Conformal Mappings :
Print this page
 
First   |   Last   |   Prev   |   Next
  

Angle between two curves: Let $\gamma_{1}$ and $\gamma_{2}$ be two smooth curves that intersect at the point MATH. Let $T_{1}$ and $T_{2}$ be the tangent vectors of $\gamma_{1}$ and $\gamma_{2}$ respectively at the point $z_{0}$. We define the angle between $\gamma_{1}$ and $\gamma_{2}$ at $z_{0}$ to be the angle $\theta$ measured counterclockwise from the tangent vector $T_{1}$ to the tangent vector $T_{2}$.

Now, suppose that $f$ is a non-constant function (need not be analytic), defined in the neighborhood of $z_{0}$. Let $\Gamma_{1}$ and $\Gamma_{2}$ be the images of the two curves $\gamma_{1}$ and $\gamma_{2}$ under the function $w = f(z)$. Since $\gamma_{1}$ and $\gamma_{2}$ intersect at the point $z_{0}$, the image curves $\Gamma_{1}$ and $\Gamma_{2}$ intersect at the point $w_{0} = f(z_{0})$. Let us denote the tangents at $w_{0}$ of $\Gamma_{1}$ and $\Gamma_{2}$ respectively by $T_{1}^{\prime}$ and $T_{2}^{\prime}$.

Conformal Map: We say that the mapping $w = f(z)$ is a conformal mapping at $z_{0}$ if, whenever two smooth curves $\gamma_{1}$ and $\gamma_{2}$ that intersect at $z_{0}$, the angle between these two curves $\gamma_{1}$ and $\gamma_{2}$ is the same as the angle between the image curves $\Gamma_{1}$ and $\Gamma_{2}$ (under the map $w = f(z)$) at the point $w_{0} = f(z_{0})$, and the sense of the angle between the curves and their images is also preserved under the mapping.

Here, by preservation of the sense of an angle under a mapping, we mean that if at $z_{0}$ the tangent $T_{2}$ is obtained from the tangent $T_{1}$ by a counterclockwise rotation through an angle $\alpha$, then at $w_{0} = f(z_{0})$ the tangent $T_{2}^{\prime}$ is obtained from the tangent $T_{1}^{\prime}$ in precisely the same manner. Thus, a conformal mapping is a mapping that preserves the angle between intersecting curves together with the sense in which the angle is measured .

 
First   |   Last   |   Prev   |   Next