Schwarz–Christoffel Mappings for Nonpolygonal Regions

• An approximate conformal mapping for an arbitrary region $\varOmega$ bounded by a smooth curve $\varGamma$ is constructed using the Schwarz–Christoffel mapping for a polygonal region in which $\varOmega$ is embedded. An algorithm for finding this so-called outer polygon is presented. The resulting function is a conformal mapping from the upper half-plane or the unit disk to a region $R$, approximately equal to $\varOmega$. $R$ is bounded by a $C^\infty$ curve, and since the mapping function originates from the Schwarz–Christoffel mapping and tangent polygons are used to determine it, important properties of $\Gamma$ such as direction, linear asymptotes, and inflexion points are preserved in the boundary of $R$. The method makes extensive use of existing Schwarz–Christoffel software in both the determination of outer polygons and the calculation of function values. By the use suggested here, the capabilities of such well-written software are extended.

Ämnesord

NATURVETENSKAP  -- Matematik -- Beräkningsmatematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Computational Mathematics (hsv//eng)

Nyckelord

numerical conformal mapping

Schwarz-Christoffel mapping

tangent polygon

inner region

outer polygon

Applied mathematics

Tillämpad matematik

Numerical analysis

Numerisk analys

Matematik

Mathematics

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