textbook: chapter 11 part : 3d-plotting cse 1010

Textbook: Chapter 11 Part : 3D-Plotting

CSE 1010

Demos … before we start … 1/3

3D-Plot and What Matlab Can Do …

>> teapotdemo

Demos … before we start … 2/3The Earth …

load topo.mat

[x,y,z] = sphere(50);

cla reset

axis square off

props.AmbientStrength = 0.1;

props.DiffuseStrength = 1;

props.SpecularColorReflectance = .5;

props.SpecularExponent = 20;

props.SpecularStrength = 1;

props.FaceColor= 'texture';

props.EdgeColor = 'none';

props.FaceLighting = 'phong';

props.Cdata = topo;

surface(x,y,z,props);

light('position',[-1 0 1]);

light('position',[-1.5 0.5 -0.5], 'color', [.6 .2 .2]);

view(3)

Demos … before we start … 3/3Matlab Logo …

L = 40*membrane(1,25);

logoFig = figure('Color',[0 0 0]);

logoax = axes('CameraPosition', [-193.4013 -265.1546 220.4819],...

'CameraTarget',[26 26 10], ...

'CameraUpVector',[0 0 1], ...

'CameraViewAngle',9.5, ...

'DataAspectRatio', [1 1 .9],...

'Position',[0 0 1 1], ...

'Visible','off', ...

'XLim',[1 51], ...

'YLim',[1 51], ...

'ZLim',[-13 40], ...

'parent',logoFig);

s = surface(L, ...

'EdgeColor','none', ...

'FaceColor',[0.9 0.2 0.2], ...

'FaceLighting','phong', ...

'AmbientStrength',0.3, ...

'DiffuseStrength',0.6, ...

'Clipping','off',...BackFaceLighting','lit', ...'SpecularStrength',1.1, ...'SpecularColorReflectance',1, ...'SpecularExponent',7, ...'Tag','TheMathWorksLogo', ...'parent',logoax);l1 = light('Position',[40 100 20], ...'Style','local', ...'Color',[0 0.8 0.8], ...'parent',logoax);l2 = light('Position',[.5 -1 .4], ...'Color',[0.8 0.8 0], ...'parent',logoax);

Demos … before we start … 3/3

Contents

Part 1 - 2D Plotting = 3D Plotting

Part 2 - Linear 3D Plotting

- Direct Linear Plotting

- Parametric Linear Plotting

Part 3 - 3D Mesh Plotting

Part 4 - 3D Surface Plotting

3D – Plotting

Part 1

When 2D Potting = 3D-Plotting

Linear Plotting

3D – Plotting – Part 1 When 2D Plotting = 3D Plotting

1 – Create a 2D Plot

2 – Click icon “Rotate 3D”

3 – Woow!

We can extend 2-plots by adding a set of z values

Syntax: Plot3 ( x, y, z, str)

str is an optional parameter,

It specifies color/line style

The default is “solid blue”

… Examples

Example

Plot3 ( x, y, z, str)str is an optional parameter,

specifies color/line styledefault solid blue

… Examples

3D-PLotting

Part 2

Linear 3D Plotting

3D – Plotting – Part 2Linear 3D Plotting

Linear 3-D Plotting looks like …

How do we do that …?

Linear 3D Plottingx = 0 : 0.1 : 3.* pi; % each plot has the same set of x values y1 = zeros(size(x)); % The y values for the first plot are all 0 z1 = sin(x); %

z2 = sin(2.*x); % The 2nd and 3rd plots are sin(x) at different freq.z3 = sin(3.*x); %

y3 = ones(size(x)); % The y values of the 2nd and 3rd plots y2 = y3./2; % are all 0.5 and 1 respectively

plot3(x,y1,z1, 'r', x,y2,z2, 'b', x, y3, z3, 'g')

grid on % activates the grid

xlabel('x-axis')ylabel('y-axis')zlabel('z-axis')

NOTE: .* and ./ :

3D-PLooting

Part 3

Parametric Plotting

3D – Plotting – Part 3Linear Parametric 3D Plot

3-D parametric plots allow the variables on each axis to be dependent on a separate, independent variable that defines a path in the plotting space

Two examples - The Spiral - 3D Motion of a particle

3D – Plotting – Part 3: Linear Parametric 3DExample 1 – The Spiral

The spiral as an example of a 3-D plot where two of the dimensions, x and y, are dependent on the third, independent parameter.

