About

Conformal mapping

Simulation window

In the z-plane (left) there is a quadratic point array of predefined width and position. Its points are differentiated in color according to their imaginary value. The lower left edge point is accentuated in red;  it is connected to the origin by a green vector arrow. The array can be shifted in the plane, drawing the red point with the mouse. Its coordinates can be varied independently by two sliders x, y. Exact values can be written into the x,y number fields; they may be beyond the slider range. The array width can be changed by another slider; it can be contracted to a point.

In addition there is a circular color coded point array around the origin. Its center is highlighted in magenta. The two symmetric points at the real axis are accentuated in red and yellow. The circular array can be shifted with the mouse by drawing at its center. The diameter of the circular array can be varied by a radius slider and can also be contracted to a point.

In both planes there are black circles with radii characteristic for the special function (1, e , π). In the z-plane red lines define the limits of periodic regions or of mapping strips

The arrays of the z-plane are mapped into the w-plane by the complex function. The color coding is conserved in this process to identify specific rows of points. When the arrays are contracted to a point, one sees the mapping of this single point. Exact coordinates are shown when a point is marked with the mouse.

The play button starts an animation that shifts the quadratic array by a raster unit per second along a line that is interesting for the special function (real axis, imaginary axis, unit circle). When the array reaches the end of the scale, it jumps to the opposite limit. One can shift the array with the mouse, sliders or coordinate number fields while the animation is running. This way the plane can be quickly rastered. The pause button stops the animation.

The color coding is most distinct when the window is blown up to full screen size.

Complex functions

Let z = x + iy = e^(iθ) = cos(θ) +  i sin(θ),

where x,y are real numbers and i^2 = -1

The graph on the left shows z as plotted in the complex plane.

The graph on the right shows a complex number w, which is mapped from z using one of the functions provided.

The available functions provided in the Combobox, along with their equations, are as follows:

- Exponential:  w = e^(nz), where n is a real number

- Sine:  w = sin(z)

- Cosine: w = cos(z)

- Tangent:  w = tan(z)

- Logarithmic:  w = ln(z)

- Power:  w = z^n, where n is a real number

this file was created by Dieter Roess in July 2008

This simulation is part of

“Learning and Teaching Mathematics using Simulations

– Plus 2000 Examples from Physics”

ISBN 978-3-11-025005-3, Walter de Gruyter GmbH & Co. KG

 

Translations

Code	Language		Translator	Run

Credits

Dieter Roess - WEH- Foundation; Tan Wei Chiong; Loo Kang Wee

Executive Summary:

This document reviews a JavaScript simulation applet designed for learning and teaching mathematics, specifically focusing on conformal mapping of complex functions. The applet visualizes how various complex functions transform points and arrays in the complex plane (z-plane) to another complex plane (w-plane). It offers interactive tools to manipulate the input in the z-plane and observe the corresponding transformations in the w-plane, aiding in the understanding of the behavior and properties of complex functions. The simulation is part of a larger collection aimed at enhancing learning through interactive simulations.

Main Themes and Important Ideas/Facts:

1. Visualizing Complex Functions: The core purpose of the applet is to provide a visual representation of complex functions, which is challenging due to the two-dimensional nature of complex numbers. As stated, "this is not possible with complex numbers, which are represented with a 2-dimensional plane. Thus, we map a complex number from one Argand plane to another Argand plane. By seeing how each function transforms a square array of points (red to blue) and the unit circle (black), we can reveal some very interesting properties of how complex numbers behave under a function."
2. Interactive Exploration: The applet emphasizes interactive learning. Users can manipulate objects in the z-plane (input) and observe the real-time transformation in the w-plane (output). Key interactive features include:

• Quadratic Point Array: A grid of colored points that can be shifted using the mouse or adjusted via sliders (x, y coordinates and array width).
• Circular Point Array: A color-coded circle around the origin that can be shifted and resized using the mouse and a radius slider.
• Number Fields: Allow for precise input of x, y coordinates, potentially beyond the slider range.
• Animation: A play button initiates an animation that automatically shifts the quadratic array along specified lines (real axis, imaginary axis, unit circle) to quickly raster the plane. The animation can be paused.
• Point Mapping: Contracting the arrays to a single point allows observation of the mapping of individual complex numbers. Marking a point with the mouse displays its exact coordinates.

