
Exploring a vector field
-------------------------

In this example, we create a vector field from the gradient of a scalar
field and explore it interactively. This example shows you how to do some
operations similar to the previous example, but interactively, using the
filters and modules. This approach requires a better knowledge of Mayavi
and the VTK filters than the previous example, in which `numpy` is used
for most of the operations. The big gain is that the resulting
visualization can be explored interactively.

First, let us create the same scalar field as the previous example:. We
open Mayavi and enter the following code in the Python shell::

    from mayavi import mlab
    import numpy as np

    def V(x, y, z):
        """ A 3D sinusoidal lattice with a parabolic confinement. """
        return np.cos(10*x) + np.cos(10*y) + np.cos(10*z) + 2*(x**2 + y**2 + z**2)
    X, Y, Z = np.mgrid[-2:2:100j, -2:2:100j, -2:2:100j]
    mlab.contour3d(X, Y, Z, V)

As in the previous example, we can change the color map and the values
chosen in the isosurfaces.

We want to take the gradient of the scalar field, to create a vector
field. To do this we are going to use the `CellDerivatives` filter, that
takes derivatives of the data located in the cells (that is, between the
points, see :ref:`data-structures-used-by-mayavi`). For this, we first
need to interpolate the data from the points where it is located to the
cells, using a `PointToCellData` filter. We can then apply our
`CellDerivatives` filter, and then a `CellToPointData` filter to get
point data back.

To visualize the vector field, we can use a `VectorCutPlane` module. The
resulting vectors are too large, and we can go to the `Glyph` tab,
(and the `Glyph` tab in this tab), to reduce the scale factor to 0.2. The
vector field is still too dense, therefore we go to the `Masking` tab to 
enable masking, mask with an `on ratio` of 6 (one arrow out of 6 is
masked) and turn off the random mode.

.. image:: example_vector_cut_plane.png
    :scale: 50


To have nice colors, we also changed the color map of the vector field by
going to the `Colors and legend` node just above the `VectorCutPlane`,
and choosing a look up table **in the VectorLUT** tab, as there can be
different color maps for vector data and scalar data.

Unlike the previous example, we can play with all the parameters in the
dialog box, like masking, or select `color_by_scalar` in the `Glyph` tab,
to display the value of the potential. We can also move the cut plane
used to display the vectors by dragging it.

Now that we have a 3D vector field, we can also use Mayavi to integrate
the trajectory of a particle in it. For this we can use the streamline
module. It displays trajectories starting from the vertices of a seed
surface. We choose (in the `Seed` tab) a `Point Widget` as a seed. We can
then move the seed point by dragging it along in the 3D scene. This
allows us to explore the trajectories in the potential created by the
initial scalar field. In our case, all the trajectories end up in a local
potential minimum, and moving the seed point along lets us see in which
minimum each point will fall into, in other world the basin of attraction
of each local minimum.

.. image:: example_streamline.jpg