The independent parameter is the rotation angle, θ, varying from 0 to 10 π (five complete revolutions).

The x and y values are mapped as sin(θ) and cos(θ)—the classic means of describing a circle.

The spiral effect is accomplished by plotting θ on the z-axis.

theta = 0 : 0.1 : 10.*pi; % theta = θplot3(sin(theta),cos(theta),theta)title( 'parametric curve based on angle');grid on

Example 1 = The Spiral

This is a fully parametric plot:

- The values of ALL three coordinates are mappings of an independent parameter, t.

- This is a plot of the 3D motion of a particle receiving random impulses in all three axes.

Note the use of text anchored in x-y-z space to label points on the graph.

Script …

Example 2: 3D Motion of a Particle

N = 20;dvx = rand(1, N) - 0.5 % random v changesdvy = rand(1, N) - 0.5dvz = rand(1, N) - 0.5

vx = cumsum(dvx); % integrate to get vvy = cumsum(dvy);vz = cumsum(dvz);

x = cumsum(vx); % integrate to get posy = cumsum(vy);z = cumsum(vz);

plot3(x,y,z)grid ontitle(‘All 3 axes varying with parameter t')text(0,0,0, ‘Start');text(x(N),y(N),z(N), ‘End');

Part 4

Other Plot Capabilities

bar3(x, y)

bar3h(x, y)

stem3(x, y ,z)

scatter3(x, y, z)

pie3(y)

Other 3D Plot Capabilitie – Part 4bar3(x, y)

x = [ ]y = [ ]

for k = -5:5 T = 1 + k + (k * 2) + (k * 3) x = [x k] % stores values for x-axis y = [y T] % stores values for y-axisend

bar3(x, y)


Other 3D Plot Capabilitie – Part 4bar3h(x, y)

x = [ ]y = [ ]

for k = -5:5 T = 1 + k + (k * 2) + (k * 3) x = [x k] % stores values for x-axis y = [y T] % stores values for y-axisend

bar3h(x, y) % Does not work well!!!


Other 3D Plot Capabilitie – Part 4stem3(x, y,z,’fill’)

t = 0:0.2:10x = ty = sin(t)z = t .* (1.5) stem3(x,y,z,'fill')grid on xlabel('x-axis')ylabel('y-axis')zlabel('z-axis')

Other 3D Plot CapabilitiePart 4

scatter3(x, y,z,’filled’)[x,y,z] = sphere(16);X = [x(:)*.5 x(:)*.75 x(:)];Y = [y(:)*.5 y(:)*.75 y(:)];Z = [z(:)*.5 z(:)*.75 z(:)];S = repmat([1 .75 .5]*10,prod(size(x)),1);C = repmat([1 2 3],prod(size(x)),1);scatter3(X(:),Y(:),Z(:),S(:),C(:),'filled‘);view(-60,60) % view is explained later

% sphere: built-in function% repmat: replicate a matrix% prod(x): product of all the% items in x% sphere See next slide

sphere and peaks

sphere and peaks are 2 built-in Matlab functions

Their main use is to test the visualization (graphics) effects

sphere(n)The sphere function generates the x-, y-, and z-coordinates of a unit sphere for use with surf and mesh.

sphere generates a sphere consisting of 20-by-20 faces.sphere(n) draws a surf plot of an n-by-n sphere in the current figure. [X,Y,Z] = sphere(n) returns the coordinates of a sphere in three matrices

that are (n+1)-by-(n+1) in size.

You draw the sphere with surf(X,Y,Z) or mesh(X,Y,Z).

% Matlab script % From the Matlab website

% sphere(16) optional parm sphere axis equal

Peaks in brief Details

peaks is a function of two variables, obtained by translating and scaling Gaussian distributions, which is useful for demonstrating mesh, surf, pcolor, contour, etc.

