Plotting 1.Overview 2.plot in 2D 3.Plot in 3D 4.Other possible charts 5.Engineers: label your plots! 6.Plots & Polynomial

1. Plots & Charts, overview Plotting functions plot(), plot3(), polar(), meshgrid() Charting functions pie(), pie3(), bar(), bar3(), bar3h(), hist(), errorbar() Plot-related functions polyfit(), polyval(), text(), title(), xlabel(), ylabel() 2

2. Plots Create a graph of y vs. x plot(x, y) %order of arguments matters Example: 100 data-points x = linspace(-pi, pi); y = sin(x); plot(x, y) By default, plot() connects the data-points with a solid line without markers. 3 3

2. Plots, cont. Let us try with less data-points: x = linspace(-pi, pi, 10); y = sin(x); plot(x, y); Notice that the curve is less smooth This is another reason why linspace() is friendly to use: easily fixable. 4

2. Plots: line specifiers A third argument can be added in the plot function-call: plot(x,y, _____) The third argument must be a string, made of up to three components: – One component for the line’s color – One component for the line-style – And one component for the marker symbol (the symbol at each data point 5 5

2. Plots: lineSpecs - color Specify the color only: 6 plot(x, y, ‘r’); 6

2. Plots: lineSpecs – line style Color and line-style: 7 plot(x, y, ‘r:’); 7 (none): no line

2. Plots: lineSpecs - marker Color, type of line and data-marker: 8 plot(x, y, ‘r:d’);

2. Plots: line specifier, cont. If only the data-points must show, leave out the line-style: plot(x, y, 'rd') Forgot all the options? >> doc plot 9 9

2. Plots: multiple plots Repeat the series of 3 arguments to combine multiple plots on 1 graph. Example: X = linspace(-2*pi,2*pi,50); Ysine = sin(X); Ycosine = cos(X); plot(X,Ysine,'r:o',X,Ycosine,'b--d') The string argument is unnecessary. MATLAB will rotate through the default colors to make sure each plot has a different color. The other 4 arguments are MANDATORY. 10

2. Plots: hold on/off At default mode, the plot() command erases the previous plots before drawing new plots. If subsequent plots are meant to add to the existing graph, use: hold on, which holds the current plot. When you are done, use hold off to return to default mode. hold, by itself, toggles the hold modes. 11

>> hold on >> plot(x1,sin(2*x1),'g’) 2. Plots: hold on/off, cont. 12 >> x1 = linspace(0,pi,25); >> plot(x1,sin(x1),'r') * Range of the axis will be automatically adjusted.

Example: Airfoil Lift Vortex Panel Method + Boundary Layer analysis – A numerical Computational fluid dynamics – Models the lifting surface of a wing as an infinitely thin sheet of discrete vortices – Used to compute lift and induced drag & predict separation/stall 13

3. Plots: 3 dimensions plot3() makes a 3D plot – requires x, y, and z data. 14

Meshgrid() 15 Given: x = [1 2 3]; y = [4 5]; Evaluate: f(x,y) = x + y In other words, I need to evaluate (x+y) at every pair of points between x and y.

Meshgrid(), cont. 16 To make a useful plot, it is necessary to match up each x with each y before computing z. The function meshgrid() makes this easy to do.

3. Plots: 3 dimensions, cont. x = linspace(-pi, pi); y = linspace(-pi, pi); [X, Y] = meshgrid(x, y); Z = sin(X).^3 - cos(Y).^2; plot3(X, Y, Z) Creates two arrays X, Y where each value of x is matched with each value of y 17

4. Other Possible Charts polar() creates polar coordinate plots: x = linspace(-pi, pi); y = cos(x) - sin(x).^2; polar(x, y) 18

4. Other Possible Charts, cont. pie(), pie3(), bar(), bar3(), bar3h(), hist(), errorbar() 19 pie3() 19 Much like Excel offers: >> x = >> pie3(x)

4. Other Possible Charts, cont. pie(), pie3(), bar(), bar3(), bar3h(), hist(), errorbar() 20 Much like Excel offers: bar3h() x = >> bar3h(x)

4. Other Possible Charts, cont. pie(), pie3(), bar(), bar3(), bar3h(), hist(), errorbar() 21 errorbar()

Example: Flight Delays 1.Percentage of flights delayed 1.Percentage of those flights delayed due to carrier’s fault 2.Percentage of those flights delayed due to weather 3.Percentage of those flights delayed due to NAS 4.Percentage of those flights delayed due to security 5.Percentage of those flights delayed due to late flights 2.Percentage of flights arrived on time 3.Present result in table 4.If time, present results in pie chart 22

Example: Plot results Plot the results for 2010 and

4. Other Possible Charts, cont. As with all the MATLAB possibilities, use… 24 F1 = Help 24

5. Engineers: Complete Plots The following built-in functions should be applied to ALL graphs created: title() %title on figure xlabel() %x-axis label ylabel() %y-axis label zlabel() %z-axis label Each function takes 1 ‘string’ argument only and has no return-value. 25

5. Engineers: Complete Plots Additional built-in function: text() Example: text(1, -2, -2, 'Cool plot!') 26

5. Engineers: Complete Plots Built-in function: grid on COMMAND line typed in the script, after a plot command. It stands alone on one line, requires no arguments, and returns no value. 27

5. Engineers: Complete Plots Additional built-in functions: legend(), xlim, ylim, axis 28

Plots All plots MUST have a title and axis labels! All plots with more than 1 line should have a legend with appropriate labeling. Grids should be appropriate for the problem Plot limits should be appropriate for the problem – Is there a “0” value? As with using fprintf and input, a happy grader is a generous grader – make your plots easy to read/understand and your grader will be happier. If you see the phrase “suitable for technical presentation,” assume that means title, axis labels, grid (where appropriate), legend (where appropriate), and appropriate axis limits.

6. Plotting & Polynomials Plot-related functions that try to find an equation that links data-points: polyfit(), polyval() polyfit() is like linear regression which finds the curve that best fits the data-points. polyfit() attempts to fit a polynomial – not a line. It mainly finds the coefficients of the polynomial that best fits the data given. polyval() is used to evaluate the polynomial at specified points. It makes use of the coefficients generated by polyfit(). – It is frequently used to generate a plot. 30

6. Plotting & Polynomials, cont. clear clc %generate tables of x, and y data = [1, 50; 4, 4900; 7, 4600; 10, 3800; 70, 1300; 100, 850; 300, 0.2e9; 700, 1.2e9; 1000, 1.2e9]; x = data(:, 1)'; y = data(:, 2)'; %plot data points, omit line plot(x, y, 'd') hold on %combine future plots on this one %find best-fit polyn. of order 3 coeff = polyfit(x, y, 3) px = linspace(min(x), max(x), 100); py = polyval(coeff, px); plot(px, py) 31 Remember that fitting a curve does NOT mean hitting every data-point! 31

Example: Beam Deflection Cantilever Beam Deflection – Determine the equation of the polynomial from experimental data. 32

Key Ideas Plotting any kind of graphs mostly requires vectors of identical dimensions: (x,y) (x, y, z) (r, theta, z)... hold on allows multiple plots to be combined together. The independent variable is the first argument. IT IS A COMMON MISTAKE TO SWAP THEM. All functions are easily explained in the help, usually with examples that show how to place arguments. As engineer, remember all graphs must be labeled correctly. For mathematical analysis, polyfit() and polyval() allow to fit a curve through data-points. 33