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Re: Help: how to disp. 3-D vectors

  • To: mathgroup at
  • Subject: Re: Help: how to disp. 3-D vectors
  • From: villegas
  • Date: Wed, 6 Oct 93 08:01:07 -0500

> I am trying to display three data sets, a 3-D vector (v) and
> two scalars (a and b), all three given for a set of points in
> 3-D space. My data looks something like:

> data = { {{x0,y0,z0}, {vx0,vy0,vz0}, a0, b0},
>          {{x1,y1,z1}, {vx1,vy1,vz1}, a1, b1},
>          ...
>          {{xi,yi,zi}, {vxi,vyi,vzi}, ai, bi},
>          ... }

> I would like to have a 3-D display with an arrow at each point
> {xi, yi, zi}, the arrow being parallel with vi and its length
> and colour representing the values of ai and bi respectively. 

> I have tried using ListPlotVectorField3D which almost does the
> job, but not quite. Am I missing something, or perhaps just asking
> for too much?

New in 2.2 is the package Graphics`Arrow`, which defines a new graphics
primitive called 'Arrow' (analogous to the familiar 'Line', 'Polygon',
et al) using some PostScript features introduced in 2.2.  After
loading the package


we can look up the usage of Arrow:

? Arrow

Arrow[start, finish, (opts)] is a graphics primitive representing an
   arrow starting at start and ending at finish.

Since Arrow takes the starting and ending point of the arrow, we can
apply a function across your list of arrow-specifications to produce
the Arrow objects.  If we use Mathematica's pure functions, then we
can represent the different parts of a general arrow-specification
{{xi,yi,zi}, {vxi,vyi,vzi}, ai, bi} with Slot symbols #1 through #4:

#1 = {xi, yi, zi}
#2 = {vxi,vyi,vzi}
#3 = ai
#4 = bi

Using this notation, the desired Arrow object would be

Arrow[#1, #1 + #3 * #2/(#2 . #2)]   (* assuming #2 is not a unit vector *)

and the color, as determined by element 4, bi, is put into a list with the
Arrow.  Say bi is a value intended for the Mathematica Hue function:

{ Hue[#4], Arrow[#1, #1 + #3 * #2/(#2 . #2)] }

If we want, we can give options inside the Arrow to control features of
the arrowhead.  The options are described in the Technical Report:  _Guide
to Standard Mathematica Packages_, and are summarized in the usage messages
for the package.

   Let's make a sample list of arrow-specifications and show how to
form and render the Arrows.  In this example, I used direction vectors,
so I did not need to divide #2 by the norm (#2 . #2) as above.  But your
data list might need the more general form above.


vectorSpecs =
  Table[{{Cos[t], Sin[t]}, {Cos[t], Sin[t]}, 1/3 Sin[8 t],
    1/2 + 1/3 Sin[8 t]}, {t, 0, (2Pi - Pi/50), Pi/50}];

arrows =
  Apply[ {GrayLevel[#4],
    Arrow[#1, #1 + #3 * #2, HeadLength->1/50] }&, vectorSpecs, {1} ];

Show[Graphics[arrows, Axes->True, AspectRatio->Automatic]]

If you have a color screen, then here is a somewhat prettier variation:

vectorSpecsColor =
  Table[{{Cos[t], Sin[t]}, {Cos[t], Sin[t]}, 1/3 Sin[8 t],
    2/3 + 1/3 Sin[8 t]}, {t, 0, (2Pi - Pi/50), Pi/50}];

arrowsColor =
  Apply[ {Hue[#4],
    Arrow[#1, #1 + #3 * #2, HeadLength->1/50] }&, vectorSpecsColor, {1} ];

Show[Graphics[arrowsColor, Axes->True, AspectRatio->Automatic]]


Robby Villegas

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