Example of use:

Z = peaks; % returns a 49-by-49 matrix

% Matlab script

% From the Matlab website

[xx,yy,zz] = peaks(120);

surf(xx,yy,zz)

colormap 'default'

shading faceted

axis tight

title('peaks')

xlabel('x'),ylabel('y'),zlabel('z')

Matlab built-in function: peaks

Function Syntax

Z = peaks;

Z = peaks(n);

Z = peaks(V);

Z = peaks(X,Y);

peaks;

peaks(N);

peaks(V);

peaks(X,Y);

[X,Y,Z] = peaks;

[X,Y,Z] = peaks(n);

[X,Y,Z] = peaks(V);

Description

peaks is a function of two variables, obtained by translating and scaling Gaussian distributions, which is useful for demonstrating mesh, surf, pcolor, contour, etc.

Z = peaks; returns a 49-by-49 matrix.Z = peaks(n); returns an n-by-n matrix.Z = peaks(V); returns an n-by-n matrix, where n = length(V).Z = peaks(X,Y); evaluates peaks at the given X and Y (which must be the same size) and returns a matrix the same size.peaks(...) (with no output argument) plots the peaks function with surf.

> > help peaks … or go to:http://www.mathworks.com/access/helpdesk/help/techdoc/ref/peaks.html

Other 3D Plot Capabilitie – Part 4pie3(y)

y = [ 23 56 87 12 9 ]

pie3(y)

Surface Plots

Matlab Rendering Capabilities

3D-Plot: Surface Plots

4 Basic concepts:1 – plaid2 – meshgrid3 – mesh4 - surf

Plaid

The “plaid” is a Matlab concept, NOT a Matlab functionThe underlying surface is referred to as plaid because of its

conceptual similarity to a Scottish tartan pattem.To design such a pattern, one needs only to specify the color

sequence of the horizontal and vertical threads.In the same way, we specify a plaid by defining vectors of the row

and column data configurations.The simplest surface plots are obtained by defining a z value for

each point on an x-y plaid.

meshgrid(x,y)

meshgrid(x,y)

… accepts the x1xm and y1xm vectors that bound the edges of the plaid, and replicates the rows and columns appropriately to produce xxnxm and yy1xm, containing the x and y values (respectively) of the complete plaid

This enables us in general to compute mappings for the 3-D coordinates of the figure we want to plot

mesh(xx,yy)

mesh(xx,yy) plots the surface as white facets outlined by colored lines

The line coloring uses one of many color maps (listed in Appendix A of the textbook), where the color is selected in proportion to the parameter

You can turn the white facets transparent with the command “hidden off”.

surf(xx,yy,zz)

surf (xx, yy, zz)

plots the surface as colored facets outlined by black lines

The line coloring by default is selected in proportion to the zz parameter

The lines can be removed by using one of a number of shading commands listed in Appendix A of the textbook

Also on the next slide

Shading Commands

shading < value >: shades a surface plot with one color per grid section

SHADING controls the color shading of SURFACE and PATCH objects.

SURFACE and PATCH objects are created by the functions SURF, MESH, PCOLOR, FILL, and FILL3

shading flat sets the shading of the current graph to flat. shading interp sets the shading to interpolated. shading faceted sets the shading to faceted, which is the default.

Read: > > >> help shading

More on Mesh and Surface PlotsCommon Rendering tools Available in Matlab

Mesh mesh(x,y,z)

Mesh Curtain meshz(x,y,z)

Mesh Contour meshc(x,y,z)

Waterfal waterfall(x,y,z)

3D Contour Plot contour3(x,y,z,n)

2D Contour Plot contour(x,y,z,n)

Surf surf(x,y,z)

Surface and contour surfc(x,y,z)

Surface and lighting surfl(x,y,z)

Details and Examples

More on Mesh and Surface PlotsCommon Rendering tools Available in

Matlab

The MESH Series

In the following slides we use the same script to plot:

x = -3:0.25:3;y = -3:0.25:3; [X, Y] = meshgrid(x,y) % build the plaid Z = (1.8) .^ (-1.4) * sqrt((X .^2) + (Y .^2)) .* cos (0.5 * Y ) .* sin(X);

The only line of code that is different is the Matlab function used to plot our [X,Y] values in the red box

Mesh mesh(x,y,z)x = -3:0.25:3;

y = -3:0.25:3;

[X, Y] = meshgrid(x,y) % build the plaid

Z = (1.8) .^(-1.4) * sqrt((X.^2) + (Y .^2)) .* cos (0.5 * Y) .* sin(X);

mesh(X,Y,Z)

xlabel('x-coordinate');

ylabel('y-coordinate');

zlabel('z-coordinate');

