Computational Methods (PHYS 2030)

Transcription

1 Computational Methods (PHYS 2030) Instructors: Prof. Christopher Bergevin Schedule: Lecture: MWF 11:30-12:30 (CLH M) Website: York University Winter 2018 Lecture 10

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3 Ex. Classic curves EXclassicCurves.m Ø There are many useful classic curves from mathematics that are relevant in physics, engineering, and biology contexts y x Devries (1994)

4 Ex. Classic curves y x Gray s Anatomy

5 Ex. Classic curves Ø Not to mention the dreaded uzumaki...

6 Ex. Classic curves Devries (1994)

7 EXclassicCurves.m % ### EXclassicCurves.m ### clear % three-leaved rose if 1==1 a= 1; N= 200; theta= linspace(0,2*pi,n); % polar coord. r= a*sin(3*theta); x= r.*cos(theta); y= r.*sin(theta); % convert to Cartesian % spiral of Galileo if 1==0 a= 1; N= 2000; theta= linspace(0,10*pi,n); % polar coord. r= a*theta.^2; x= r.*cos(theta); y= r.*sin(theta); % convert to Cartesian % Witch of Agnesi if 1==0 a= 1; N= 2000; x= linspace(-20,20,n); % polar coord. y= (8*a^3)./(x.^2+4*a^2); % Involute of a circle if 1==0 a= 1; N= 200; theta= linspace(0,4*pi,n); % polar coord. x= a*cos(theta)+a*theta.*sin(theta); y= a*sin(theta)- a*theta.*cos(theta); % Cissoid of Diocles if 1==0 a= 1; N= 100; theta= linspace(0,2*pi,n); % polar coord. r= a*sin(theta).*tan(theta); x= r.*cos(theta); y= r.*sin(theta); % convert to Cartesian % Cochleoid if 1==0 a= 1; N= 2000; theta= linspace(pi,16*pi,n); % polar coord. r= a*sin(theta)./theta; x= r.*cos(theta); y= r.*sin(theta); % convert to Cartesian % Folium of Descartes if 1==0 a= 1; N= 2000; theta= linspace(0.8*pi,1.7*pi,n); % polar coord. r= (3*a*sin(theta).*cos(theta))./((cos(theta)).^3 + (sin(theta)).^3); x= r.*cos(theta); y= r.*sin(theta); % convert to Cartesian % Lemniscate of Bernoulli if 1==0 a= 1; N= 1000; theta= linspace(0,2*pi,n); % polar coord. r= sqrt(a^2*cos(2*theta)); x= r.*cos(theta); y= r.*sin(theta); % convert to Cartesian % Logairthmic spiral if 1==0 a= 0.25; N= 1000; theta= linspace(0,6*pi,n); % polar coord. r= exp(a*theta); x= r.*cos(theta); y= r.*sin(theta); % convert to Cartesian % Fey's butterfly if 1==0 N= 5000; theta= linspace(0,24*pi,n); % polar coord. r= exp(cos(theta))-2*cos(4*theta)+ (sin(theta/12)).^5; x= r.*cos(theta+pi/2); y= r.*sin(theta+pi/2); % convert to Cartesian figure(1); clf; plot(x,y,'k-','linewidth',2); xlabel('x'); ylabel('y');

8 EXclassicCurves.m Three-leaved rose y x

9 EXclassicCurves.m Fey s butterfly y x

10 % ### EXclassicCurves.m ### CB % Motivated by DeVries (1994), this code plots a handful of 'classic' curves % (simply turn on/off below) EXclassicCurves.m clear % three-leaved rose if 1==0 a= 1; N= 200; theta= linspace(0,2*pi,n); % polar coord. r= a*sin(3*theta); x= r.*cos(theta); y= r.*sin(theta); % convert to Cartesian % spiral of Galileo if 1==0 a= 1; N= 2000; theta= linspace(0,10*pi,n); % polar coord. r= a*theta.^2; x= r.*cos(theta); y= r.*sin(theta); % convert to Cartesian % Witch of Agnesi if 1==0 a= 1; N= 2000; x= linspace(-20,20,n); % polar coord. y= (8*a^3)./(x.^2+4*a^2); % Involute of a circle if 1==0 a= 1; N= 200; theta= linspace(0,4*pi,n); % polar coord. x= a*cos(theta)+a*theta.*sin(theta); y= a*sin(theta)-a*theta.*cos(theta); % Cissoid of Diocles if 1==0 a= 1; N= 100; theta= linspace(0,2*pi,n); % polar coord. r= a*sin(theta).*tan(theta); x= r.*cos(theta); y= r.*sin(theta); % convert to Cartesian % Cochleoid if 1==0 a= 1; N= 2000; theta= linspace(pi,16*pi,n); % polar coord. r= a*sin(theta)./theta; x= r.*cos(theta); y= r.*sin(theta); % convert to Cartesian

