DOCSLIB.ORG

Real-Time Visualization of Geometric Singularities Master’S Thesis by Stefan Kranich

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Real-Time Visualization of Geometric Singularities Master’S Thesis by Stefan Kranich TECHNISCHE UNIVERSITÄT MÜNCHEN Department of Mathematics Real-time Visualization of Geometric Singularities Master’s thesis by Stefan Kranich Supervisor: Prof. Dr. Dr. J¨urgen Richter-Gebert Advisor: Prof. Dr. Dr. J¨urgen Richter-Gebert Submission date: September 17, 2012 Abstract Dynamic geometry is a branch of geometry concerned with movement of elements in geometric constructions. If we move an element of a construction and update the positions of the other elements accordingly, we traverse a continuum of instances of the construction, its configuration space. In many geometric constructions, interesting effects occur in the neighbourhood of geometric singularities, i.e. around points in configuration space at which ambiguous geometric operations of the construction branch. For example, at a geometric singularity two points may exchange their role of being the upper/lower point of intersection of two circles. On the other hand, we frequently see ambiguous elements jump unexpectedly from one branch to another in the majority of dynamic geometry software. If we want to explain such effects, we must consider complex coordinates for the elements of geometric constructions. Consequently, what often hinders us to fully comprehend the dynamics of geometric constructions is that our view is limited to the real perspective. In particular, the two-dimensional real plane exposed in dynamic geometry software is really just a section of the multidimensional complex configuration space which determines the dynamic behaviour of a construction. The goal of this thesis is to contribute to a better understanding of dynamic geom- etry by making the complex configuration space of certain constructions accessible to visual exploration in real-time. In Section 1, we elaborate on our motivation. To that end, we describe effects observable in dynamic geometry software, a mathe- matical model of dynamic geometry, and complex tracing (a concept for avoiding jumping elements). Section 2 introduces mathematical prerequisites from the the- ory of Riemann surfaces. In Section 3, we present our method for visualizing the complex configuration space underlying certain geometric constructions and discuss its implementation using OpenGL. We make use of our visualization method in Section 4, where it is applied to three elementary geometric constructions. There, we analyze the intersection of a circle with a line, the intersection of two circles, and angle bisectors of a triangle, particularly w.r.t. the involved geometric singularities. The thesis concludes with an outlook on possible directions for further research. I herewith declare that I independently prepared the master’s thesis titled Real-time Visualization of Geometric Singularities using only the specified sources. Munich, September 17, 2012 Stefan Kranich ii Acknowledgements First and foremost, I want to express my deepest gratitude to my wonderful wife Sabine. Your constant support, encouragement, and love has been absolutely invaluable for me! While I was writing this thesis, we often had little time for each other. Also, I could not take enough care of our beloved children Eva and Quirin. I am truly sorry about that and am giving my best to make up for it again. Thanks to my mom Lore, for being a great mother and grandmother. I am greatly indebted to J¨urgen Richter-Gebert. Not only do I owe much of my knowledge of geometry to you. You have also introduced me to the topic of this thesis, which I find both delightful and demanding, and have been a committed supervisor and advisor. Despite a tightly packed schedule, you always took time for my questions and problems. Thank you very much! Thank you everyone at the Chair of Geometry and Visualization. You make work very enjoyable. Thanks for feedback and comments regarding my thesis go to Susanne Apel, Martin von Gagern, Johann Hartl, Tim Hoffmann, and Hermann Vogel. iii Contents 1 Motivation 1 1.1 Dynamic geometry software . 1 1.2 A mathematical model of dynamic geometry . 3 1.3 Complex detours and complex tracing . 5 2 Some mathematical prerequisites 7 2.1 Riemannsurfaces................................ 7 2.2 Analytic continuation . 11 2.3 Monodromy theorem . 13 3 Implementation of the visualization software 14 3.1 Domain colouring . 15 3.2 Camera model and transformations . 19 3.3 Usage of OpenGL and GLSL shaders . 25 3.4 The visualization algorithm . 27 4 Analysis of three elementary geometric constructions 32 4.1 Intersectionofacirclewithaline. 33 4.2 Intersectionoftwocircles . 35 4.3 Angle bisectors of a triangle . 