Mesh Curtain meshz(x,y,z)x = -3:0.25:3;y = -3:0.25:3;



meshz(X,Y,Z)

xlabel('x-coordinate');ylabel('y-coordinate');zlabel('z-coordinate');

%Draws a curtain around%the mesh

Mesh Contour meshc(x,y,z)x = -3:0.25:3;y = -3:0.25:3;



meshc(X,Y,Z)


% Draws a contour plot% beneath the mesh

Mesh Waterfall waterfall(x,y,z)x = -3:0.25:3;

y = -3:0.25:3;



waterfall(X,Y,Z)




% Draws a mesh

% in one direction

% only

3D Contour contour3(x,y,z,n)x = -3:0.25:3;

y = -3:0.25:3;



contour3(X,Y,Z,50)




% n is the number of

% contour levels

% optional

2D Contour contour(x,y,z,n)x = -3:0.25:3;

y = -3:0.25:3;



contour(X,Y,Z,20)




% n is the number of

% contour levels

% optional

% input is 3D

% output is 2D

More on Mesh and Surface PlotsCommon Rendering tools Available in

Matlab

The SURF (Surface) Series

Surface – Simple Version surf(x,y,z)x = -3:0.25:3;y = -3:0.25:3;



surf(X,Y,Z)


% % % %

Surface with parameters surf(x,y,z,c)

% Color a sphere with the pattern of +1s and -1s

% in a Hadamard matrix

k = 5;

n = 2^k-1;

[x,y,z] = sphere(n);

c = hadamard(2^k);

surf(x,y,z,c);

colormap([1 1 0;0 1 1])

axis equal

% Example from the

% Matlab website

The “sinc” function (sin(r)/r) where r = radius we need to add eps to avoid inf when dividing by zero

» [xx,yy]= meshgrid(-4.*pi:pi./5:4.*pi);

» R=sqrt(xx.^2 + yy.^2)+eps; % radius

» zz=sin(R)./R;

» surf(xx,yy,zz)

» axis tight

eps: smallest possible difference between 2 floating point numbers

inf: infinity

mesh(X,Y,Z) versus surf(X,Y,Z)

Surface and Contour Plot surfc(x,y,z)x = -3:0.25:3;y = -3:0.25:3;



surfc(X,Y,Z)


% % % %

Surface Plot with Lighting surfl(x,y,z)x = -3:0.25:3;y = -3:0.25:3;



surfl(X,Y,Z)


% % % %

Surface Plot with Triangles trisurf(tri,x,y,z)

[x,y]=meshgrid(1:15,1:15);

tri = delaunay(x,y); % Matlab built-in function

z = peaks(15);

trisurf(tri,x,y,z)

% Example from the

% Matlab website

Misc. Examples

The Cube - Introduction

Example 1

Uses the surf function

Listing

The Cube – Listing (textbook, p. 264)

xx = [-1 -1 1 1 -1 % A-B-C-D-A

-1 -1 1 1 -1] % D-E-F-G-D

yy = [ 1 -1 -1 1 1 % A-B-C-D-A

1 -1 -1 1 1] % D-E-F-G-D

zz = [ 1 1 1 1 1 % A-B-C-D-A

-1 -1 -1 -1 -1] % D-E-F-G-D

subplot(1, 2, 1)

surf(xx, yy, zz)

axis equal

shading interp

view(-36, 44)

axis off

xx = [ 0 0 0 0 0 % P-P-P-P-P

-1 -1 1 1 -1 % A-B-C-D-A

-1 -1 1 1 -1 % D-E-F-G-D

0 0 0 0 0] % Q-Q-Q-Q-Q

yy = [ 0 0 0 0 0 % P-P-P-P-P

1 -1 -1 1 1 % A-B-C-D-A

1 -1 -1 1 1 % D-E-F-G-D

0 0 0 0 0] % Q-Q-Q-Q-Q

zz = [ 1 1 1 1 1 % P-P-P-P-P

1 1 1 1 1 % A-B-C-D-A

-1 -1 -1 -1 -1 % D-E-F-G-D

-1 -1 -1 -1 -1] % Q-Q-Q-Q-Q

subplot(1, 2, 2)

surf(xx, yy, zz)

axis equal

shading interp

view(-36, 44)

