Chapter 8 Complex Numbers & 3-D Plots

Transcription

1 EGR115 Introduction to Computing for Engineers Complex Numbers & 3-D Plots from: S.J. Chapman, MATLAB Programming for Engineers, 5 th Ed Cengage Learning Topics Introduction: Complex Numbers & 3-D Plots 8.1 Complex Data 8.2 Multi-Dimensional Arrays D Plots

2 Slide 1 of 8 Introduction: Complex Numbers & 3-D Plots Complex Numbers Complex numbers are numbers with both a real and an imaginary component (occur in many problems in science and engineering: for example, flight mechanics & control analysis). A complex number has the general form of: c = a + bi Complex Number Representations Complex numbers can be represented either in rectangular coordinate system or polar coordinate system: c = a + bi (rectangular coordinate) = z (polar coordinate)

3 Slide 2 of Complex Data MATLAB Complex Variables In MATLAB, complex variable is created automatically when a complex value is assigned to a variable name. The easiest way to create a complex value is to use the intrinsic values i or j, both of which are pre-defined to be. >> c1 = 4 + i*3 C1 = i >> c1 = 4 + 3i C1 = i Complex Numbers with Relational Operators It is possible to compare two complex numbers with == relational operator to see if they are equal to each other and compare them with ~= operator to see if they are not equal to each other. Comparisons with the >, <, >=, or <= operator are applicable to the real parts of the complex numbers ONLY. MATLAB Complex Functions Type conversion functions: convert data from the complex data type to the real data type. The MATLAB function real returns the real part of a complex number as a double precision (double) data type (discard the imaginary part). The MATLAB function imag returns imaginary part of a complex number as a double precision (double) data type. Absolute value and angle functions: convert a complex number to its polar representation. The MATLAB function abs(c) calculates the absolute value of a complex number using the equation: abs(c) = a 2 + b 2. The MATLAB function angle(c) calculates the angle of a complex number using the equation: angle(c) = atan2(imag(c), real(c)) producing an answer in the range of π θ π. Mathematical functions: include exponential functions, logarithms, trigonometric functions, and square roots. The MATLAB functions sin, cos, log, sqrt, and so forth will work with complex data (just like they work with real data).

4 Slide 3 of Complex Data EXAMPLE 8-1 (from EXAMPLE 4-4) Write a program to solve for the roots of a quadratic equation, regardless of type. 2 (NOTE) the solution of the quadratic equation of the form of: ax bx c 0 can be given by: 2 b b 4ac x 2a Rather than checking the discriminant, we can solve all roots, using the complex variables. MATLAB PSEUDOCODE Prompt the user for the coefficients a, b, and c. Read a, b, and c Discriminant <- b^2 4 * a *c x1 <- ( -b + sqrt(discriminant))/(2 * a) x2 <- ( -b - sqrt(discriminant))/(2 * a) Print The roots of this equation are: Print x1 =, real(x1), +i, imag(x1) Print x2 =, real(x2), +i, imag(x2) calc_roots2.m % calc_roots2.m disp ('This program solves for the roots of a quadratic '); disp ('equation of the form A*X^2 + B*X + C = 0. '); a = input ('Enter the coefficient A: '); b = input ('Enter the coefficient B: '); c = input ('Enter the coefficient C: '); % Calculate discriminant discriminant = b^2-4 * a * c; % Solve for the roots x1 = ( -b + sqrt(discriminant) ) / ( 2 * a ); x2 = ( -b - sqrt(discriminant) ) / ( 2 * a ); % Display results disp ('The roots of this equation are:'); fprintf ('x1 = (%f) +i (%f)\n', real(x1), imag(x1)); fprintf ('x2 = (%f) +i (%f)\n', real(x2), imag(x2)); >> calc_roots2 Command Window This program solves for the roots for a quadratic equation... Enter the coefficient A: 1 Enter the coefficient B: 2 Enter the coefficient C: 5 The roots of this equation are: x1 = ( ) + i ( ) x2 = ( ) + i ( )

5 Slide 4 of Complex Data Plot the function: EXAMPLE t cost i sint y t e Plotting Complex Data % plot_complex_a.m % The conventional plot function will % plot the real data part of the % function only (imaginary part % ignored) t = 0:pi/20:4*pi; y = exp(-0.2*t).*(cos(t)+i*sin(t)); plot(t,y, LineWidth,2); title( \bfplot of Complex Function ) xlabel( \bf\itt ); ylabel( \bf\ity(t) ); % plot_complex_b.m % Both real and imaginary parts can be plotted % on the same axes using hold on-off t = 0:pi/20:4*pi; y = exp(-0.2*t).*(cos(t)+i*sin(t)); plot(t,real(y), b-, LineWidth,2); hold on plot(t,imag(y), r--, LineWidth,2); title( \bfplot of Complex Function ) xlabel( \bf\itt ); ylabel( \bf\ity(t) ); legend ( real, imaginary ); hold off % plot_complex_c.m % The real part of the function is % plotted against imaginary part t = 0:pi/20:4*pi; y = exp(-0.2*t).*(cos(t)+i*sin(t)); plot(y, b-, LineWidth,2); title( \bfplot of Complex Function ) xlabel( \bfreal Part ); ylabel( \bfimaginary Part ); % plot_complex_d.m % The function is plotted as a polar plot % showing magnitude versus angle t = 0:pi/20:4*pi; y = exp(-0.2*t).*(cos(t)+i*sin(t)); polar(angle(y),abs(y)); title( \bfplot of Complex Function )

