In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space. Why do we want to preserve angle not the area or size?

It will be messy and CGA is efficient, put into algorithm. inserting into it mobius transformation, also known as a linear fractional transformation, is a function of a complex variable that has the following form: f(z) = az+b/cz+d. where a,b,c, and d are complex numbers, and ad−bc !=0 (to ensure the transformation is non-degenerate). The variable z is a complex number.

inserting into Cauchy-Riemann equation, The Cauchy-Riemann equations are a set of two partial differential equations that provide a necessary (and, under certain conditions, sufficient) condition for a function to be holomorphic (i.e., complex differentiable) in a domain of the complex plane. provide a bridge between the properties of complex functions and the calculus of real-valued functions.

Matrix/linear algebra is no longer the best math language to deal with rotation, reflection etc., complex number or more specifically geometric algebra is.

Least Squares Conformal Maps (LSCM) is a method predominantly used in computer graphics and geometry processing to produce UV maps. UV mapping is the process of projecting a 3D surface onto a 2D plane, and it’s essential for texture mapping in computer graphics.

The LSCM method aims to produce a UV map that is as conformal as possible. A conformal mapping is one that preserves angles, which means that the mapped (or “flattened”) representation of the 3D surface won’t have skewed or distorted textures. However, producing a perfectly conformal map in many cases is impossible or impractical, so the “least squares” part of LSCM aims to minimize the overall distortion.

Here’s a more detailed breakdown:

1. Conformality: A map is conformal if it preserves angles. In the context of UV mapping, this means that the textures will appear “natural” and undistorted on the 3D surface.
2. Least Squares: In many cases, it’s impossible to produce a UV map that is perfectly conformal due to the inherent differences in geometry between the 3D surface and the 2D plane. The “least squares” method is a mathematical approach to minimize the overall distortion in the map.
3. Application: The LSCM method is often used in 3D modeling software for UV unwrapping. It provides a good balance between minimizing distortion and computational efficiency.
4. Mathematics: The LSCM approach can be understood as solving a certain kind of optimization problem. The goal is to find the UV coordinates for each vertex of the 3D mesh that minimize a specific energy function. This energy function measures the distortion between the 3D surface and its 2D representation. The problem can be cast as a sparse linear system, which can be efficiently solved using various numerical methods.