Transform: Conformal Mapping Algorithms

The methods described in this document use conformal mapping to transform points in a circular sector, circle, ellipse, or nearly circular region to points in a circle or rectangle.

Background

A conformal mapping, is a transformation w=f(z) that preserves local angles. An analytic function is conformal at any point where it has a nonzero first derivative. A complex function is analytic on a region R if it is complex differentiable at every point in R. In a conformal mapping, in any small neighborhood the relative angle and shape are preserved.

Circular Sector to Rectangle

The algorithm uses a 2D conformal mapping to convert a circular sector, defined by four user points at the sector corners, to a rectangle of the user specified size. The circular sector has an inner radius <math>r_{min}</math>, an outer radius <math>r_{max}</math>, and extends over an angle <math>theta = \alpha* \pi</math> radians, with <math>0 \le \alpha \le 1</math>.

Let, <math>z_1</math> and <math>z_2</math> belong to <math>r_{max}</math>, and <math>z_3</math> and <math>z_4</math> belong to <math>r_{min}</math>. The mapping to the user specified rectangle is performed in the following steps (1-6):

The maximum radius can be facing to the top, bottom, right, or left. So the algorithm, first, determines which of 4 orientations is present. Calculating the center of the circular segment. The center is calculated as a point <nath>(x_c, y_c)</math> where two lines (z3, z2) and (z4, z1) cross. Therefore,

Calculating the angle of the sector in radians, see Figure 8. The angle is calculated as follows theta = theta2 - theta1.

For the maximum radius facing to the left, we must take into account the discontinuity of the angle at the negative x axis. The angle changes from <math>-\pi to \pi</math>, so the equation for <math>\theta</math> is now:

theta = abs(theta1) + abs(theta2) - 2.0 * Pi

The angle theta1 is along the line from center to z4 to z1 in - Pi to Pi radians:

tan(theta1) = ((y1 - y4)/(x1 - x4)) => theta1 = arctan((y1 - y4)/(x1 - x4))

the angle theta2 is along the line from center to z3 to z2 in - Pi to Pi radians:

tan(theta2)= ((y2 - y3)/(x2 - x3)) => theta2 = arctan((y2 - y3)/(x2 - x3))

See also:

For the time being, please refer to the MIPAV HTML help Algorithms/TransformConformalMapping.html