week 11 part c matlab: 2d and 3d plotsen1811/17s2/lectures/week11/week11c... · 2017-03-01 · week...

ENGG1811 © UNSW, CRICOS Provider No: 00098G1 W11 slide 1

Week 11 Part C Matlab: 2D and 3D plots

ENGG1811 Computing for Engineers

More on plotting

•  Matlab has a lot of plotting features •  Won’t go through them in details •  Good to know what’s possible and when you want

to use it, look up the documentation •  Some of these slides (if marked with ®) are for

references and we won’t be going through them in details in the lecture

ENGG1811 © UNSW, CRICOS Provider No: 00098G W9 slide 2


Plot appearance

•  Line and marker styles –  values encoded in a short optional string following each

x, y pair in a plot-type command

•  See Matlab documentation

>> plot(x, y, 'b*-‐'); % more x, y, stylestrings can follow

Line colour, r=red, b=blue, g=green, w=white, c=cyan, m=magenta, y=yellow, k=black

Marker style, .=point, o=circle, x=x, *=star, s=square, v=down triangle, ^=up triangle, etc

Line style, -‐=solid, -‐-‐=dashed, :=dotted, -‐.=dash dot

®


Additional control

•  Additional characteristics of the lines and markers are specified by extra pairs of arguments to plot of the form

'propertyname', value,

Properties include LineWidth – in pixels (integer) MarkerSize – in pixels MarkerEdgeColor – string, same codes as for lines MarkerFaceColor – string, same codes as for lines

®


LaTeX formatting

•  Labels, titles and legend text can be formatted using a scheme used for typesetting mathematical documents

•  LaTeX (pronounced ‘lay-teck’) embeds codes or escape sequences using backslash and other special characters \it{text} \bf{text} – italic, boldface* _{text} ^{text} – subscript, superscript \lambda \SIGMA – Greek lowercase (λ), upper (Σ) \circ \pm \neq \infty – symbols ° ± ≠ ∞ \rightarrow \uparrow – arrow symbols (also left, down) \_ \^ \{ \} \\ – literal _ ^ { } or \

•  Can omit { } if a single character, for example \itx •  See Chapman Table 3-2 * Font style changes do not properly terminate in Matlab versions to 2012b

®


LaTeX equation examples

gives this result (though equations are images, not scalable text):

If you are really interested, try help latex (part of the syms package, converts a symbolic expression to a LaTeX equation)

%% % Einstein's famous equation is $E = m c^2$. % % $$\sum_{i=1}^{\infty} \frac{1}{2^i} = 1$$ % % $$\int_{0}^{\infty} x^2 e^{-‐x^2} dx = \frac{\sqrt{\pi}}{4}$$

®


Polar plots

•  Polar plots have angles 0 to 2π in place of x, and magnitude (distance from the origin) in place of y

•  Chapman Example 3.3 – cardiod microphone response –  gain (relative sensitivity) varies with angle θ (0 = directly

in front) according to the formula

>> g = 0.5; % gain coefficient, characteristic of mic >> theta = linspace(0, 2*pi, 121); % 360/(121-‐1) = 3 degrees >> gain = 2*g * (1 + cos(theta)); >> polar(theta, gain, 'r-‐'); % red, solid, 1px line >> title('\bfCardiod microphone gain versus angle \it{\theta}');

)cos1(2 θ+= ggain

®


Annotations

Not a full graphics editor, so results are only approximate

Figure window has a toolbar to add elements and save

Plot browser Add/edit legend Select tool View – Plot Edit Toolbar to

access annotation tools such as arrows, shapes and text boxes

File – Save As (*.fig) See Chapman section 3.3

®


Other plot types

•  Chart types similar to OpenOffice Calc/Excel §  bar(x,y) – vertical bar chart §  barh(x,y) – horizontal bar chart §  stem(x,y) – marker and vertical line §  stairs(x,y) – like bar, but only top of skyline shown §  pie(x, explode) – use with caution §  compass(x,y) – polar with arrow to each Cartesian point

•  If y is a matrix instead of a vector, each column is a separate plot –  applies to all plot types that accept y values as second

argument

•  See Matlab documentation

®


3D line plots

•  3 dimensional line plots can be generated by passing three equal sized vectors to plot3

•  Example: decaying oscillations in a mechanical system in two dimensions (Chapman 8.3.1)

t = linspace(0, 10, 200); x = exp(-‐0.2*t) .* cos(2*t); y = exp(-‐0.2*t) .* sin(2*t); plot(x, y, 'Linewidth', 2); % 2D plot title (’2D Line Plot'); xlabel(’x'); ylabel(’y'); grid on; … plot3(x, y, t, 'Linewidth', 2); % 3D plot, time is z (up) title (’3D Line Plot'); xlabel(’x'); ylabel(’y'); zlabel(’t'); grid on;

tetytetx tt 2sin)(2cos)( 2.02.0 −− ==Code in plot3d_demo.m


3D time-based plot

•  3D plot shows the effect of time arguably better than as distance along the 2D plot

•  Even better, the 3D view has interactive rotation


3D mesh and surface plots

•  Data that has two independent variables (for example, temperature measured at many coordinates in 2D space) can be visualised as a 3D plot –  x and y are normally the independent variables –  z is normally the dependent variable –  can be displayed in three ways

1.  as a mesh or wireframe of individual line plots 2.  as a continuous surface, with colouring to highlight

the slope at each point 3.  as a series of contours, slices parallel to the x-y plane

–  The x, y and z matrices have exactly the same shape

mesh(x, y, z); surf(x, y, z); contour(x, y, z);

Preparing to plot

•  For 2D plot, for each value of x, you need the corresponding value in y


•  For 3D plot, for each pair of (x,y), you need the corresponding value of z

x = linspace(0, 10, 200); y = exp(-‐0.2*t) .* cos(2*t);


3D plot arrays Example: evaluated at x = 0,1,2 and y = -3,-2,-1,0

z(x, y) = xexp(−x2 − y2 )

x =

0 1 20 1 20 1 20 1 2

!