6 Slide 5 of Complex Data Do-It-Yourself (DIY) EXERCISE What is the value of result in the following statements? (a) x = 12 + i*5; y = 5 i*13; result = x > y; (b) x = 12 + i*5; y = 5 i*13; result = abs(x) > abs(y); (c) x = 12 + i*5; y = 5 i*13; result = real(x) - imag(y); 2. If array is a complex array, what does the function plot(array) do? MATLAB Functions for Handling Complex Numbers 1. exercise_8_1.m Command Window % exercise_8_1.m >> exercise_8_1 x = 12 + i*5; y = 5 - i*13; result_a = 0 result_a = x > y result_b = 0 result_b = abs(x) > abs(y) result_c = 25 result_c = real(x) - imag(y) 2. If array is a complex array, the MATLAB function plot(array) will plot only the real part of the array data (the imaginary part of the array data will be ignored).

7 Subarrays: Slide 6 of Multi-Dimensional Arrays EXAMPLE 8-3 MATLAB supports arrays with more than two dimensions (multi-dimensional arrays). Multi-dimensional arrays are natural extension of two-dimensional arrays and easily be created in MATLAB. >> a = [ ; ] a = >> b = ones(4,4,2) b(:,:,1) = b(:,:,2) = >> a(:,:,2) = [ ; ] >> a(:,:,3) = [ ; ] a(:,:,1) = a(:,:,2) = a(:,:,3) = >> c = randn(2,2,3) c(:,:,1) = c(:,:,2) = c(:,:,3) = MATLAB Multi-Dimensional Arrays MATLAB supports arrays with more than two dimensions (2-D arrays are called the matrix ). The multi-dimensional (more than 2-D) arrays are important for displaying the data intrinsically (if the data has more than 2-D in size) or for displaying multiple versions of 2-D data sets. Multi-dimensional arrays are a natural extension of two dimensional arrays (matrix). Each additional dimension is represented by one additional subscript used to address the data location within the multi-dimensional array data. It is very simple to create a multi-dimensional array in MATLAB. It can be created either by: (i) Assigning values directly in assignment statement, or (ii) By using the same functions that are used to create one- and two-dimensional arrays. Multi-Dimensional Arrays:

8 Slide 7 of D Plots EXAMPLE 8-4 MATLAB includes a rich variety of three-dimensional plots that can be useful for displaying certain types of data. Consider plotting the following functions: 0.2 t cos2 0.2 t sin2 x t e t y t e t 3-D Plot % line_plot_2d.m % The conventional plot function will % create a line plot on x-y plane, % but it will not show time-dependent % nature of the function t = 0:0.1:10; x = exp(-0.2*t).*cos(2*t); y = exp(-0.2*t).*sin(2*t); plot(x,y) title( \bftwo-dimensional Line Plot ); xlabel( \bfx ); ylabel( \bfy ); grid on; % line_plot_3d.m % The plot3 function will create a 3-D line % plot, clearly showing time-dependent nature % of the function t = 0:0.1:10; x = exp(-0.2*t).*cos(2*t); y = exp(-0.2*t).*sin(2*t); plot3(x,y,t) title( \bfthree-dimensional Line Plot ); xlabel( \bfx ); ylabel( \bfy ); zlabel( \bftime ); grid on;

9 Slide 8 of D Plots EXAMPLE 8-5 Consider a simple object, like a sphere. A sphere can be defined as the locus of all points that are a given distance r from the center, regardless of azimuth angle and elevation angle f. The equation is: r = a (where, a is any positive number). In Cartesian space, the points on the surface of the sphere are defined by the following equations: x rcosfcos y rcosfsin z rsinf, where the radius r is constant, the elevation angle f varies from π 2 to π 2, and the azimuth angle varies from to. sphere.m % Script file: sphere.m % Purpose: % This program plots the sphere using the surf function. % Define variables: % n -- Number of points in az and el to plot % r -- Radius of sphere % phi -- meshgrid list of elevation values % Phi -- Array of elevation values to plot % theta -- meshgrid list of azimuth values % Theta -- Array of azimuth values to plot % x -- Array of x point to plot % y -- Array of y point to plot % z -- Array of z point to plot % Define the number of angles on the sphere to plot % points at n = 20; % Calculate the points on the surface of the sphere r = 1; theta = linspace(-pi,pi,n); phi = linspace(-pi/2,pi/2,n); [theta,phi] = meshgrid(theta,phi); % Convert to (x,y,z) values x = r * cos(phi).* cos(theta); y = r * cos(phi).* sin(theta); z = r * sin(phi); % Plot the sphere figure(1) surf (x,y,z); title ('\bfsphere');