42 5 Outlook 50 References 52 iv 1 Motivation Dynamic geometry is a branch of geometry concerned with movement of elements in geometric constructions. If we carry out a ruler-and-compass construction on a sheet of paper, the resulting image is static. However, in the construction, we can distinguish free and dependent elements. Free elements are those that do not depend on other elements, and can therefore, in principle, be moved around freely. The free elements could e.g. be two circles and the dependent elements their points of intersection. Imagine that we move a free element of a construction and continuously update the dependent elements accordingly. In the process, we traverse a continuum of instances of the construction, the so-called configuration space. In some instances, a geometric singularity occurs, which means that two or more branches of an ambiguous geometric operation used as a construction step touch each other. In other words, two or more elements resulting from the ambiguous geometric operation are indistinguishable. For example, two points of intersection can be indistinguishable w.r.t. their position (because they coincide) or w.r.t. their role of being the upper/lower point of intersection (because they have the same height). The goal of this thesis is to visualize what happens in the neighbourhood of geometric singularities. Using the algorithm presented in Section 3.4, the complex configuration space underlying certain geometric constructions can be explored visually in real-time. The visualization can help explain some effects arising in dynamic geometry. On the one hand, these effects can be observed while working with dynamic geometry software. On the other hand, they are inherent effects of the respective constructions themselves and can be described using a mathematical model of dynamic geometry. Therefore, the visualization can contribute to a better understanding of the dynamics of some constructions from both a practical and a theoretical perspective. 1.1 Dynamic geometry software In practice, geometric constructions are often carried out and analyzed using dynamic geometry software. Such programs allow us to interactively manipulate geometric constructions by letting us move free elements, e.g. a circle, and updating the positions of dependent elements, e.g. its intersections with another circle, appropriately. A priori, it is not clear, what “appropriately” really means in this context. Before we discuss it, please consider the following advice: It is highly recommended that the reader tries and experiments with the constructions presented in this thesis using several dynamic geometry systems. You will notice that in most cases they behave very differently, since their underlying models of dynamic geometry differ significantly. In particular, compare the behaviour of Cinderella 2 [12, http://www.cinderella.de] with that of Cabri II Plus (http://www.cabri.com), GeoGebra (http://www.geogebra.org), or The Geometer’s Sketchpad (http://www. dynamicgeometry.com). In what follows, we contrast Cinderella 2 and GeoGebra. For example, let us consider two intersecting circles, a configuration which we examine in more detail in Section 4.2. 1 direction of movement Figure 1: An experiment in dynamic geometry: We move a circle (blue) along a line through the midpoint of another circle of equal radius (black) and observe what happens to the intersections of the two circles (red and green points). Two singular situations occur – the first, when both circles overlap exactly, is removable; the second at the touching point, where both intersections collapse. Beyond that point, the intersections become complex (real projections of their positions are indicated using transparent points). Depending on the position of the circles relative to each other, there may be none, one, two, or infinitely many real points of intersection. Assume that we begin with an instance with two real intersections and two circles of equal radius. If we start to move one circle across the other along a line through their midpoints, we observe what follows (cf. Figure 1): The two points of intersection move continuously on the resting circle until the two midpoints (and thus also the two circles) coincide. In this situation, where both circles have infinitely many points in common, the position of the previous two intersections suddenly becomes ill-defined – we run into a geometric singularity. On the other hand, in accordance with their earlier behaviour, it seems reasonable to expect that they move continuously through the singularity. (More evidence to justify this assumption is presented in Section 4.2.) Hence we also obtain two distinct (imagined) positions for the situation in which the two circles overlap. These allow us to remove the discontinuity and let the intersections proceed continuously afterwards. In Cinderella 2, the construction behaves as we have foreseen. GeoGebra swaps the two points when we cross the singularity. Let us move the two circles further apart, such that there are no real intersections anymore and then bring them together again. In doing so, we are crossing the singularity at the touching point twice. In GeoGebra, both points reappear as the upper intersection. Cinderella 2 assigns exactly one of the points to each intersection. Actually, you may notice that in Cinderella 2 the two points exchange their role of being either the upper or lower intersection each time you carry out the movement. At first sight, this comes as a surprise, but it does make sense, as we will see later.