% axis off

The Cube – 3D - Plot (p. 264)

Here we use the function meshgrid(x,y) to define the plaid

This plot is created in three parts:1. Develop the underlying plaid specifying the x-y location of every point on the x-y plane2. Calculate the z values from the plaid3. Call some function that will accept the plaid and

these z values to produce the required plot

meshgrid(x,y,z,C)

Example: The Parabolic Dish (pp. 264-266)

% Parabolic dish

x = -3:3;

y = x;

[xx,yy]= meshgrid(x,y); % generates the plaid

zz = xx.^2 + yy.^2; % mapping of the z coordinates

mesh(xx,yy,zz) % color is proportional to mesh height

axis tight

title('z = x^2 + y^2')


NOTE:

Many more possibilities

Read: >> help mesh

Recommended Exercises

pp. 266-267: Exercise 11.2

Working with Colors

3D – Basic PeaksColors (p. 267)

[xx,yy,zz] = peaks(30); % Matlab peaks

surf(xx,yy,zz)

colormap 'default'

shading interp

axis tight

title('peaks')


More on Colors

Colors in Matlab (for 2D and 3D)

pcolor

colormap

shading

alpha

pcolor

% Example

n = 6;r = (0:n)'/n;

theta = pi*(-n:n)/n;

X = r*cos(theta);

Y = r*sin(theta);

C = r*cos(2*theta);

pcolor(X,Y,C)axis equal tight

In Matlab, pcolor - pseudocolor plot - is a rectangular array of cells with colors determined by C. MATLAB creates a pseudocolor plot using each set of four adjacent points in C to define a surface rectangle (i.e., cell).

Complete description at:

http://www.mathworks.com/access/helpdesk/help/techdoc/ref/pcolor.html

Syntax:

pcolor(C)pcolor(X,Y,C)pcolor(axes_handles,...)h = pcolor(...)

Explanations

More about Colormap (Matlab web page):http://www.mathworks.com/access/helpdesk/help/techdoc/ref/colormap.html

Built-in Colors

User Defined Colors

Next Slide

User Defined Colors

See Presentation:

CSE1010_2009Fall_W-13_Ch11=ColorsInMatlab.ppt

Shading% From the Matlab website% Compare flat, faceted, interpolated-shaded shape

subplot(3,1,1)sphere(16)axis squareshading flattitle('Flat Shading')

subplot(3,1,2)sphere(16)axis squareshading facetedtitle('Faceted Shading')

subplot(3,1,3)sphere(16)axis equalshading interptitle('Interpolated Shading')

alpha

% Set transparency properties for objects in current axes

surf(peaks);

alpha(0.5);

More on … alpha

plot(get(gcf,'Alphamap'))

plot(alphamap('vup'))alphamap('increase',.4)plot(get(gcf,'Alphamap'))

[x,y,z] = meshgrid(-1.25:.1:-.25,-2:.2:2,-2:.1:2);v = x.*exp(-x.^2-y.^2-z.^2);

h = slice(x,y,z,v,[-1 -.75 -.5],[],[0]);alpha('color')set(h,'EdgeColor','none','FaceColor',... 'interp', 'FaceAlpha','interp')

alphamap('rampdown')alphamap('increase',.1)colormap(hsv)

comet3: animation of a linear plot

t = -10*pi:pi/250:10*pi;

comet3((cos(2*t).^2).*sin(t),(sin(2*t).^2).*cos(t),t);

Drawnow: animationof a surface

x = 0:pi/100:4*pi;y = x

[X,Y]= meshgrid(x,y);

z = 3*sin(X) + cos(Y);

h = surf (z);

axis tight; set (gca, 'nextplot', 'replacechildren');

shading interp; colormap (jet)

m = 1;

for k = 0 : pi/100 : 2* pi

z = (sin(X) + cos (Y) ) .* sin (k);

set (h, 'Zdata',z);

M(m) = getframe; m = m + 1;

movie(M,2); drawnow

End

% Note: File: animation02.m

A better 3D Plot: Color and Lighting of the Peaks

Textbook: p. 268

Let’s change the parameter to peaks to 120, and add the line light angle (60, 45) at the bottom of the script.

This illuminates the surface with a light at the specified azimuth and elevation angle (in degrees).

