11.1: Geometric Definition of Conformal Mappings

We start with a somewhat hand-wavy definition:

Informal Definition: Conformal Maps

Conformal maps are functions on $C$ that preserve the angles between curves.

More precisely: Suppose $f(z)$ is differentiable at $z_0$ and $\gamma (t)$ is a smooth curve through $z_0$. To be concrete, let's suppose $\gamma (t_0) = z_0$. The function maps the point $z_0$ to $w_0 = f(z_0)$ and the curve $\gamma$ to

$$\tilde{\gamma} (t) = f(\gamma (t)). \nonumber$$

Under this map, the tangent vector $\gamma ' (t_0)$ at $z_0$ is mapped to the tangent vector

$$\tilde{\gamma} ' (t_0) = (f \circ \gamma)' (t_0) \nonumber$$

at $w_0$. With these notations we have the following definition.

Definition: Conformal Functions

The function $f(z)$ is conformal at $z_0$ if there is an angle $\phi$ and a scale $a > 0$ such that for any smooth curve $\gamma (t)$ through $z_0$ the map $f$ rotates the tangent vector at $z_0$ by $\phi$ and scales it by $a$. That is, for any $\gamma$, the tangent vector $(f \circ \gamma)' (t_0)$ is found by rotating $\gamma '(t_0)$ by $\phi$ and scaling it by $a$.

If $f(z)$ is defined on a region $A$, we say it is a conformal map on $A$ if it is conformal at each point $z$ in $A$.

Note

The scale factor $a$ and rotation angle $\phi$ depends on the point $z$, but not on any of the curves through $z$.

Example $\PageIndex{1}$

Figure $\PageIndex{1}$ below shows a conformal map $f(z)$ mapping two curves through $z_0$ to two curves through $w_0 = f(z_0)$. The tangent vectors to each of the original curves are both rotated and scaled by the same amount.

Figure $\PageIndex{1}$: A conformal map rotates and scales all tangent vectors at $z_0$ by the same amount. (CC BY-NC; Ümit Kaya)

Remark 1. Conformality is a local phenomenon. At a different point $z_1$ the rotation angle and scale factor might be different.

Remark 2. Since rotations preserve the angles between vectors, a key property of conformal maps is that they preserve the angles between curves.

Example $\PageIndex{2}$

Recall that way back in Topic 1 we saw that $f(z) = z^2$ maps horizontal and vertical grid lines to mutually orthogonal parabolas. We will see that $f(z)$ is conformal. So, the orthogonality of the parabolas is no accident. The conformal map preserves the right angles between the grid lines.

Figure $\PageIndex{1}$: A conformal map of a rectangular grid. (CC BY-NC; Ümit Kaya)