Load more

Recommended publications
  • GPU-Based Visualization of Domain-Coloured Algebraic Riemann Surfaces
    GPU-based visualization of domain-coloured algebraic Riemann surfaces Stefan Kranich∗ 11 November 2015 Abstract We examine an algorithm for the visualization of domain-coloured Riemann surfaces of plane algebraic curves. The approach faithfully reproduces the topology and the holomorphic structure of the Riemann surface. We discuss how the algorithm can be implemented efficiently in OpenGL with geometry shaders, and (less efficiently) even in WebGL with multiple render targets and floating point textures. While the generation of the surface takes noticeable time in both implementations, the visualization of a cached Riemann surface mesh is possible with interactive performance. This allows us to visually explore otherwise almost unimaginable mathematical objects. As examples, we look at the complex square root and the folium of Descartes. For the folium of Descartes, the visualization reveals features of the algebraic curve that are not obvious from its equation. 1 Introduction 1.1 Mathematical background The following basic example illustrates what we would like to visualize. Example 1.1. Let y be the square root of x, p y = x: If x is a non-negative real number, we typically define y as the non-negative real arXiv:1507.04571v3 [cs.GR] 10 Nov 2015 number whose square equals x, i.e. we always choose the non-negative solution of the equation y2 − x = 0 (1) p as y = x. For negative real numbers x, no real number y solves Equation 1. However, if we define the imaginary unit i as a number with the property that i2 = −1 then the square root of x becomes the purely imaginary number y = ipjxj: ∗Zentrum Mathematik (M10), Technische Universit¨atM¨unchen, 85747 Garching, Germany; E-mail address: [email protected] 1 Im Im Re Re Figure 1.2: When a complex number (black points) runs along a circle centred at the origin of the complex plane, its square roots (white points) move at half the angular velocity (left image).
    [Show full text]
  • Visualization of Complex Function Graphs in Augmented Reality
    M A G I S T E R A R B E I T Visualization of Complex Function Graphs in Augmented Reality ausgeführt am Institut für Softwaretechnik und Interaktive Systeme der Technischen Universität Wien unter der Anleitung von Univ.Ass. Mag. Dr. Hannes Kaufmann durch Robert Liebo Brahmsplatz 7/11 1040 Wien _________ ____________________________ Datum Unterschrift Abstract Understanding the properties of a function over complex numbers can be much more difficult than with a function over real numbers. This work provides one approach in the area of visualization and augmented reality to gain insight into these properties. The applied visualization techniques use the full palette of a 3D scene graph's basic elements, the complex function can be seen and understood through the location, the shape, the color and even the animation of a resulting visual object. The proper usage of these visual mappings provides an intuitive graphical representation of the function graph and reveals the important features of a specific function. Augmented reality (AR) combines the real world with virtual objects generated by a computer. Using multi user AR for mathematical visualization enables sophisticated educational solutions for studies dealing with complex functions. A software framework that has been implemented will be explained in detail, it is tailored to provide an optimal solution for complex function graph visualization, but shows as well an approach to visualize general data sets with more than 3 dimensions. The framework can be used in a variety of environments, a desktop setup and an immersive setup will be shown as examples. Finally some common tasks involving complex functions will be shown in connection with this framework as example usage possibilities.