[xx,yy,zz] = peaks(120);

surf(xx,yy,zz)

lightangle(60, 45)

colormap 'default'

shading interp

axis tight

title('peaks')


% Illumination of the surface with

% a light at the specified azimuth

% and elevation angle (degrees)

view(azimuth,elevation)

The position of the viewer (the viewpoint) determines the orientation of the axes. You specify the viewpoint in terms of azimuth and elevation, or by a point in three-dimensional space.

view(az,el) and view([az,el]) set the viewing angle for a three-dimensional plot.

The default view angles are:

az = 20º

el = 30º

Surface Plotting – Parametric Surfaces

Working with polar coordinates Examples: The cylinder (uses ONE angle) and the Sphere (uses TWO angles)

Listing

The Cylinder (p. 270)

facets = 120; len = 2; radius = 1; % Constants to define the smoothness of the cylinder thr = linspace(0, 2*pi, facets); % Definition of a plaid using x and theta xr = [0 len]; [x, th] = meshgrid( xr, thr );

y = radius * cos(th);

z = radius * sin(th);

surf(x, y, z);

shading interp

axis equal,axis tight,axis off

lightangle(60, 45) % Illuminate the figure alpha(0.8) % Sets the transparency of the surface so that view(-50, 35) % a portion of the hidden details can show through

The Sphere (textbook p. 271)

facets = 120; radius = 1;

thr = linspace(0, 2*pi, facets); % range of theta

phir = linspace(0, pi, facets); % range of phi

[th, phi] = meshgrid( thr, phir ); % builds the plaid in theta and phi

x = radius * cos(phi);

y = radius * sin(phi) .* cos(th);

z = radius * sin(phi) .* sin(th);

surf(x, y, z);

shading interp

colormap copper

axis equal, axis tight, axis off

lightangle(60, 45)

The Ellipsoid

Effects:

Meshing / No MeshingShading parameters

Ellipsoid – mesh view


[x, y, z] = ellipsoid(0,0,0,5.9,3.25,3.25,30);

surfl(x, y, z)

% shading = not used

colormap spring

axis equal

Ellipsoid – faceted flat% From the Matlab website

[x, y, z] = ellipsoid(0,0,0,5.9,3.25,3.25,30);

surfl(x, y, z)

shading flat

colormap spring

axis equal

Ellipsoid - Interpolation


[x, y, z] = ellipsoid(0,0,0,5.9,3.25,3.25,30);

surfl(x, y, z)

shading interp

colormap spring

axis equal

Ellipsoid – mesh only

% From the Matlab websiteclc; clear;

[x, y, z] = ellipsoid(0,0,0,5.9,3.25,3.25,30);

surfl(x, y, z)

mesh(x, y, z)

colormap spring

axis equal

Bodies of Rotation - pp. 272 - 276

Rotating Continuous Functions

Bodies of rotation are created by rotating a linear curve about a specified axis, that is, by rotating a general function v = f (u) defined over a range of u values about the x

or z axes.

Rotating Continuous Functions

Rotating the v = u { 2 } about the x and z axes

% set up the plaid

facets = 100;

x = linspace(0, 5, facets);

th = linspace(0, 2*pi, facets);

[xx tth] = meshgrid(x, th);

% rotate about the x axis

subplot(1, 2, 1)

rr = xx.^2;

yy = rr .* cos(tth);

zz = rr .* sin(tth);

surf(xx, yy, zz, xx);

shading interp, axis tight

xlabel('x'), ylabel('y'), zlabel('z')

title('x^2 rotated about the x axis')

% rotate about the z axis

subplot(1, 2, 2)

rr = xx;

zz = rr.^2;

xx = rr .* cos(tth);

yy = rr .* sin(tth);

surf(xx, yy, zz);

shading interp, axis tight

xlabel('x'), ylabel('y'), zlabel('z')

title('x^2 rotated about the z axis')

Rotating Discrete Functions

“2D profile” of a fictitious machine part and the picture created when that profile is rotated about the x-axis.

The figure is generated by the code shown on the next slide

x = [0 0 3 3 1.75 1.75 2 2 1.75 1.75 3 4 ...

5.25 5.25 5 5 5.25 5.25 3 3 6 6];

y = [0 .5 .5 .502 .502 .55 .55 1.75 1.75 ...

