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 –2  x  2 and 2  y  8. 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:

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= [ ]; y= [ ]; [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= *X.^ *Y.^ *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 arms for 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; end subplot(3,1,1) fill(x,y,'y'); axis('square') subplot(3,1,2) % Set color to the point’s y height fill(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 = none Color = [0 0 0] EdgeColor = none EraseMode = normal Editing = off Extent = [ ] FontAngle = normal FontName = Helvetica FontSize = [10] FontUnits = points FontWeight = normal HorizontalAlignment = center LineStyle = - LineWidth = [0.5] Margin = [2] Position = [ ] Rotation = [0] String = Peaks in 3D Units = data Interpreter = tex VerticalAlignment = 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')