1. Available Complex Functions: The applet provides a selection of fundamental complex functions that can be chosen from a Combobox:

• Exponential: w = e^(nz), where n is a real number
• Sine: w = sin(z)
• Cosine: w = cos(z)
• Tangent: w = tan(z)
• Logarithmic: w = ln(z)
• Power: w = z^n, where n is a real number

1. Visual Cues for Understanding Transformations: The simulation uses several visual aids to help users understand the effects of the complex functions:

• Color Coding: Points in the z-plane are colored according to their imaginary value, and this color coding is conserved in the w-plane, allowing for easy identification of corresponding points and rows.
• Special Features in the z-plane: A red accentuated point connected to the origin by a green vector, a magenta highlighted center of the circular array with red and yellow points on the real axis, and black circles with radii 1, e, π.
• Mapping Strips: Red lines in the z-plane indicate the boundaries of periodic regions or mapping strips, which are crucial for understanding the behavior of functions like exponential and trigonometric functions in the complex plane.

1. Educational Purpose: The applet is designed as an open educational resource for "Learning and Teaching Mathematics using Simulations." It aims to help students visualize abstract concepts related to complex numbers and functions. The "For Teachers" section explicitly states the difficulty of visualizing complex functions using traditional 2D graphs and highlights the value of mapping between two Argand planes.
2. Technical Details: The simulation is a "JavaScript Simulation Applet HTML5," indicating its web-based and interactive nature. It was created by Dieter Roess in July 2008 and is part of a larger work titled "Learning and Teaching Mathematics using Simulations – Plus 2000 Examples from Physics."
3. Embedding Capability: The applet can be easily embedded into webpages using the provided <iframe> code, making it readily accessible for educational purposes.
4. Related Resources: The webpage also includes links to other interactive simulations developed under the Open Educational Resources / Open Source Physics @ Singapore project, showcasing a wide range of topics in physics and mathematics.

Key Quotes:

• "By seeing how each function transforms a square array of points (red to blue) and the unit circle (black), we can reveal some very interesting properties of how complex numbers behave under a function."
• "In the z-plane (left) there is a quadratic point array of predefined width and position. Its points are differentiated in color according to their imaginary value."
• "In addition there is a circular color coded point array around the origin. Its center is highlighted in magenta."
• "The arrays of the z-plane are mapped into the w-plane by the complex function. The color coding is conserved in this process to identify specific rows of points."
• "The available functions provided in the Combobox, along with their equations, are as follow s: - Exponential: w = e^(nz), where n is a real number - Sine: w = sin(z) - Cosine: w = cos(z) - Tangent: w = tan(z) - Logarithmic: w = ln(z) - Power: w = z^n, where n is a real number"
• "However, this is not possible with complex numbers, which are represented with a 2-dimensional plane. Thus, we map a complex number from one Argand plane to another Argand plane."

Conclusion:

The Conformal Mapping JavaScript Simulation Applet is a valuable tool for educators and students seeking to understand complex functions visually and interactively. Its features allow for dynamic exploration of how different transformations affect points and regions in the complex plane, enhancing intuition and comprehension of abstract mathematical concepts. The availability as an embeddable HTML5 applet further increases its accessibility and utility in online learning environments.

