MATLAB by rudra pratap

6.3 3-D Plots
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6.3.4 Vector field and volumetric plots
One of the most crucial needs of visualization in scientific computation is for data
that is essentially volumetric, i.e., defined over a 3-D space. For example, we may
have temperature or pressure defined over each (x, y, z) triple in a bounded 3-D
space. How do we display this data graphically? If we have a function z = f ( x, y)
defined over a finite region of the xy-plane, we can display z or f as a 3-D surface.
But we have f(x, y, z)! So, we need a 4-D hypersurface. That is the basic problem.
We display volumetric data by slicing it along several planes in 3-D and plotting
the data on those planes, either with graded color maps or with contours. Such
displays are still an area of active research. However, we can do fairly well with the
tools currently available. One of the most common applications is in the area of
3-D vector fields. A vector field defines a vector quantity as a function of the space
variables ( x, y, z) . Fortunately, in this case we can display the data (the vector) with
an arrow drawn at each (x, y, z) triple, with the magnitude of the vector represented
by the length of the arrow, and the direction represented by the orientation of the
arrow. This concept is used extensively in dynamical systems in various ways.
MATLAB provides extensive tools for visualizing vector fields and volumetric
data. Unfortunately, these tools are beyond the scope of this book_ Therefore, we
merely mention the tools here and give examples of only the "most likely to be
used" tools.
Plotting vector fields
The plotting functions available in MATLAB for vector field visualization include
quiver, quiver3, stream2, stream3, streamline, streamtube, streamribbon,
streamslice, streamparticles, coneplot, divergence, curl, etc.
If u(x, y) and v(x, y) are given as vector components in x- and y-directions,
respectively, then the vector field can be easily drawn with quiver (qui ver3 in
3-D). An example of quiver appears on page 188 in the table of 2-D plots. The
stream functions are an extension of the same concept; they draw streamlines or
trajectories from user-specified points in the specified vector field. This suite of
functions has been a welcome addition in MATLAB (version 6 onward).
The function streamline is useful for drawing solution trajectories in 2-D and
3-D vector fields defined by ODEs. You need not solve the ODEs!
Example: Let
x y + x - x(x 2 + y 2 ) and
Y
-x + Y - y(x2 + y2).
These two ODEs define a vector field (u = x and v = y). Let u::; use streamline to
draw a few solution trajectories starting from various points in the xy-plane (initial
conditions in the phase plane). The general syntax of streamline is
streamline (x,y,z, u,v,w, xO ,yO,zO)