T1 is applied first to the vector v, because it is written closest to it :) ... just kidding. There exists a mathematically profound explanation. Somehow I remember even a proof in linear algebra lesson, but I could not find.
However, the matrix multiplication is carried out from left to right, such that
v_t = T * v, with T = T2 * T1