11 % Folium of Descartes if 1==0 a= 1; N= 2000; theta= linspace(0.8*pi,1.7*pi,n); % polar coord. r= (3*a*sin(theta).*cos(theta))./((cos(theta)).^3 + (sin(theta)).^3); x= r.*cos(theta); y= r.*sin(theta); % convert to Cartesian % Lemniscate of Bernoulli if 1==0 a= 1; N= 1000; theta= linspace(0,2*pi,n); % polar coord. r= sqrt(a^2*cos(2*theta)); x= r.*cos(theta); y= r.*sin(theta); % convert to Cartesian % Logairthmic spiral if 1==0 a= 0.25; N= 1000; theta= linspace(0,6*pi,n); % polar coord. r= exp(a*theta); x= r.*cos(theta); y= r.*sin(theta); % convert to Cartesian % Fey's butterfly if 1==1 N= 5000; theta= linspace(0,24*pi,n); % polar coord. r= exp(cos(theta))-2*cos(4*theta)+ (sin(theta/12)).^5; x= r.*cos(theta+pi/2); y= r.*sin(theta+pi/2); % convert to Cartesian figure(1); clf; plot(x,y,'k.','linewidth',1); xlabel('x'); ylabel('y'); EXclassicCurves.m (cont)

12 What about 3-D plots? (e.g., plotting multivariable functions) Solution to diffusion equation (or heat eqn.) Aside: This equation is directly tied back to the development of Fourier analysis (a topic we will return to in some detail a bit later on)

13 EXcreate3D.m % ### EXcreate3D.m ### % code to demonstrate/fiddle w/ 3-D plots in Matlab % Notes % o once you make the plot, in the figure window, look up top for the icon % that is a cube w/ a circular arrow around it ("Rotate 3D"). Click on that % and then have some fun... % o type "help peaks" to see where/how the multivariable function (i.e., z) % is generated clear % N= 25; % # of axis points in xy-plane M= 20; % # of lines for contour plot % % ==== % surface plot [X,Y,Z] = peaks(n); % use built-in Matlab function to create "z" figure(1); clf; surf(x,y,z); h= colorbar; xlabel('x'); ylabel('y'); zlabel('z'); ylabel(h,'z'); % ==== % contour plot figure(2); clf; contour(x,y,z,m,'linewidth',2); h= colorbar; grid on; xlabel('x'); ylabel('y'); zlabel('z'); ylabel(h,'z');

14 EXcreate3D.m

15 % ### EXcreate3D2.m ### CB EXcreate3D2.m % code to demonstrate/fiddle w/ 3-D plots in Matlab and function handles % % Notes % o once you make the plot, in the figure window, look up top for the icon % that is a cube w/ a circular arrow around it ("Rotate 3D"). Click on that % and then have some fun... % % Different functions to try for % o (1./sqrt(Y)).*exp(-(X.^2)./Y) [sol. to Diffusion Eq. re point source] % o Y.*cos(2*pi*X) [slanted sinusoid] % o (1-Y).*tanh(8*X-3) [slanted asympotote] % o exp((-(x-0.5).^2 + -(Y-0.5).^2)/0.05) [3-D Gaussian] % o others?? clear % =================== % function handle to express function (see options above) exp((-(x-0.5).^2 + -(Y-0.5).^2)/0.05); % other plotting params. N= 40; % grid resolution xb= [0 1]; % min/max values along x yb= [0 1]; % min/max values along y vp= [-62 36]; % view angle (vals. between 0 and 90) usepeaks= 0; % eschew function above and use Matlab's built-in peaks.m) {0} % =================== % % create x and y axes x= linspace(xb(1),xb(2),n); y= linspace(yb(1),yb(2),n); % from those, create a base "grid" for f [X,Y]= meshgrid(x,y); % multi-dimensionlize as required % Note: this is a trick I learned from Hanselman & Littlefield ch.27.2 % % now determine F if usepeaks==0 F= f(x,y); else [X,Y,F] = peaks(n); % use built-in Matlab function to create "z" % figure(1); clf; surf(x,y,f); view(vp); colormap jet; hcb= colorbar; ylabel(hcb,'f'); xlabel('x'); ylabel('y'); zlabel('f'); if (1==0); shading interp; % turn on to "smooth" coloring {0} if (1==0); shading flat; % turn on to eschew grid lines {0}

16 % function handle to express function (see options above) exp((-(x-0.5).^2 + -(Y-0.5).^2)/0.05); EXcreate3D2.m

17 EXcreate3D2.m Matlab provides an easy means to rotate 3-D plots around (remember, what is on the screen is a 2-D projection!)