 

Conformal Mapping Simulation Study Guide

Key Concepts

• Complex Numbers: Numbers of the form z = x + iy, where x and y are real numbers and i is the imaginary unit (i² = -1). They can also be represented in polar form as z = r e^(iθ) = r(cos(θ) + i sin(θ)).
• Complex Plane (z-plane): A two-dimensional plane where the horizontal axis represents the real part of a complex number and the vertical axis represents the imaginary part.
• Mapping: A transformation that takes points from one set (in this case, the z-plane) to another set (the w-plane) according to a specific rule, often defined by a function.
• Conformal Mapping: A type of transformation that preserves the angles between intersecting curves at each point. While not explicitly detailed in the provided text, the title suggests the applet demonstrates such mappings.
• Complex Function: A function whose input and/or output are complex numbers. In this context, a complex function w = f(z) maps points from the z-plane to the w-plane.
• z-plane Simulation Window: The left side of the applet interface where a quadratic array and a circular array of points are displayed in the complex plane.
• Quadratic Point Array: A grid of points in the z-plane, color-coded based on their imaginary values, with adjustable width, position, and the ability to be contracted to a single point.
• Circular Point Array: A set of points arranged in a circle around the origin in the z-plane, color-coded, with adjustable center position (via mouse drag) and radius (via slider), and the ability to be contracted to a single point.
• w-plane: The right side of the applet interface where the mapped images of the points and arrays from the z-plane are displayed under the transformation of a selected complex function.
• Color Coding: A visual technique used in the applet to track the transformation of specific rows of points from the z-plane to the w-plane.
• Sliders and Number Fields: Interactive elements in the applet that allow users to adjust parameters such as the position and width/radius of the point arrays.
• Play/Pause Buttons: Controls for an animation that automatically shifts the quadratic array along predefined paths (real axis, imaginary axis, unit circle) in the z-plane.
• Combobox: A drop-down menu in the applet that allows users to select from a list of predefined complex functions for mapping.
• Predefined Complex Functions: The list of available functions in the applet, including Exponential (w = e^(nz) or w = e^z), Sine (w = sin(z)), Cosine (w = cos(z)), Tangent (w = tan(z)), Logarithmic (w = ln(z)), and Power (w = z^n).
• Mapping Strips/Periodic Regions: Areas in the z-plane, bounded by red lines, that might correspond to repeating patterns or fundamental domains under certain complex mappings.
• Special Function Radii: Black circles in both the z-plane and w-plane with radii of 1, e, π, likely serving as visual references.

Quiz

1. Describe the purpose of the quadratic and circular point arrays in the z-plane of the conformal mapping applet. How can these arrays be manipulated by the user?
2. Explain how the color coding of the points in the z-plane helps in understanding the mapping to the w-plane. What happens to the color of a specific row of points during the transformation?
3. What are the controls available in the applet that allow a user to change the position and size of the point arrays in the z-plane? Provide at least three examples.
4. Describe the functionality of the "play" button in the applet. What are the predefined paths along which the quadratic array can be animated?
5. What happens in the w-plane when the user selects a complex function from the combobox? How is the relationship between the z-plane and the w-plane visualized?
6. List at least three of the complex functions available in the combobox of the applet and provide their corresponding equations as given in the source.
7. Why is it stated that plotting a graph of y against x is not possible for visualizing functions of complex numbers? What alternative method is used, as demonstrated by the applet?
8. What is the significance of the black circles with radii 1, e, π present in both the z-plane and the w-plane?
9. Explain what happens in the w-plane when a point array in the z-plane is contracted to a single point using the respective sliders.
10. What visual cues in the z-plane might indicate the presence of periodic regions or mapping strips related to the chosen complex function?