"

####

$

%

&&&&

y =

−3 −3 −3−2 −2 −2−1 −1 −10 0 0

!

"

####

$

%

&&&&

z =

0 0.0000 0.0000 0 0.0067 0.0007 0 0.1353 0.0135 0 0.3679 0.0366

!

"

####

$

%

&&&&

(0,-3)

(0,-2)

(0,-1)

(0 , 0)

x=0 1 2

(2,-3)

(2,-2)

(2,-1)

(2,0)

y = -3 -2 -1 0

(1,-3) (1,-2)

z(2,-1)

Correction: This arrow should be pointing at the number 0.0135 or the (3,3) element


3D plot arrays, continued

•  Having the arrays structured this way allows you to calculate the z value using array operators

•  Fortunately you can easily construct the x and y arrays from their respective vectors using meshgrid:

>> [x,y] = meshgrid([0,1,2], [-‐3,-‐2,-‐1,0]); >> z = x .* exp(-‐x.^2 -‐ y.^2); >> surf(x,y,z); >> xlabel(’x'); >> ylabel(’y'); >> zlabel(’z');

Mesh doesn’t have surface

shading:

Code in vis_demo.m Also, includes heatmap.

Automatic screen rotation

•  How can a smartphone tell how the user is holding the phone?

http://allaboutgalaxynote.com/galaxy-note-3-online-user-guide/control-motions/

•  We will use 3D plot to understand how this is done

Smart phone and accelerometer (1) •  Smartphones have

accelerometers for measuring acceleration in 3 directions

êAxes of accelerometer è Accelerometer readings. The phone was moved in the +ve y direction. That caused an acceleration in +ve y direction (green line)

http://www.mathworks.com.au/help/simulink/slref/samsunggalaxyandroidaccelerometer.html

Smart phone and accelerometer (2)

•  An important part of problem solving is to understand how things work.

•  Visualisation (charts, graphs etc) is extremely helpful.

•  Demo: We will use the Android Physics Toolbox to see how accelerometers react to motion. This toolbox can be obtained from: –  https://play.google.com/store/apps/details?

id=com.chrystianvieyra.physicstoolboxsuite&hl=en –  Similar apps are also available for iPhones

•  If you are interested to find out how the accelerometers inside smartphones are manufactured, see here: –  http://www.engineerguy.com/elements/videos/video-accelerometer.htm

4 orientations We label the orientation according to which edge is pointing down

–  MX (for minus x), MY (for minus y) –  PX (for plus x), PY (for plus Y)

MY PY

PX

MX

Let us start simple

•  We considered two orientations: MY and MX •  We assumed the phone is vertical •  We wanted to find a condition based on the

accelerations to classify the orientation –  Basically a if-else statement, but what is the condition?

cond?

MY MX

MY

MX True False

This condition depends on the acceleration

Let us plot acceleration versus time

•  Can you find some characteristics on the acceleration to differentiate between the orientations MX and MY?


Plot y-acc versus x-acc


•  Orientation MX –  x-acceleration big –  y-acceleration small

•  Orientation MY –  x-acceleration small –  y-acceleration big

Pseudo-code: If y-acc > x-acc

orientation MY else

orientation MX

A different perspective


line x = y

y > x

x > y

Finding the Boolean expression is the same as finding a line that can separate the acceleration of different orientations into separate regions

Classify 4 orientations

•  We want to classify 4 orientations •  We no longer assume that the phone is vertical •  Experiment

–  Put the phone in each orientation –  Tilt the phone but keeping one of +x, -x, +y, -y

down –  Measure the acceleration in 3-directions

•  Time plot on the next page

•  Let us use 3D plot instead


Time plot


MY MY

PY PX

Matlab feature: subplot

Only need x- and y-acceleration


The surprise is that acceleration in the z-direction is not needed at all!


An exercise for you: boat hull*

A simple model for the hull of a boat is given by

where y is the width of the hull from the centre line, x is the distance along the centre line, and z is the depth of the hull. B is the beam (max width), L is the length and D is the draft (max depth). This hull is 8m long, with 3m beam and 1.5m draft.

* Based on Holloway, p.144 Project 15.

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛−⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛−=22

1212 D

zLxBy


Implementing the model

•  x and z are the independent variables, and range from –L/2 to L/2, and –D to 0 respectively –  use linspace and meshgrid to generate these arrays

•  The first half of the y values (the starboard side) are produced by the formula, the second set of y values has the opposite sign –  to display both halves (thus avoiding a massive maritime

disaster!), use the command hold on between plot commands

–  axis equal maintains aspect ratio (uniform scales)

•  You can try –  use the 3D rotation tool to examine the shape –  does mesh or surf give the better view? –  experiment with shading (faceted, flat or interp)

week 11 part c matlab: 2d and 3d plotsen1811/17s2/lectures/week11/week11c... · 2017-03-01 · week...

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