In the context of surfaces the strict analog of arclength parametrized curves is an *isometric immersion*

\[f: M\to \mathbb{R}^3\]

of a standard surface into $\mathbb{R}^3$. Here “isometric” means that lengths of curves and intersection angles of curves on the surface remain the same as they were on the standard surface $M$. As an example, here is an isometric immersion of an annulus in $\mathbb{R}^2$:

Here we used in a `Point Wrangle`

node the following VEX code that wraps the $z,x$-plane isometrically around a cone:

#include "Math.h" float alpha = PI/180 * ch("alpha"); float ca = cos(alpha); float sa = sin(alpha); float z = @P.z; float x = @P.x; float r = sqrt(z*z + x*x); float phi = atan2(z,x)/ca; @P.z = ca * r * cos(phi); @P.x = ca * r * sin(phi); @P.y = sa * r;

If the parameter domain is a subset $M\subset \mathbb{R}^2$, the standard coordinates of $\mathbb{R}^2$ are labeled $u,v$ and subscripts denote partial derivatives then $f$ is isometric if and only if

\begin{align*}\left|f_u\right|=\left|f_v\right|&=1 \\\\ \langle f_u,f_v\rangle &=0.\end{align*}

Very few surfaces admit such an isometric parametrizations. Also the possible isometric deformations of a given surface in $\mathbb{R}^3$ are quite limited and rigid. We therefore look at the larger class of *conformal deformations*, which preserve intersection angles of curves on the surface but not the length. So local features are enlarged or shrunk but not distorted.

A parametrization $f$ defined on a domain $M\subset \mathbb{R}^2$ is conformal if and only if the exists local scalings $e^\phi$ defined by a function $\phi: M \to \mathbb{R}$ such that

\begin{align*}\left|f_u\right|=\left|f_v\right|&=e^\phi \\\\ \langle f_u,f_v\rangle &=0.\end{align*}

For many applications (including numerical computations) it is desirable that the triangles of a simplicial surface are as equilateral as possible. Similarly often the quadrilaterals of a quadrilateral mesh should preferably as square-shaped as possible. Conformal deformations have the pleasant feature that they do not deteriorate the quality of the mesh (when measured in this sense).