Interested in experimenting with the bilinear transformation of a rectangle to a quad? Check out the details here: on page 4.
The square coordinates are:
(500,900)(599,900)(599,999)(500,999)
While the quad coordinates are:
(454,945)(558,951)(598,999)(499,999)
Simply align the corresponding corners for both shapes. Then create matrices based on these values:
[1 500 900 450000][a0] [454]
[1 599 900 539100][a1]=[558]
[1 599 999 598401][a2] [598]
[1 500 999 499500][a3] [499][1 500 900 450000][b0] [945]
[1 599 900 539100][b1]=[951]
[1 599 999 598401][b2] [999]
[1 500 999 499500][b3] [999]Solving for these matrices gives:
a0=-709.911845730028
a1=1.50964187327824
a2=0.709621467197225
a3=-0.000510152025303541
b0=148.305785123967
b1=0.611570247933884
b2=0.85154576063667
b3=-0.000612182430364249However, running a Python script for manual simulation results in different output. For instance, converting point (454, 945) on the quad to (500,900) on the square yields (442.90822654, 1024.0)...
X = 454
Y = 945
a0=-709.911845730028
a1=1.50964187327824
a2=0.709621467197225
a3=-0.000510152025303541
b0=148.305785123967
b1=0.611570247933884
b2=0.85154576063667
b3=-0.000612182430364249
A = b2*a3 - b3*a2
C_one = (b0*a1 - b1*a0)
C = C_one + (b1*X - a1*Y)
B_one = (b0*a3 - b3*a0) + (b2*a1 - b1*a2)
B = B_one + (b3*X - a3*Y)
V = (-B + (B*B - 4*A*C)**0.5 ) / (2*A)
U = (X - a0 - a2*V) / (a1 + a3*V)
print U,V
Seeking advice on what may be causing this discrepancy.