Answer Key

1. The quadratic and circular point arrays in the z-plane serve as visual tools to observe how different regions and specific points in the complex plane are transformed by a selected complex function. Users can manipulate these arrays by dragging them with the mouse, using sliders to adjust their position and size (width or radius), and by entering exact coordinates in number fields.
2. The color coding of points, based on their imaginary value in the z-plane, allows users to track how specific rows of points are deformed and relocated in the w-plane after the mapping. The color coding is conserved during the transformation, making it easier to identify the image of a particular set of points.
3. Users can change the position and size of the point arrays using several controls. For the quadratic array, there are x and y sliders and corresponding number fields to adjust its position, and an "array width" slider to change its size. For the circular array, the center can be shifted by dragging with the mouse, and its diameter is controlled by a "radius slider."
4. The "play" button starts an animation that automatically shifts the quadratic array by a raster unit per second along a line. The predefined paths for this animation are the real axis, the imaginary axis, and the unit circle in the z-plane.
5. When a complex function is selected from the combobox, the applet applies this function to each point in the z-plane (both the quadratic and circular arrays) and plots the resulting complex numbers in the w-plane. This visualizes how the chosen function transforms the shapes and positions of the original point arrays.
6. Three of the available complex functions are: Exponential (w = e^(nz) or w = e^z), Sine (w = sin(z)), and Logarithmic (w = ln(z)). The variable n in the exponential and power functions represents a real number that can be adjusted.
7. Visualizing functions of complex numbers as a graph of y against x is not possible because both the input (z) and the output (w) are two-dimensional (having a real and an imaginary part), requiring four dimensions for a complete graph. The applet uses an alternative method by mapping points and shapes from one 2D complex plane (z-plane) to another 2D complex plane (w-plane).
8. The black circles with radii 1, e, π likely serve as reference scales in both complex planes, allowing users to observe how distances and magnitudes are affected by the chosen conformal mapping. They provide a visual context related to important mathematical constants.
9. When a point array in the z-plane is contracted to a single point, the applet shows the mapping of that single complex number under the selected function. This allows users to observe the transformation of individual points between the two planes.
10. The red lines in the z-plane define the limits of periodic regions or mapping strips. These lines visually segment the z-plane into areas that might be mapped in a repeating or characteristic way by the selected complex function.

Essay Format Questions

1. Discuss the advantages of using a visual simulation, like the conformal mapping applet, for understanding the behavior of complex functions compared to purely analytical methods. How does the applet facilitate intuition about these transformations?
2. Explain how the interactive features of the conformal mapping applet, such as the movable point arrays, sliders, and animation, contribute to a deeper understanding of the concept of mapping in the complex plane. Provide specific examples from the applet's functionality.
3. Consider the set of predefined complex functions available in the applet. How might exploring the transformations generated by different functions (e.g., exponential, sine, logarithmic) reveal key properties of these functions in the context of complex numbers?
4. The applet displays mappings between the z-plane and the w-plane. Discuss the significance of observing how geometric shapes (like the quadratic grid and the circle) are distorted or transformed by various complex functions. What information can be gained from these visual deformations?
5. The description mentions that the applet can reveal "very interesting properties of how complex numbers behave under a function." Based on the functionality described, elaborate on the types of behaviors or properties that users might be able to observe and learn about through this simulation.

Glossary of Key Terms

• Argand Plane: Another name for the complex plane, used to geometrically represent complex numbers.
• Conformal: Preserving angles locally between intersecting curves.
• Domain (of a function): The set of all possible input values for a function (in this case, the z-plane).
• Image (of a point or set): The point or set in the w-plane that results from applying a complex function to a point or set in the z-plane.
• Imaginary Unit (i): Defined as the square root of -1 (i² = -1).
• Magnitude (of a complex number): The distance of a complex number from the origin in the complex plane, denoted as |z| or r. For z = x + iy, |z| = √(x² + y²).
• Polar Form (of a complex number): Representing a complex number by its magnitude (r) and the angle (θ) it makes with the positive real axis: z = r(cos(θ) + i sin(θ)) = r e^(iθ).
• Range (of a function): The set of all possible output values of a function (in this case, the w-plane).
• Real Axis: The horizontal axis in the complex plane, representing the real part of complex numbers.
• Imaginary Axis: The vertical axis in the complex plane, representing the imaginary part of complex numbers.
• Unit Circle: A circle in the complex plane centered at the origin with a radius of 1.

Sample Learning Goals

[text]

For Teachers

In the real numbers, we can express a relationship between two sets of numbers by mapping elements from one set to elements in another. We call this mapping a function.

To visualize this, we can plot out a graph of y against x, where y = f(x).

