3-D Plotting and Surfaces

plot3(x,y,z) is plot with a third coordinate, and it accepts the same line specifications and name-value arguments. It draws a curve or a set of points in space, for instance a particle’s path. zlabel, zlim and grid on complete the axes.

Not run here. The chart shows a helix that winds three times around the z-axis. axis equal keeps a circle round in the x-y projection.

Source: MATLAB documentation: plot3

A surface needs the height at every point of a grid. [X,Y] = meshgrid(x,y) repeats the vector x along the rows and y down the columns, so both matrices have numel(y) rows and numel(x) columns. ndgrid is the n-dimensional counterpart with the opposite orientation: its first output varies along the first dimension, so the result has numel(x) rows. Functions for plotting (surf, mesh, contour) expect the meshgrid orientation, where rows follow y; functions that interpolate on gridded data (griddedInterpolant) use ndgrid.

The heatflow plate is 1 m wide and 0.5 m high. With its top edge held at the temperature 100*sin(pi*x) degrees Celsius and the other edges at 0, the steady temperature is known in closed form:

\[ T(x,y) = 100\,\sin(\pi x)\,\frac{\sinh(\pi y)}{\sinh(\pi/2)}. \]

The second example below uses it in place of the numerical solution that the heatflow project computes with a sparse system, so that the figures have an exact reference.

Octave 11.3.0, a coarse grid of 5 by 3 points (the code uses fprintf and mat2str, which exist in both MATLAB and Octave):

Octave 11.3.0 output (MATLAB lays the numbers out with a different column format):

The matrix T has one row per y value and one column per x value, and its last row (the top edge, y = 0.5) is the heated edge with the peak 100 at x = 0.5.

Source: MATLAB documentation: meshgrid

A 3-D surf is created in the 3-D view and a contour in 2-D, both with parula as the default colormap (the default since R2014b).

Not run here. The first tile is a surface that rises from 0 along three edges to a ridge of height 100 along the middle of the top edge. The second tile is the plate seen from above with filled bands every 20 degrees and four labelled black contour lines at 20, 40, 60 and 80 degrees, which are arches around the hot middle of the top edge.

The figure draws the same temperature field from the Octave-computed data, as a contour map: the shades are the field and the four black lines are the contours at 20, 40, 60 and 80 degrees. It is an original drawing, not a MATLAB export.

Source: MATLAB documentation: surf

Octave 11.3.0, checking the sizes and the peak on the coarse 5-by-3 grid (code run as written below):

This continues the earlier snippet (the variables X, Y and T come from it). Octave 11.3.0 output:

The surface keeps Z as a matrix of the same size (3 by 5), the peak is at the middle of the top edge, and view(3) gives the default 3-D view, an azimuth of -37.5 and an elevation of 30 degrees. In MATLAB get(hs,"type") returns "surface" as well, and hs is a Surface object, not a number as in Octave.

quiver(X,Y,U,V) draws an arrow with components U and V at each point (X,Y). The natural vector field of the heatflow plate is the heat flux, minus the conductivity times the gradient of the temperature. gradient(T,hx,hy) returns the numerical derivatives with respect to the column and row directions of the matrix, hx and hy being the grid spacings.

Octave 11.3.0 output of the qy matrix (rounded to one decimal):

The vertical heat flux is negative everywhere: heat flows from the hot top edge (large y) toward the cold bottom edge. The three rows are y = 0, 0.25 and 0.5. The values at the boundary rows are one-sided differences, which are less accurate. The chart shows arrows that point downward and are longest under the middle of the top edge. quiver3, streamslice and streamline are the 3-D and streamline versions.

Source: MATLAB documentation: quiver

Not run here. The surface is smoothly coloured from blue (0 degrees) to red (100 degrees) with a headlight that makes the ridge look three-dimensional. shading interp and lighting gouraud are what turn the faceted grid into a smooth shape.

Pick a colormap that suits the data: a sequential map for temperature, a diverging one for deviations from a reference, and never one that depends on colour alone for people with colour-vision deficiency (a lightness ramp such as gray or parula survives monochrome printing better than jet, the pre-R2014b default).

Source: MATLAB documentation: colormap

fsurf(f,[xmin xmax ymin ymax]) plots a function handle of two variables with adaptive sampling, and fcontour and fmesh are its siblings. They replace ezsurf and ezcontour, which the documentation marks as not recommended, and need no grid. They were added in R2016a.

Not run here. The first chart is the same surface as above and the second shows the four labelled contour lines. The handle must be vectorized (use .* and ./), because it is called with arrays.

Source: MATLAB documentation: fsurf

[X,Y,Z] = sphere(n) returns the coordinates of a unit sphere as (n+1)-by-(n+1) matrices, and [X,Y,Z] = cylinder(r,n) the coordinates of a surface of revolution whose radii are given by r, with 2 rows (bottom and top) and n+1 columns. Both are for passing to surf, scaling and translating.

Octave 11.3.0 (continuing the gradient snippet; the code uses only functions that exist in MATLAB):

Octave 11.3.0 output:

The sphere has 21 by 21 points for n = 20, and the cylinder call with the radii [1 0.5] gives a cone frustum whose bottom radius is 1 and top radius 0.5.

Source: MATLAB documentation: sphere

The exact messages were not run here; the causes are documented behaviour.