11.1: Geometric Definition of Conformal Mappings

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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.

001 - (Example 11.1.1).svg — 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.

002 - (Example 11.1.2).svg — Figure \(\PageIndex{1}\): A conformal map of a rectangular grid. (CC BY-NC; Ümit Kaya)