    [Show full text]
  • Domain Coloring of Complex Functions
    Domain Coloring of Complex Functions Contents 1 Introduction Not only in mathematic we are confronted with graphs of all possible data sets. In economics and science we want to present the results by plotting their function graph inside a optimal coordinate system. In complex analysis we work with holomorphic functions which are complex differen- tiable. Like in real analysis we are interesting to plot such functions as graph inside a optimal coordinate systems. The graph of holomorphic functions is a subset of the four-dimensional coordinate system. But we are limited to three dimension, because we do not know how to draw objects in higher dimensional spaces. Therefore we are forced to construct a methode to visualize such functions. In this sec- tion we give a short overview of the domain coloring which allows us to visualize the graph of complex funtions by using colors. First we give a overview of complex analysis and holomorphic function. After we have presented the main idea behind the domain coloring we discuss a example to understand the main concept of domain coloring. 1 2 Introduction to Complex Analysis 2 Introduction to Complex Analysis In this section we give a short review of the main idea of complex analysis which we need to understand the methode of domain coloring. The reader should familiar with the construction of complex numbers and the representation of such numbers with polar coordinates. In complex anlysis we work with complex function which consist of three parts: first, a set D ⊂ C of input values, which is called the domain of the function, second, the range of f in C and third, for every input value x 2 D, a unique function value f(x) in the range of f.
    [Show full text]
  • Domain Coloring of Complex Functions
    Domain Coloring of Complex Functions Konstantin Poelke and Konrad Polthier 1 Introduction 2 What is a Function? Let us briefly recap the definition of a Visualizing functions is an omnipresent function to fix terminology. A function f task in many sciences and almost every day consists of three parts: first, a set D of in- we are confronted with diagrams in news- put values, which is called the domain of the papers and magazines showing functions of function, second, a set Y called the range of all possible flavours. Usually such func- f and third, for every input value x ∈ D, tions are visualized by plotting their func- a unique value y ∈ Y , called the function tion graph inside an appropriate coordinate value of f at x, denoted f(x). The set Γ(f) system, with the probably most prominent of all pairs (a, f(a)), a ∈ D, is a subset of choice being the cartesian coordinate sys- the product set D × Y and called the func- tem. This allows us to get an overall im- tion graph of f. pression of the function’s behaviour as well One particular type of functions that are as to detect certain distinctive features such widely used in engineering and physics are as minimal or maximal points or points complex functions, i.e. functions f : D ⊆ where the direction of curvature changes. C → Y ⊆ C whose domain and range In particular, we can “see” the dependence are subsets of the complex numbers, and between input and output. However, this we will focus on complex functions in the technique is limited to three dimensions, following.
    [Show full text]
  • Una Introducción Al Método De Dominio Colorado Con Geogebra Para
    101 Una introducción al método de dominio colorado con GeoGebra para la visualización y estudio de funciones complejas Uma introdução ao método do domínio colorado com GeoGebra para visualizar e estudar funções complexas An introduction the method domain coloring with GeoGebra for visualizing and studying complex JUAN CARLOS PONCE CAMPUZANO1 0000-0003-4402-1332 researchgate.net/profile/Juan_Ponce_Campuzano geogebra.org/u/jcponce http://dx.doi.org/10.23925/2237-9657.2020.v9i1p101-119 RESUMEN Existen diversos métodos para visualizar funciones complejas, tales como graficar por separado sus componentes reales e imaginarios, mapear o transformar una región, el método de superficies analíticas y el método de dominio coloreado. Este último es uno de los métodos más recientes y aprovecha ciertas características del color y su procesamiento digital. La idea básica es usar colores, luminosidad y sombras como dimensiones adicionales, y para visualizar números complejos se usa una función real que asocia a cada número complejo un color determinado. El plano complejo puede entonces visualizarse como una paleta de colores construida a partir del esquema HSV (del inglés Hue, Saturation, Value – Matiz, Saturación, Valor). Como resultado, el método de dominio coloreado permite visualizar ceros y polos de funciones, ramas de funciones multivaluadas, el comportamiento de singularidades aisladas, entre otras propiedades. Debido a las características de GeoGebra en cuanto a los colores dinámicos, es posible implementar en el software el método de dominio coloreado para visualizar y estudiar funciones complejas, lo cual se explica en detalle en el presente artículo. Palabras claves: funciones complejas, método de dominio coloreado, colores dinámicos. RESUMO Existem vários métodos para visualizar funções complexas, como plotar seus componentes reais e imaginários separadamente, mapear ou transformar uma região, o método de superfície analítica e o método de domínio colorido.
