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MATLAB GRAPHICS - PART IIADVANCED PLOTTING

June 4, ‘99

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3-D PLOTTING

There are numerous ways to display a function in 3-D in black and white as well as color

One way to interpret 3D data is a series of points in space given by (x,y,z) coordinates

The direct extension of the 2-D plot function, plot(x,y), to 3-D is plot3(x,y,z)

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PLOTTING A CORKSCREW

How would you model a corkscrew?

Corkscrew, or spiral, is the 3-D equivalent of a spiral

It goes around a circle but it also rises from the ground plane. So, what is its equation?

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3-D EQUATION

A circle can be parametrically described by

– x=cos(t)

– y=sin(t)

To make it rise from the ground plane, let z=t and run t from 0 to 10*pi.

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Try it!

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Try it!

Another interesting plot is the same as corkscrew but you are going up around a cone rather than a cylinder. Can you write the code?

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DISPLAYING A FUNCTION AS A SET OF HEIGHTS

A 3-D plot can be interpreted as heights above the ground plane. These heights are evaluated at some predefined grid points

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mesh and meshgrid

To plot a function in 3D we need to understand mesh and meshgrid.

meshgrid samples the ground plane into a grid of points

– x=- 8:0.5:8;

– y=x;

– [x,y]=meshgrid(x,y)

x and y are now matrices

mesh evaluates the function over the grid

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EXMAPLE: SOMBRERO

Sombrero, looking like a Mexican hat, is defined by sin(r)/r.

r is the distance of a point (x,y) to the origin, i.e.

– r2=x2+y2

r

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Try it!

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SOMBRERO

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GENERATING TRUE AXIS UNITS

Use of mesh (z) plots z vs. index positions not actual x or y values

To plot z vs. actual units of x and y, just use x and y in the mesh command like

– mesh(x,y,z)

Note that x and y can come from the output of meshgrid

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Try it!

MATLAB has a built-in function called peaks

How can you make the plot look smoother?

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What to do in the following slides

Each slide shows a variation of the mesh command on a function z

Since you already have z defined for both sombrero and peaks, for each slide duplicate the command shown and see the result for yourself

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CONTOUR PLOT

Contours are slices of constant height that are then projected onto the ground plane

In its simplest form meshc (z) does the job

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CURTAIN PLOT

You can put your plot on a ‘pedestal’ by using meshz (Z)

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CONTROLLING VIEWPOINT

Viewpoint is controlled by two angles: azimuth and elevation

Azimuth is rotation around the Z-axis

Elevation is rising above the ground plane

z

x

yelaz

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DEFAULT VIEWPOINTS

In MATLAB, default viewpoints are az=- 37.5 and el=30 degrees

Zero degrees azimuth is like looking up the x-axis shown in the previous slide .

90 degrees of elevation is like looking directly down on the the surface

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Working with viewpoint

The best way to understand viewpoint is to play around

To understand the effect of elevation, fix your azimuth at 0 then change your elevation:

– view(0,10),view(0,30),view(0,60)

Or fix your elevation at 30 degrees and change your azimuth

– Compare view(30,60) with view(-20, 60)

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INTERPRETING SIGNS OF VIEWPOINT ANGLES

Increasingly negative azimuth angle corresponds to holding the object in front of you and rotating it counterclockwise.

Equivalently, it corresponds to keeping the object stationary and moving around it clockwise

Positive elevation angle mean rising above the object. Elevation of +90 degrees means being directly overhead and looking down

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Homework#1

For the following function

– do a mesh plot then title and label all axis

– visually find out how deep the hole is?

– what is happening inside(looking underneath)?

– generate 3D contours(30 of them)

– Write a procedure that would cap the plot to 70% of its peak value then plot it. Your plot should show a flat top

z =sin x2 + y2( )

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GENERATING SHADED PLOTS

mesh generates wiremesh plots(can see lines)

To generate surfaces with solid shading, surf and its variations are used

These variations are

– surf

– surfl (this is surf followed by lower case L)

– surfc

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Using surf

Usage:

– surf(x,y,z,C)

(x,y) is generated via meshgrid and z is the height of the function.

z-to-color mapping is done according to the entries into “colormap” via C. More on this later

If surf(z) is used, color is proportional to height z.

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Plotting peaks using surf

Generate the peaks function in the range (-4 to 4) in increments of .5

Then use

– surf(x,y,z)

For comparison, use mesh and display it in a second window

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Try it!

Look closely and see if you can tell the difference between surf and mesh

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mesh vs. surf

Display mesh and surf side-by-side

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SOLID SHADING- shading

To plot solid looking shapes, as opposed to wiremeshes, shading command comes in

– shading flat

– shading faceted

– shading interp

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Try it!

Display one of your favorite 3D shapes you have done so far and in the command window type and observe

– shading flat

– shading faceted

– shading interp

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FLAT vs. FACETED vs. INTERPOLATED SHADING

Flat shading assigns constant colors to surface patches

Faceted shading assigns constant colors but also shows the wiremeshes

Interpolated shading assigns colors proportional to height of the function

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3D CONTOURS

contour projects 3D contours of a surface onto the ground plane

contour3 shows the true 3D contours

Usages are

– contour3(x,y,z,N)

This command generates N contours of the function Z. Default is N=10

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Try it!

Display the peaks function over x and y ranging over - 5 to 5 in increments of 0.1

Then do the following

– Display 50 2D contours using contour

– Display 50 3D contours using contour3d

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FEW EXAMPLES

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Can you get this?

Hint: contour displays its plots on the z=0 plane

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CONTROLLING LIGHTING DIRECTION-surfl

You can shine light on a surface from a desired direction

Shading is based on a combination of diffuse, specular and ambient lighting models

Usage:

– surfl(x,y,z,s)

– s=lighting direction=[az,el] where az and el are azimuth and elevation angles previously defined

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Lighting example

Keep changing the s parameter and watch

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WATERFALL PLOTS

An interesting effect can be generated by just plotting the rows of the Z matrix using

– waterfall(x,y,z)

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Putting several plots on a single page

subplot(mnp) divides the page into m(rows)xn(columns) tiles then selects the pth tile for the current plot

This tile would be Referred to by subplot (235)

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Seeing subplot at work

Let’s say we want to partition the page into 3x2=6 tiles. Simply type the following in the command window and see what happens

– subplot(232)

– subplot(235)

– subplot(233)

– subplot(234)

– subplot(236)

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Homework #2: placing 4 plots on a page

Let’s say we have 4 plots (choose your own) and want to arrange them on paper in the following styles

– Across the page in one row

– Vertical in one column

– In a matrix, 2x2 tiles on a page

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Special surfaces: cylinder and sphere

sphere(n) will generate a plot of unit sphere using (n+1)^2 points. Another usage is

– [x,y,z]=sphere(25);

– surf(x,y,z)

Similarly, we can generate a cylinder of radius 1 using cylinder.

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Generalized cylinder

Think of a cylinder with changing cross section

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How to do it?

Usage:

– Cylinder(radius) where radius is the growing cross sectional radius described by a vector

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Homework #3

Plot z=sin(sqrt(x^2+y^2)). Plot it using

– mesh

– surf, surfl, surfc

– Experiment with shading: flat, faceted, interpolated

– Experiment with lighting directions. For good effect type colormap(copper) after bringing up the plot.