2d transformations

we use a $\text{[math]}$ vector to represent a point in 2d space. a general form of linear transformations (or more precisely, affine transformations) in $\text{[math]}$ can be written as: $\text{[math]}$ or $\text{[math]}$ or, in matrix form, using homogeneous coordinates: $\text{[math]}$ matrices allow arbitrary linear transformations to be displayed in a consistent format, suitable for computation.

2d translation

given a point $\text{[math]}$ , we wanna reposition it with the translation distance $\text{[math]}$
$\text{[math]}$ this is an affine transformation, we deduce an affine transformation matrix:
$\text{[math]}$

[[blk:1672403653][2d rotation about the origin]]

a rotation about the origin by angle $\text{[math]}$ can be done with the matrix:
$\text{[math]}$

2d scaling

scaling is just multiplying the variables by some constants, it is defined using the hadamard product of the point by the scaling vector $\text{[math]}$
$\text{[math]}$ we deduce the transformation matrix:
$\text{[math]}$ using homogeneous coordinates:
$\text{[math]}$

mapping a grid onto another

this is a special case of the formula for linearly mapping an interval onto another, where the number of dimensions is 2.
mapping the point $\text{[math]}$ from the intervals $\text{[math]}$ , respectively, to the intervals $\text{[math]}$ , respectively:
$\text{[math]}$