2.5 2.5 1.5 1.5 1.4 1.4 ...

.55 .55 .502 .502 .5 .5 0];

subplot(1, 2, 1)

plot(x, y)

axis ([-1 7 -1 5]), axis off

title('2-D profile')

facets = 200;

subplot(1, 2, 2)

[xx tth] = meshgrid( x, linspace(0, 2*pi, facets) );

radius = meshgrid( y, 1:facets);

yy = radius .* cos(tth);

zz = radius .* sin(tth);

surf(xx, yy, zz);

shading interp

axis square, axis tight, axis off

colormap copper

lightangle(60, 45)

alpha(0.8)

title('rotated object')

Other 3-D Surface Plot Capabilities

Other 3-D Surface Plot Capabilities 1/2

MATLAB can also create special-purpose plots with the following functions:

alpha (x) sets the transparency of the surfaces: 0 <= x <= 1, where 0 means completely transparent and 1 is opaque.

The options available can be studied with >> help alpha

contour (z) produces a contour plot of the plaid surface defined by z. The options available can be studied with >> help bar3

[x,y, z] = cylinder(n) constructs the meshgrid for a cylinder with n facets in each direction. For more options, see >> help cylinder

[x,y,z] = ellipsoid(n) constructs the meshgrid for an ellipsoid with n facets in each direction. For more options, see >> help ellipsoid

lightangle (az , el) sets the angle of a light source (angles in degrees). For more options, see >> help lightangle

Other 3-D Surface Plot Capabilities 2/2 meshc (x, y, z) makes a mesh plot with contours below. For more options, see >> help meshc.

meshz (x, y, z) makes a mesh plot with vertical line extensions For more options, see >> help meshz.

pie3 (y) makes a 3-D pie chart of the values in y. For more options, see >> help pie3.

[x,y,z] = sphere(n) constructs the meshgrid for a sphere with n facets in each direction. For more options, see >> help sphere.

surfc (x, y, z) makes a surface plot with contours below. For more options, see >> help surfc

surf z (x, y, z) makes a surface plot with vertical line extensions. For more options, see >> help surf z.

waterfall (x, y, z) makes a mesh plot with vertical line extensions only in the x direction. For more options, see» help surf z.

Assembling Compound Surfaces

We can assemble more complex solid bodies by constructing simple surfaces and concatenate the data before submitting it to the rendering machine.

Example: A Solid Disk (from the textbook)

%basic parameters facets = 200; len = 2; radius = 3; radial = [0, radius]; th = linspace(0, 2*pi, facets); along = [0 len]; % build the front face [r1, tth] = meshgrid( radial, th ); x1 = zeros(size(r1)); y1 = r1.*cos(tth); z1 = r1.*sin(tth); % build the curved surface [l, tth] = meshgrid(along, th); x2 = l; y2 = radius*cos(tth); z2 = radius*sin(tth);

% build the back face x3 = len*ones(size(r1)); [r3, tth] = meshgrid(radial(end:-1:1), th); y3 = r3.*cos(tth); z3 = r3.*sin(tth); % assemble and draw the three parts x = [x1 x2 x3]; y = [y1 y2 y3]; z = [z1 z2 z3]; surf(x, y, z); shading interp colormap winter axis equal, axis tight, axis off

The Klein Bottle (textbook, p. 279)

http://www.mathworks.com/products/matlab/demos.html?file=/products/demos/shipping/matlab/xpklein.html

Visualizing Geographic Data (p. 283)

Example (from the textbook):

You have been given two files of data:

atlanta.txt, which presents the streets of Atlanta in graphical form, and

ttimes.txt, which gives the travel times between Atlanta suburbs and the city center.

You have been asked to present these data sets in a manner that will help to visualize and validate the data

Analyzing the Data

First we proceed to determine the nature of the data by opening the files in a text editor and examining their format and content.

1. Determine the file format: The first step is to open the data files in a plain text editor (the MATLAB editor would work fine). The format appears to be consistent with that of a text file delimited by tab characters. Since there are no strings in the file, it should be suitable to be read using MATLAB's built-in dlmread(...) function.