    [Show full text]
  • Arxiv:2002.05234V1 [Cs.GR] 12 Feb 2020 Have Emerged
    VISUALIZING MODULAR FORMS DAVID LOWRY-DUDA Abstract. We describe practical methods to visualize modular forms. We survey several current visualizations. We then introduce an approach that can take advantage of colormaps in python's matplotlib library and describe an implementation. 1. Introduction 1.1. Motivation. Graphs of real-valued functions are ubiquitous and com- monly serve as a source of mathematical insight. But graphs of complex functions are simultaneously less common and more challenging to make. The reason is that the graph of a complex function is naturally a surface in four dimensions, and there are not many intuitive embeddings available to capture this surface within a 2d plot. In this article, we examine different methods for visualizing plots of mod- ular forms on congruent subgroups of SL(2; Z). These forms are highly symmetric functions and we should expect their plots to capture many dis- tinctive, highly symmetric features. In addition, we wish to take advantage of the broader capabilities that exist in the python/SageMath data visualization ecosystem. There are a vast number of color choices and colormaps implemented in terms of python's matplotlib library [Hun07]. While many of these are purely aesthetic, some offer color choices friendly to color blind viewers. Further, some are designed with knowledge of color theory and human cognition to be perceptually uniform with respect to human vision. We describe this further in x3. 1.2. Broad Overview of Complex Function Plotting. Over the last 20 years, different approaches towards representing graphs of complex functions arXiv:2002.05234v1 [cs.GR] 12 Feb 2020 have emerged.
    [Show full text]
  • An Introduction to the Riemann Hypothesis
    An introduction to the Riemann hypothesis Author: Alexander Bielik [email protected] Supervisor: P¨arKurlberg SA104X { Degree Project in Engineering Physics Royal Institute of Technology (KTH) Department of Mathematics September 13, 2014 Abstract This paper exhibits the intertwinement between the prime numbers and the zeros of the Riemann zeta function, drawing upon existing literature by Davenport, Ahlfors, et al. We begin with the meromorphic continuation of the Riemann zeta function ζ and the gamma function Γ. We then derive a functional equation that relates these functions and formulate the Riemann hypothesis. We move on to the topic of finite-ordered functions and their Hadamard products. We show that the xi function ξ is of finite order, whence we obtain many useful properties. We then use these properties to find a zero-free region for ζ in the critical strip. We also determine the vertical distribution of the non-trivial zeros. We finally use Perron's formula to derive von Mangoldt's explicit formula, which is an approximation of the Cheby- shev function . Using this approximation, we prove the prime number theorem and conclude with an implication of the Riemann hypothesis. Contents Introduction 2 1 The statement of the Riemann hypothesis3 1.1 The Riemann zeta function ζ .........................................3 1.2 The gamma function Γ.............................................4 1.3 The functional equation............................................7 1.4 The critical strip................................................8 2 Zeros in the critical strip 10 2.1 Functions of finite order............................................ 10 2.2 The Hadamard product for functions of order 1............................... 11 2.3 Proving that ξ has order at most 1.....................................
