VectorScale Explanation

Question

I'm having great difficulty understanding VectorScale. Here are the definitions from the documentation.

With the explicit specification VectorScale -> {unitlen,aratio,sfun} each field vector is constrained to fit in a vector box oriented in the direction of the field, centered at the location of the vector.

The unitlen is given as a fraction of the diagonal of the overall bounding box and is used as a local scale for vector boxes, aratio is the aspect ratio for vector boxes, sfun is a scaling function that determines the width of vector boxes.

I've used it with VectorPlot and have gotten the desired result, but I still just can't understand the above definitions. I've even read some authors saying "don't worry about how this works, but here is how you get a good picture." I just think I need to be a little more ready than that to explain things to my students.

Is it possible to draw a picture, maybe just a few arrows, and show their vector boxes surrounding them? And then explain the numbers used in the VectorScale option and how they relate to the size and shape of the vector boxes?

Answer 1

The option VectorPoints determines how many vector boxes there are. The plot domain is subdivided into a grid, whose grid points span the plot domain in each direction. Equal-size rectangular boxes surround each grid point so that the boxes are adjacent and tile the plot range (ignoring any padding).

Here is a picture with VectorPoints -> 9 (the default is 15):

VectorPlot[{x, y}, {x, -1, 1}, {y, -1, 1},
 VectorPoints -> 9, 
 GridLines -> With[{n = 8}, {#, #} &@Range[-1 - 1/n, 1 + 1/n, 2/n]]]

Here is a utility function I'll use to explore various settings:

vsplot[unitlen_, aratio_: Automatic, sfun_: Automatic, vf_: {x, y}] :=
  VectorPlot[vf, {x, -1, 1}, {y, -1, 1},
  VectorPoints -> 9, 
  GridLines -> With[{n = 8}, {#, #} &@Range[-1 - 1/n, 1 + 1/n, 2/n]], 
  VectorScale -> {unitlen, aratio, sfun}]

unitlen

With unitlen = 1/9 (middle), the longest vector just fits in the box and the rest are scaled proportionally. Here the length is 1/9 the overall width of the domain, where the 9 is the number of PlotPoints. If using the default, the PlotPoints are 15 and the corresponding unitlen should be 1/15. Other settings are scaled versions of this, either exceeding the boundaries of the boxes or staying well within the box.

GraphicsRow@Table[vsplot[l], {l, {1/5, 1/9, 1/12}}]

aratio

A numeric setting makes all arrowheads an absolute constant size. With Scaled[a], the size of the arrowhead depends on the length of the vector. @xzczd points out, aratio probably should be read as standing for "arrowhead ratio" and not "aspect ratio" as stated in the documentation for VectorScale.

GraphicsRow@Table[vsplot[1/9, a], {a, {1/2, 1, 2}}]

GraphicsRow@Table[vsplot[1/9, Scaled[a]], {a, {1/2, 1, 2}}]

sfun

The length of the vector is proportional to the value of sfun, which according to the documentation is to be

a function of x, y, fx, fy, Norm[{fx,fy}]

I'm not sure why one would want to change the vector length, other than to make them all the same length or perhaps use things like ArcTan[#5] & or Log[#5] & to attenuate the rapid growth of the length of the vectors of a vector field.

The columns below each show the same vector field scaled according to one of the arguments. The last pair are roughly inversely proportional to the length of the vector, which give somewhat misleading views of the actual vector fields.

GraphicsGrid@
 Table[Show[vsplot[1/9, Automatic, f, vf], 
   PlotLabel -> Row[{sfun, "=", f, ", ", {fx, fy}, "=", vf}], 
   ImageSize -> 200],
  {f, {#1 &, #2 &, #3 &, #4 &, 1/(1 + #5) &}},
  {vf, {{x, y}, {-y + x, y + x}}}]

Answer 2

Get intuition from this:

Manipulate[
     VectorPlot[{Sin[x], Cos[y]}, {x, -Pi, Pi}, {y, -Pi, Pi}, 
     VectorScale -> {mysize, aratio}],
         {mysize, {Small, Medium, Large}},
          {aratio, .1, 1, .1}]

Here, mysize merely states how long a vector is (by gving a size of a bounding box). Here aratio ($0<aratio<1$) merely states the ratio of the size of the arrowhead to the length of the arrow line. My version (10.0.0) does not support sfun.

Or try this to get a box:

    Manipulate[
 VectorPlot[{-1 - x^2 + y, 1 + x - y^2},
  {x, -3, 3},
  {y, -3, 3},
  PlotRange -> {{-3, 3}, {-3, 3}},
  StreamPoints -> Coarse,
  StreamScale -> Full,
  VectorScale -> {unitlen, aratio, sfun},
  StreamStyle -> 
   Arrowheads[{{0.03, Automatic, 
      Graphics[{FaceForm[Yellow], EdgeForm[Directive[Thin, Red]], 
        Rectangle[1/unitlen {-sfun/2, -aratio/2}, 
         1/unitlen {sfun/2, aratio/2}]}]}}]],
 {unitlen, .1, 1},
 {aratio, .1, .5},
 {sfun, .1, 10}]