plotting – 3-dimensional. 3d plots versus 2d plots 3-dimensional plots, in contrast to...

Plotting – 3-Dimensional

3D Plots versus 2D Plots

3-dimensional plots, in contrast to 2-dimensional ones, has a third axis (often called the z-axis).

We plot both 2D and 3D charts on a flat surface (screen, paper, etc.). In 3D charts, the third dimension gives the visual impression of depth.

One dimensional data

One dimensional data (a single vector) can be plotted either in 2D or 3D.

One dimensional data

cont= {'Asia', 'Europe', 'Africa', 'N.America', 'S.America'};

pop= [3332; 696; 694; 437; 307];

ti= 'World population 1992';

pie(pop, cont);

title(ti);

figure;

pie3(pop, cont);

title(ti);

One dimensional data

pop= [3332; 696; 694; 437; 307];

subplot(2,2,1); bar(pop);

subplot(2,2,2); bar3(pop);

subplot(2,2,3); barh(pop);

subplot(2,2,4); bar3h(pop);

Two dimensional data

Two dimensional data (a pair of vectors, typically in the form y=f(x)) can be plotted either in 2D or 3D.

Plotting multiple vector pairs is also possible, such as

y1=f(x), y2=g(x), y3=h(x)

Each pair is usually plotted with a different color for easy discriminitation.

Two dimensional data

x= linspace(0, 6*pi, 200);

y1= sin(x);

y2= cos(x);

y3= 0.5*sin(x).*sin(12*x);

plot(x,y1, x,y2, x,y3);

legend('sin(x)', 'cos(x)', 'sin(x)sin(12x)/2');

figure;

% 3D ribbon plot with 0.4 ribbon thickness

ribbon(x', [y1',y2',y3'], 0.4);

legend('sin(x)', 'cos(x)', 'sin(x)sin(12x)/2');

3D graph viewpoint

The angle from which an observer sees a 3D plot can be adjusted using the VIEW function.

view(azimuth, elevation) Azimuth is the horizontal rotation, and elevation is

the vertical angle from the x-y plane (both in degrees).

Azimuth revolves about the z-axis, with positive values indicating counter-clockwise rotation of the viewpoint.

Positive values of elevation correspond to moving above the object; negative values move below.

View command formats

Different ways to use the view command: Most used:

view(azimuth, elevation) Look at the plot from the angle of a point in

cartesian space:view([x, y, z])

Default 3D view (azimuth: -37.5°, elevation: 30°):view(3)

Convert to 2D view (look from the top):view(2)

View command examples

view(100,15) view([5,5,5])

view(3) view(2)

Three dimensional data

We almost always need to use a 3D chart to represent three dimensional data.

Three dimensional data is often available in one of two formats: Three vectors, each holding x, y, and z

components of points which, when interconnected, form a trajectory (path) in space

Three matrices, each holding x, y, and z components of points which, when interconnected, form a surface in space

Trajectory plots in 3D

Trajectory plots in 3D are based on three vectors. They are generally characterized by one free and

two dependent, or no free and three dependent vectors.

[y, z]= f(x)

or

[x, y, z]= g(t)

Trajectory plots in 3D

z= linspace(0,4,1000);

x= z.*cos(exp(z));

y= z.*sin(exp(z));

plot3(x,y,z,'r')

grid on

Trajectory plots in 3D

t= linspace(0,10,4000);

z= -sin(t);

y= sqrt(1-z.^2).*cos(20*t);

x= sqrt(1-z.^2).*sin(20*t);

comet3(x,y,z)

Surface plots in 3D

Surface plots are based on three matrices. They are generally characterized by two

independent (free) matrices and one dependent matrix.

Z= f(X, Y) The independent matrices generally contain a

rectangular grid of coordinates on the X-Y plane. If we are only interested in the shape, rather than

the coordinates of the surface, then we can build the plot by using only the Z matrix.

Building the X-Y grid (1)

Assume that a surface chart is to be drawn in the range –2x2 and 2y8.

We want data to be plotted for all integer x in the range, and only even numbers in y.

This implies a rectangular grid whose x-coordinates are [-2, -1, 0, 1, 2].

The y-coordinates of this grid are[2, 4, 6, 8].

Building the X-Y grid (2)

We can represent this 4x5= 20-point grid by putting the x-coordinates of every point into a X matrix, and the corresponding y-coordinates into a Y matrix.

The resulting matrices would then look like this:

21012

21012

21012

21012

X

88888

66666

44444

22222

Y

Building the X-Y grid (3)

A function named MESHGRID is very useful in automatically creating the X and Y matrices out of x and y vectors of the grid.

x= [-2 -1 0 1 2];

y= [2 4 6 8];

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

Building the Z matrix

Once the X-Y grid is ready, building the actual surface coordinates into a matrix is a matter of writing an equation of X and Y:

Z= 0.4 - 0.4*X.^2 - 0.1*Y.^2 + 1.2*Y;

surf(X,Y,Z)

Surface plots

We often want a large number of points in the grid for seeing a better rendered surface.