    [Show full text]
  • Visualizing Complex Functions Using Gpus
    Visualizing Complex Functions Using GPUs Khaldoon Ghanem RWTH Aachen University German Research School for Simulation Sciences Wilhelm-Johnen-Straße 52428 Jülich E-mail: [email protected] Abstract: This document explains some common methods of visualizing complex functions and how to imple- ment them on the GPU. Using the fragment shader, we visualize complex functions in the complex plane with the domain coloring method. Then using the vertex shader, we visualize complex func- tions defined on a unit sphere like spherical harmonics. Finally, we redesign the marching tetrahedra algorithm to work on the GPGPU frameworks and use it for visualizing complex scaler fields in 3D space. 1 Introduction GPUs are becoming more and more attractive for solving computationally demanding problems. This is because they are cheaper than CPUs in two senses. First, they provide more performance for less cost .i.e. cheaper GFLOPs. Second, they are more energy efficient .i.e. cheaper running times. This reduced cost comes at the expense of less general purpose architecture and hence a different program- ming model. There are two main ways of programming GPUs. The first one is used in the graphics community using shading languages like HLSL, GLSL and Cg. In these languages, the programmer deals with vertices and fragments and processes them with the so called vertex and fragment shaders, respectively. Actu- ally, this has been the only way of programming GPUs for a while. Fortunately, in the recent years, frameworks for general programming have been developed like CUDA and OpenCL. They are much more suited for expressing the problem more abstractly in terms of threads.
    [Show full text]
  • Vizualiz´Acia Komplexn´Ych Funkci´I Pomocou Riemannov
    UNIVERZITA KOMENSKEHO´ V BRATISLAVE FAKULTA MATEMATIKY, FYZIKY A INFORMATIKY RNDr. Miroslava Val´ıkova´ Autorefer´atdizertaˇcnejpr´ace VIZUALIZACIA´ KOMPLEXNYCH´ FUNKCI´I POMOCOU RIEMANNOVYCH´ PLOCH^ na z´ıskanie akademick´ehotitulu philosophiae doctor v odbore doktorandsk´ehoˇst´udia 9.1.7 Geometria a topol´ogia Bratislava 2014 Dizertaˇcn´apr´acabola vypracovan´av dennej forme doktorandsk´ehoˇst´udiana Katedre algebry, geometrie a didaktiky matematiky Fakulty matematiky, fyziky a informatiky Univerzity Komensk´ehov Bratislave. Predkladateˇl: RNDr. Miroslava Val´ıkov´a Katedra algebry, geometrie a didaktiky matematiky FMFI UK, Mlynsk´adolina, 842 48 Bratislava Skoliteˇ ˇl: doc. RNDr. Pavel Chalmoviansk´y,PhD. Katedra algebry, geometrie a didaktiky matematiky FMFI UK, Mlynsk´adolina, 842 48 Bratislava Oponenti: ................................................................... ................................................................... ................................................................... ................................................................... ................................................................... ................................................................... Obhajoba dizertaˇcnejpr´acesa kon´a..................... o ............. h pred komisiou pre obhajobu dizertaˇcnejpr´acev odbore doktorandsk´ehoˇst´udiavy- menovanou predsedom odborovej komisie prof. RNDr. J. Korbaˇs,CSc. Geometria and topol´ogia{ 9.1.7 Geometria a topol´ogia na Fakulte matematiky, fyziky a informatiky Univerzity
    [Show full text]
  • Extending Riemann Mapping Capabilities for the Sage Mathematics Package
    Extending Riemann Mapping Capabilities for the Sage Mathematics Package Ethan Van Andel Calvin College 2011 1 !4#04'#5' This senior honor project focused on expanding and improving the riemann module that I developed as part of research project at Calvin College in the summer of 2009. riemann is part of the Sage software system and provides computation and visualization tools for Riemann mapping—an important component of complex analysis. The riemann module is the only publicly available implementation of Riemann mapping. Its primary use is educational and explorational, although it could be applied to scientific research as well. This project involved a great deal of mathematical and computer science research to develop and implement the techniques needed. I have attempted to provide a thorough explanation of the relevant mathematical background. Nevertheless, readers without a background in complex analysis may be lost at times. Generally, if the reader finds terminology that they want defined more thorougly, Wikipedia is an effective resource for filling that gap. In addition, since this project is highly interwoven with the 2009 research, some of the work described in this report was actually done two years ago. I have tried to keep the focus on the work for this project, discussing previous work only when it is relevant to an understanding of the whole. To avoid unnecessary complications, strict distinctions will not be made as to the exact place of every bit of work. As a rough guide, the work that was done for this project includes some general and usability improvements, substantial optimization, the exterior Riemann map, the Ahlfors spiderweb, and analytic error testing.