However, this is not possible with complex numbers, which are represented with a 2-dimensional plane. Thus, we map a complex number from one Argand plane to another Argand plane. By seeing how each function transforms a square array of points (red to blue) and the unit circle (black), we can reveal some very interesting properties of how complex numbers behave under a function.

The available functions are as follows:
- Exponential (w = e^z)
- Sine (w = sin(z))
- Cosine (w = cos(z))
- Tangent (w = tan(z))
- Logarithmic (w = ln(z))
- Power (w = z^n)

Research

[text]

Video

[text]

 Version:

1. http://weelookang.blogspot.sg/2016/02/vector-addition-b-c-model-with.html improved version with joseph chua's inputs
2. http://weelookang.blogspot.sg/2014/10/vector-addition-model.html original simulation by lookang

Other Resources

[text]

Frequently Asked Questions: Conformal Mapping Simulation

What is conformal mapping and why is it useful?

Conformal mapping is a transformation that preserves angles locally between curves. In the context of complex numbers, it's a way to map points from one complex plane (the z-plane) to another (the w-plane) using a complex function, while maintaining the angles at which curves intersect. This is particularly useful for visualizing how complex functions transform geometric shapes and understanding their properties, which has applications in various fields like fluid dynamics, electromagnetism, and heat transfer.

How does the JavaScript simulation applet demonstrate conformal mapping?

The applet provides a visual representation of conformal mapping by showing two complex planes side-by-side. In the z-plane (left), users can manipulate a quadratic array of colored points and a circular array around the origin. These arrays are then transformed by a selected complex function, and the resulting mapped points are displayed in the w-plane (right). The color-coding of the points is preserved during the mapping, allowing users to track how different regions of the z-plane are transformed.

What complex functions are available to explore in the simulation?

The simulation offers several built-in complex functions for mapping: Exponential (w = e^(nz)), Sine (w = sin(z)), Cosine (w = cos(z)), Tangent (w = tan(z)), Logarithmic (w = ln(z)), and Power (w = z^n). Users can select these functions from a combobox to observe how each one transforms the point arrays in the z-plane to the w-plane. The parameters 'n' in the Exponential and Power functions can be adjusted.

How can users interact with the simulation to understand conformal mapping better?

Users have multiple ways to interact with the simulation. They can shift the quadratic and circular point arrays in the z-plane using the mouse, adjust their position and width/radius using sliders and numerical input fields, and even contract the arrays to a single point to see the mapping of that specific point. An animation feature allows the quadratic array to automatically shift along predefined paths (real axis, imaginary axis, unit circle), effectively rastering the plane.

What visual cues does the simulation provide to aid understanding?

The simulation employs several visual cues to help users grasp the concepts of conformal mapping. The color-coding of the point arrays is crucial for identifying corresponding points and understanding how lines and regions are transformed. Black circles with radii 1, e, and π are present in both planes as reference points related to special functions. In the z-plane, red lines indicate boundaries of periodic regions or mapping strips, providing context for the transformations.

Why is it helpful to view the simulation in full screen?

The color coding of the transformed points is most distinct when the simulation window is blown up to full screen size. This enhanced visual clarity makes it easier to observe the details of the mapping and identify patterns in how the complex functions transform the point arrays.

How does this simulation relate to the teaching and learning of mathematics, particularly complex numbers?

The simulation provides an interactive and visual tool for learning about complex numbers and functions. Unlike real-valued functions that can be visualized with 2D graphs, complex functions require mapping between two 2D complex planes. By visualizing how basic geometric shapes (arrays of points, circles) are transformed by various complex functions, students can gain a more intuitive understanding of the behavior and properties of these functions. The simulation is part of a larger collection aimed at enhancing mathematics and physics education through interactive simulations.

Who created this simulation and where can I find more resources or information?

This JavaScript simulation applet was created by Dieter Roess in July 2008. It is part of the "Learning and Teaching Mathematics using Simulations – Plus 2000 Examples from Physics" project. The simulation is hosted by Open Educational Resources / Open Source Physics @ Singapore. The page provides information about the authors, translations, credits, and links to related resources and different versions of the simulation.