2D Conformal Transformation

In photogrammetry, we often need to map points from one coordinate system to another. One of the most important and widely used transformations for this purpose is the 2D conformal transformation.
This transformation preserves the shape and angles of objects while allowing for scaling, rotation, and translation.

📌 What Is a 2D Conformal Transformation?
A 2D conformal transformation connects two Cartesian coordinate systems using:
Scale (λ)
Rotation (θ)
Translation (x₀, y₀)
It is commonly used when two coordinate systems differ only by orientation, position, and size—but not by distortion.

🔢 Mathematical Formulation
The transformation is defined as:
x′ = λ · cos(θ) · x − λ · sin(θ) · y + x₀
y′ = λ · sin(θ) · x + λ · cos(θ) · y + y₀
Where:
(x, y) → coordinates in the original system
(x′, y′) → coordinates in the target system
λ → scale factor
θ → rotation angle
(x₀, y₀) → translation (shift between systems)

🧩 Simplified Form (For Computation)
To make calculations easier, we define new variables:
a = λ · cos(θ)
b = λ · sin(θ)
This simplifies the equations to:
x′ = a·x − b·y + c
y′ = b·x + a·y + d
Where:
c = x₀
d = y₀
This form is much more convenient for numerical solutions.

🧮 Solving the Transformation
To compute the unknown parameters (a, b, c, d), we use corresponding points from both coordinate systems.
For N points, we build a system of equations:
Each point gives 2 equations
Total equations = 2N
This system can be written in matrix form:
L = A · X
Where:
L → observed coordinates (target system)
A → coefficient matrix
X → unknown parameters

📊 Least Squares Solution
Since real data often contains noise, we solve the system using the least squares method:
X̂ = (AᵀA)⁻¹ AᵀL
This gives the best-fit parameters that minimize error.
After solving, we can compute residuals:
V̂ = A · X̂ − L
These residuals help evaluate the accuracy of the transformation.

⚙️ Degrees of Freedom
The 2D conformal transformation has 4 unknown parameters:
a, b → rotation + scale
c, d → translation
This means:
Minimum required points = 2 points (but more are recommended for accuracy)

📍 When to Use It
Use 2D conformal transformation when:
Shapes must remain unchanged (no distortion)
Angles must be preserved
Only rotation, scaling, and shifting are needed
Typical applications:
Image alignment
Map registration
Coordinate system conversion

⚠️ Limitations
Cannot handle skew or non-uniform scaling
Not suitable when axes are not perpendicular
Cannot model perspective effects
For more complex cases, use:
Affine transformation
Projective transformation

🚀 Summary
Preserves shape and angles
Uses scale, rotation, and translation
Simplified using secondary variables (a, b,