Schwarz-Christoffel transformation (circular version)

Schwarz-Christoffel transformation (circular version)

The complex-variables

function

f

⁢

(

z

)

=

∫

0

z

∏

k

=

1

n

(

ζ

-

z

k

)

α

k

-

1

⁢

d

⁢

ζ

,

maps the closed

unit disc

D

¯

=

{

|

z

|

≤

1

}

in the

complex plane

conformally onto a

polygon

with

n

sides,

interior angles

0

<

α

k

⁢

π

<

2

⁢

π

,
and vertices

f

⁢

(

z

k

)

. (The polygon is assumed to be not self-intersecting.)
The parameters

z

k

lie on the

unit circle

,
and depend, generally in a complicated way, on the length of the sides
of the polygon.

The

fractional powers

(

ζ

-

z

k

)

α

k

-

1

serve to clamp up an arc of the
circle into a pointy angle of measure

α

k

⁢

π

.
Indeed, the proof of the Schwarz-Christoffel formula shows that the function

f

can be decomposed near

z

k

as

f

⁢

(

z

)

=

f

⁢

(

z

k

)

+

(

z

-

z

k

)

α

k

⁢

g

k

⁢

(

z

)

,

where

g

k

is an

analytic function

with

g

k

⁢

(

z

k

)

≠

0

.
See Figure

1

.

Figure 1:

Mapping in a neighborhood of a boundary point

Note that the

exponent

is

α

k

— not

α

k

/

2

—
because the neigbourhood of a point

z

k

in the domain space
looks like a half-disc.
For the same reason, the fractional power used in the formula
is to be a single-valued branch

continuous

on the half-disc.
Finally, the extra

-

1

exponents that appear in the

integral

representation

for

f

come from the

power rule

for

differentiation

.

0.1

Example:

n

=

3

Figure

2

illustrates a mapping
from the disc to a

triangle

(

n

=

3

).
The

contours

are the approximate images, under

f

, of circles
of radius

0

<

r

≤

1

.

Figure 2:

Image of a

Schwarz-Christoffel mapping

for a triangle

Figure 3:

Corresponding contours of the domain

We describe the method used to compute the figure.
Points in the domain

D

¯

are first parameterized
as

z

=

r

⁢

e

i

⁢

θ

,
with

0

≤

r

≤

1

and

0

≤

θ

<

2

⁢

π

ranging over a discrete grid, shown schematically in Figure

4

.
The integral defining the function

f

is path-independent,
and a natural choice for the paths are
rays emanating from the origin.
When computing the integrals along each ray,
we exploit the additivity of the complex path integral:

f

⁢

(

(

r

+

Δ

⁢

r

)

⁢

e

i

⁢

θ

)

=

f

⁢

(

r

⁢

e

i

⁢

θ

)

+

∫

r

⁢

e

i

⁢

θ

(

r

+

Δ

⁢

r

)

⁢

e

i

⁢

θ

∏

k

=

1

n

(

ζ

-

z

k

)

α

k

-

1

⁢

d

⁢

ζ

,

so that

f

⁢

(

z

)

is found by summing a previously-computed value
and a new integral to be computed. And the new integral
is computed using 32-point Gauss

quadrature

after reparameterizing the path with

d

⁢

ζ

=

e

i

⁢

θ

⁢

d

⁢

r

.

Figure 4:

Parameterization of the domain for computing

f

The computation of the integrand

∏

k

=

1

n

(

ζ

-

z

k

)

α

k

-

1

=

exp

⁡

(

∑

k

=

1

n

(

α

k

-

1

)

⁢

log

⁡

(

ζ

-

z

k

)

)

is straightforward, though
we must be careful to respect the

branch cuts

prescribed above.
The

log

function in most computer languages
takes a branch cut on the negative axis.
To get the single-valued branches we need in this situation,
we must instead compute

ζ

↦

log

⁡

(

ζ

-

z

k

)

via the expression

ζ

↦

log

⁡

i

⁢

z

k

+

log

⁡

ζ

-

z

k

i

⁢

z

k

,

where

i

⁢

z

k

is the direction of the

tangent

to the circle at the point

z

k

.

Finally, after having obtained a discrete set of image points

f

⁢

(

z

)

traced along each circle

z

=

r

⁢

e

i

⁢

θ

,
the contours in the figure are obtained by interpolating a
curved Bézier spline
through the image points.

If a triangle is prescribed with the vertex locations,
it is not immediately obvious what the parameters

z

k

should be to obtain
that triangle. In the examples here, we simply avoid this difficulty
by arbitrarily choosing the parameters

z

k

=

e

2

⁢

π

⁢

i

⁢

(

k

-

1

)

/

n

to be equally spaced on the unit circle, and hope that nice figures result.

The

α

k

parameters are easily determined from the angles
of the desired figure; they are, in this example:

α

1

=

1

4

,

α

2

=

1

2

,

α

3

=

1

4

.

0.2

Example:

n

=

10

Figure

5

shows an example with

n

=

10

points.
The strategy for computing this figure is

similar

to that
of the triangle.

The parameters for this star are (rounded to four decimal places):

α

1

=

0.2422

,

α

2

=

α

1

=

1.3263

,

α

3

=

α

9

=

0.3026

,

α

4

=

α

8

=

1.3026

,

α

5

=

α

7

=

0.2754

,

α

6

=

1.3440

.

Figure 5:

Image of a Schwarz-Christoffel mapping for a star-shaped figure

0.3

Demonstration computer programs

•

http://svn.gold-saucer.org/repos/PlanetMath/SchwarzChristoffelTransformationCircularVersion/schwarz-christoffel.py

Python source code for producing images of the

Schwarz-Christoffel transformation

•

http://svn.gold-saucer.org/repos/PlanetMath/SchwarzChristoffelTransformationCircularVersion/explanation.py

Python source code for the explanatory diagrams

References

1

Lars V. Ahlfors.

Complex Analysis

, third edition. McGraw-Hill, 1979.

Title

Schwarz-Christoffel transformation (circular version)

Canonical name

SchwarzChristoffelTransformationcircularVersion

Date of creation

2013-03-22 16:52:57

Last modified on

2013-03-22 16:52:57

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

8

Author

stevecheng (10074)

Entry type

Result

Classification

msc 31A99

Classification

msc 30C20

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