Difference between revisions of "Transform: Conformal Mapping Algorithms"

Revision as of 15:38, 17 August 2012

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):

1. The maximum radius can be facing to the top, bottom, right, or left. So the algorithm, first, determines which of 4 orientations is present.

2. Calculating the center of the circular segment. The center is calculated as a point <math>(x_c, y_c)</math> where two lines <math>(z_3, z_2)</math> and <math>(z_4, z_1)</math> cross. Therefore,

Equation 1

<math> x_c=\frac {x_3 * y_2) - (x_2 * y_3)} {(x-3 -x_2)} - \frac {(x_4 * y_1) - (x_1 * y_4)} {x_4 - x_1)}

y_c = x_c \frac {y_4 -y_1} {x_4 -x_1} + \frac {(x_4 * y_1) - (x_1 * y_4)}{(x-4 - x_1)} </math>

3. 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:

<math> \theta = abs(\theta1) + abs(\theta2) - 2.0 * \pi </math>

The angle <math>\theta1</math> is along the line from center to <math>z_4 to z_1</math> in <math> -\pi to \pi </math> radians:

<math> \tan(\theta_1) = \frac{((y_1 - y_4)}{(x_1 - x4_)} => \theta_1 = \arctan{(y_1 - y_4)}{(x_1 - x_4)}</math>

the angle <math>theta_2</math> is along the line from center to <math>z_3 to z_2</math> in <math> -\pi to \pi </math> radians:

<math>\tan(\theta_2)= \frac{(y_2 - y_3)}{(x_2 - x_3)} => \theta_2 = \arctan\frac{(y_2 - y3_)}{(x_2 - x_3)}

Transformation: Circle to Rectangle

Transformation: Ellipse to Circle

Transformation: Nearly Circular Region to Circle

Applying the algorithms

Circular Sector to Rectangle

Circle to Rectangle

Ellipse to Circle

Nearly Circle to Circle

References

See also:

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