2. Discern the “street map” file content:

2.The table on the next slides shows the first few lines of the file atlanta.txt simplified by omitting certain irrelevant columns. 3.The numbers in columns 3—6 are pairs, the first of the pair being a large negative number, and the second a smaller positive number.4.Assuming that each row of this file is a street segment, these could be the x-y coordinates of the ends of a line.

2.A little thought confirms this guess when we realize that the latitude of Atlanta is —84º 42' relative to the Greenwich meridian, and its longitude is 330 65'— clearly, the values in these columns are 1,000,000 times the latitude and longitude of points within the city, probably each end of street segments.3.Column 7 contains numbers mostly in the range 1—6, which could indicate the type of street. We could explore this idea by coloring each line according to that value

3. Discern the “travel time” file content:

Table 11 .2 shows the first few lines of the file ttimes .txt simplified by omitting certain irrelevant columns.

The same latitude/longitude values occur in columns 4 and 5, but they are not repeated, suggesting that the data in this file are in a different form.

Examining the first two columns, the numbers in column 2 cycle repeatedly from 1 to 75, with column 1 counting the number of cycles up to 75.

Furthermore, the values in column 5 are the same whenever column 1 is the same, and the values in column 4 are the same whenever the value in column 2 matches.

This seems to be much like the plaid that results from a meshgrid(...) function call.

The values in column 6 then become evident—they would be the z values of the plaid, and it seems reasonable to assume that they represent the travel time in minutes.

Street Map Data (part)

… - 84546080.00 33988480.00 - 84558400.00 33995480.00 1.00

… - 84243880.00 33780010.00 - 84249980.00 33800840.00 1.00

… - 84243590.00 33780060.00 - 84249740.00 33800840.00 1.00

… - 84509920.00 33944340.00 - 84517200.00 33958190.00 1.00

… - 84510420.00 33944930.00 - 84516490.00 33957280.00 1.00

… - 84252840.00 33895840.00 - 84247360.00 33899290.00 1.00

… - 84247360.00 33899290.00 - 84240250.00 33903630.00 1.00

… - 84240250.00 33903630.00 - 84216090.00 33911010.00 1.00

… - 84216090.00 33911010.00 - 4203990.00 33913990.00 1.00

% draw the streets raw = dlmread('atlanta.txt'); streets = raw(:,3:7); [rows, cols] = size(streets) colors = 'rgbkcmo'; for in = 1: rows x = streets(in,[1 3])/1000000; y = streets(in,[2 4])/1000000; col = streets(in,5); col(col < 1) = 7; col(col > 6) = 7; plot(x,y,colors(col)); hold on end

% plot the travel times

tt = dlmread('ttimes.txt');

[rows,cols] = size(tt)

for in = 1:rows

r = tt(in, 1); c = tt(in, 2);

xc(r,c) = tt(in, 4)/1000000;

yc(r,c) = tt(in, 5)/1000000;

zc(r,c) = tt(in, 6);

end

surf(xc, yc, zc)

shading interp

alpha(.5)

grid on

axis tight

xlabel('Longitude')

ylabel('Latitude')

zlabel('Travel Time (min)')

view(-30, 45)

User Interaction with PlotsVideo at:

http://www.mathworks.com/products/demos/shipping/matlab/WhatsNewR2008a_GraphicsAndGUIBuilding.html

Brushing (textbook, p. 279)Linking a Plot to the Source Data… more: see video

Brushing Data Items

Brush Tool:

Script:

th = linspace(1, 20*pi, 500) y = sin(th)plot(th,y)

Linking a Plot to the Source Data

Summary

Basic two-dimensional line plots are accomplished by using plot(x,y)

■ Two-dimensional parametric plots are accomplished by using plot(x,y), where both x and y are dependent on another independent variable

■ Three-dimensional line and parametric plots are accomplished by using plot3(x,y,z)

■ Basic three-dimensional surface plots are accomplished by building a plaid using meshgrid(x,y), computing the zz layer as a function of xx and yy, then plotting the surface using mesh(xx, yy, zz) or surf(xx, yy, zz)

■ Parametric surface plots, like parametric line plots, are achieved by building the plaid with two independent variables and making xx, yy, and zz functions of those independent variables

■ Bodies of rotation are a special case of parametric surface plots where one of the independent variables is an angle with values between 0 and 2 (radians).

textbook: chapter 11 part : 3d-plotting cse 1010

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