When x- and y-axis grids are identical, we do not need to create separate x and y vectors.

% Egg carrier

u= linspace(-10, 10, 50);

[X,Y]= meshgrid(u, u);

Z= sin(X) + sin(Y);

surf(X, Y, Z)

Surface plot types

Contour (contour lines where the surface crosses certain z-levels)

Surf (color-filled surface patches) Mesh (displays only colored lines between Z data

points) Special shapes (spheres, cylinders, fills, etc.)

PEAKS function

PEAKS is a function of two variables, obtained by translating and scaling Gaussian distributions.

It creates a nice looking surface, which is useful for demonstrating 3D plots such as MESH, SURF, CONTOUR, etc.

For the surface charts on tnhe following slides, we will assume X, Y and Z matrices generated by the following code:

u= linspace(-3, 3, 50);[X, Y]= meshgrid(u, u);Z= peaks(X, Y);

2D CONTOUR plots

Contour plots are probably the only chart types that can successfully represent three dimensional data in 2D.

They project the cutting points of particular z-levels onto a flat surface.

The plot on the right top is a default contour plot, while the one at the bottom is a filled contour of 20 levels.contour(X, Y, Z); figure;contourf(X, Y, Z, 20);

3D CONTOUR plots

3D contour plots display contour data in their actual intersection surfaces in space.

The plot on the right top is a default 3D contour plot, while the one at the bottom is a 3D contour plot of 20 levels.contour3(X, Y, Z); figure;contour3(X, Y, Z, 20); grid off

SURF plot - faceted

This is the default SURF plot. Actual data points are at places where the x and y lines intersect.

surf(X, Y, Z);

shading('faceted');

SURF plot - flat

In this SURF plot, the lines in x and y directions are removed to show only colored rectangles.

surf(X, Y, Z);

shading('flat');

SURF plot - interpolated

In this SURF plot, the lines in x and y directions are removed, and additionally data is interpolated to display continuous colors.

surf(X, Y, Z);

shading(‘interp');

SURFC plot

In the SURFC plot, we have the SURF plot with additional contour lines projected onto the X-Y plane.

surfc(X, Y, Z);

SURFL plot

In the SURFL plot, we have the SURF plot with highlights from a light source.

surfl(X, Y, Z);

MESH plot

MESH plots give the colored parametric mesh defined by the given matrices.

mesh(X,Y,Z);

MESH plot – HIDDEN OFF

MESH plots by default hide the shapes behind any drawn mesh area. To display those, you can turn the hidden mode OFF.

mesh(X,Y,Z);

hidden off

MESHC plot

MESHC plots, in addition to the mesh plot, the contour lines projected to the X-Y plane.

meshc(X,Y,Z);

MESHZ plot

MESHZ plots give the colored parametric mesh defined by the given matrices. The small plot is with hidden mode turned off.

meshz(X,Y,Z);

WATERFALL plot

WATERFALL is similar to MESHZ, but without any lines drawn in the column direction.

waterfall(X,Y,Z);

Special types: FILL and FILL3% Five arm star, with 72° between armsfor k=1:10 a= (k-1)*36; if mod(k,2)==1 % arm x(k)= sin(a/180*pi); y(k)= cos(a/180*pi); else % trough x(k)= sin(a/180*pi)*0.382; y(k)= cos(a/180*pi)*0.382; endendsubplot(3,1,1)fill(x,y,'y'); axis('square')subplot(3,1,2)% Set color to the point’s y heightfill(x,y,y); axis('square')z=y*0.5;subplot(3,1,3)fill3(x,y,z,y); axis('square')

Special types: Sphere, etc.

There are ready functions for some known 3D shapes: SPHERE ELLIPSOID CYLINDER

[x,y,z]= ellipsoid(1,1,1,4,6,3);

surfl(x,y,z);

colormap copper

axis equal

Advanced formatting: Handles

For more advanced formatting of charts, we can obtain a handle for the plot while creating it, and modify plot properties through handle operations.

All chart creation commands return handles when we assign the command to a variable. Examples:

h_plot= surf(X, Y, Z);

h_title= title(‘Peaks in 3D'); A get command to the handle will list all properties

associated with that item.

get(h_plot);

get(h_title);

Advanced formatting: Handles

You can see the settings associated with an item, and their present values using the GET command:

>> get(h_title)BackgroundColor = noneColor = [0 0 0]EdgeColor = noneEraseMode = normalEditing = offExtent = [0 0 0 0]FontAngle = normalFontName = HelveticaFontSize = [10]FontUnits = pointsFontWeight = normalHorizontalAlignment = centerLineStyle = -LineWidth = [0.5]Margin = [2]Position = [-1.24135 -1.61775 21.2632]Rotation = [0]String = Peaks in 3DUnits = dataInterpreter = texVerticalAlignment = bottom

Advanced formatting: Handles

The SET command can be used to modify properties of the item associated with the handle:

set(h_plot, 'LineStyle', ':');

set(h_title, 'FontSize', 16, 'FontWeight', 'bold')