18 Aside: Animations Ø e.g., Standing waves on a membrane (0,1) mode (0,2) mode (0,3) mode (1,1) mode

19 Aside: Animations (1,2) mode (3,1) mode à Note clear presence of nodes (chief characteristic of standing waves)

20 % ### EXanimateMotion.m ### CB EXanimateMotion.m % demonstrates syntax to "animate" data (borrowed some elements from % EXrandom2Wwalk.m) as well as function calls clear % ================================ % most of these bits are overly complicated for the main purpose of this % code (i.e., it is "flashier" than it needs to be) xlim= [ ]; % plotting limits {[ ]} P.p= [ ]; % params for a quartic function P.dur= 100; % "duration" for animation {100} % ================================ % --- x= linspace(xlim(1),xlim(2),p.dur); % array of "reaction coord." values (for plotting) P.p(1)*x.^4+ P.p(2)*x.^3+ P.p(3)*x.^2+ P.p(4)*x+ P.p(5); % energy function y= 0.8*sin(2*pi*0.5*x)- 0.5; xp= E(y); % --- figure(1); clf; plot(x,e(x),'r-'); hold on; grid on; p= plot(x(1),y(1),'bo','linewidth',2); for nn=2:length(y) p.xdata = y(nn); p.ydata = xp(nn); drawnow

21 EXanimateMotion.m

22 Ø Consider not just how you would plot numbers created in Matlab... Ø... but also data you might collect from your research

23 Principles of Graphical Excellence Ø Useful reference: The Visual Display of Quantitative Information (E. Tufte) [ Show the data Avoid distorting what the data have to say Make large data sets coherent; present many #s in a small space Serve a reasonably clear purpose Reveal the data at several levels of detail Tell the truth about the data

24 The practical conclusions are clear. PowerPoint is a competent slide manager and projector. But rather than supplementing a presentation, it has become a substitute for it. Such misuse ignores the most important rule of speaking: Respect your audience.

25 Principles of Graphical Excellence GOAL: Give the viewer the greatest number of ideas in the shortest amount of time with the least ink in the smallest space Palmer and Russell (1986)

26 2 - Graphics Marey (~1885)

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28 2 - Graphics Marey (1880)

29 2 - Graphics Minard (1880)

30 2 - Graphics American Education (~1970)

31 2 - Graphics This may well be the worst graphic ever to find its way into print. - Edward Tufte

32 2 - Graphics Further means to improve?

33 Kiang ( The nervous system, 1975)

34 Raster plot Ø Allows for a complicated set of data to visualized very efficiently and show the data for what they are Ø This is computational in the sense that we really do need a computer to be able to efficiently deal with organizing the data in such a fashion Each row represents a repeated measurement Kiang (JASA, 1980)

35 Raster plot EXturbine.m 90 Noise Representation Trial # Time [s] Reference:

36 Data Visualization w/ Matlab Ø Lots of basic commands plotting have been introduced (indirectly) throughout 2030: Ex. Ø plot(x,f, m, Linewidth,2); Ø plot(x,f, ko, MarkerSize,6); Ø hold on; grid on; Ø subplot(211) Ø set(gca,'yscale','log'); Ø set(s,'edgecolor','none'); Ø surf(x,y,k); Ø clf; Ø close all; Ø meshgrid(x,y); Ø t= linspace(0,1,100); Ø [...] Ø For brevity s sake, we won t cover the basics further here. Some useful references though: Kutz (2013)

37 Data Visualization w/ Matlab Good rule of thumb: Matlab provides a lot of options, but you don t need to necessarily use them! Kutz (2013)

38 Data Visualization w/ Matlab Ø In many cases, 3D data is more effectively presented when projected onto 2D Kutz (2013)

39 Data Visualization w/ Matlab Ø Reduce clutter when/where possible Kutz (2013)

40 Data Visualization w/ Matlab Ø Think carefully about choosing the best option to most clearly present the data Kutz (2013)

41 Looking Ahead: Fractals EXfractal1.m

42 Post-class exercises Ø Create some of the other classic curves Ø Write down the principles of graphical excellence on paper. Repeat. Repeat. Set yourself up in a for loop with N iterations to repeat. [i.e., these are very useful to memorize!] Ø Try creating some different 3-D plots and rotate them around. Craft some of your own multivariable functions to this Contest: Create a cool 3-D image and submit it to Hugh by the of the week. We will choose the best one and put it on the course website!

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