#objective: determine what arguments should be passed to MS-Paint-compatible transformation matrices, in order to emulate an arbitrary transform.

#in other words, given the equation:
# [1 0] * [1 b] * [c 0] = [w x]
# [a 1]   [0 1]   [0 d]   [y z]
#... Solve for {a,b,c,d} in terms of {w,x,y,z}

from sympy import Matrix, nonlinsolve
from sympy.abc import a,b,c,d, w, x, y, z
paint_operation = (
    Matrix([[1, 0], [a, 1]]) *        #shear y
    Matrix([[1, b], [0, 1]]) *        #shear x
    Matrix([[c, 0], [0, d]])    #scale x and y
)
m = Matrix([[w,x],[y,z]])
solution_set = nonlinsolve(paint_operation - m, a,b,c,d)
solutions = next(iter(solution_set))
for name, expr in zip("a b c d".split(), solutions):
    print(f"{name} = {expr}")

#output:
#a = y/w
#b = -w*x/(-w*z + x*y)
#c = w
#d = -(-w*z + x*y)/w

#experimentation performed elsewhere suggests that this is the correct result