    [Show full text]
  • Domain Coloring on the Riemann Sphere María De Los Ángeles Sandoval-Romero Antonio Hernández-Garduño
    The Mathematica® Journal Domain Coloring on the Riemann Sphere María de los Ángeles Sandoval-Romero Antonio Hernández-Garduño Domain coloring is a technique for constructing a tractable visual object of the graph of a complex function. The package complexVisualize.m improves on existing domain coloring techniques by rendering a global picture on the Riemann sphere (the compactification of the complex plane). Additionally, the package allows dynamic visualization of families of Möbius transformations. In this article we discuss the implementation of the package and illustrate its usage with some examples. ■ 1. Introduction Domain coloring is a technique that uses a color spectrum to compensate for the missing dimension in the graph of a complex function. It was first introduced in [1], with a detailed description of the complex-plane case in [2]. A general definition is found in [3]. More precisely, consider a function f : U → V between two sets U, V (for example, two complex manifolds). Choose a “color" function, κ : V → HSB, where HSB denotes the Hue-Saturation-Brightness standard color space. Next, for any z ∈ U, compute f(z) and evaluate the resulting color κ ◦ f(z), assigning this color to the preimage z. The final colored domain has all the information (through the color) needed to say where the point z ∈ U gets mapped. Of course, the effectiveness of domain coloring depends on the choice of an adequate color scheme κ. Geometrically, the HSB color space is identified with an inverted solid cone. In cylin- drical coordinates, HSB is parametrized by (s b cos θ, s b sin θ, b), where θ ∈ [0, 2 π], s ∈ [0, 1], and b ∈ [0, 1].
    [Show full text]
  • Complex Representations of Functions
    6 Complex Representations of Functions “He is not a true man of science who does not bring some sympathy to his studies, and expect to learn something by behavior as well as by application. It is childish to rest in the discovery of mere coincidences, or of partial and extraneous laws. The study of geometry is a petty and idle exercise of the mind, if it is applied to no larger system than the starry one. Mathematics should be mixed not only with physics but with ethics; that is mixed mathematics. The fact which interests us most is the life of the naturalist. The purest science is still biographical.” Henry David Thoreau (1817-1862) 6.1 Complex Representations of Waves We have seen that we can seek the frequency content of a function f (t) defined on an interval [0, T] by looking for the Fourier coefficients in the Fourier series expansion ¥ a0 2pnt 2pnt f (t) = + ∑ an cos + bn sin . 2 n=1 T T The coefficients take forms like 1 Euler’s formula can be obtained using x 2 Z T 2pnt the Maclaurin expansion of e : an = f (t) cos dt. T T ¥ xn 1 xn 0 ex = = 1 + x + x2 + ··· + + ··· . ∑ n! 2! n! However, trigonometric functions can be written in a complex ex- n=0 ponential form. This is based on Euler’s formula (or, Euler’s identity):1 We formally set x = iq Then, ¥ n iq (iq) eiq = cos q + i sin q. e = ∑ n=0 n! (iq)2 (iq)3 (iq)4 The complex conjugate is found by replacing i with −i to obtain = 1 + iq + + + + ··· 2! 3! 4! −iq (q)2 (q)3 (q)4 e = cos q − i sin q.
    [Show full text]