Dual-Grid Graph Paper With Mathematica?

Question

Is there a slick way to generate the dual-grid graphs such as you can see on pages 7, 9, and 10 of this article, or this one?

I've searched, but found nothing. Thanks in advance...

Answer 1

By definition, a (real) "coordinate" is a map from a region into the Real numbers. To show a coordinate, we usually fix specific values, such as $\ldots, -3, -2, \ldots, 1, 2, 3, \ldots$, and graph the inverse images (level sets) of these values. We will want to distinguish different coordinates by varying the graphical representations of these level sets.

We should encapsulate these elements--coordinate functions, graphical attributes, and discrete sets of values--in some convenient data structure. The simplest to use is a list. For example, here are descriptions of a pair of Cartesian coordinates and a pair of Polar coordinates:

coordinates = {
   {#1 &, Gray, Range[-3, 3]},
   {#2 &, Gray, Range[-3, 3]},
   {Norm[{#1, #2}] &, Directive[Darker[Red], Dashed], Range[-3, 3]},
   {ArcTan[#1, #2] &, Directive[Darker[Red], Dashed], Range[-\[Pi] + \[Pi]/7, \[Pi], 2 \[Pi]/7]}
   };

Here's the "slick" part: merely map ContourPlot over this list and wrap it all in Show to overlay the whole bunch.

Show[ContourPlot[#1[x, y], {x, -3, 3}, {y, -3, 3}, Contours -> #3, 
    ContourStyle -> #2, ContourShading -> None] & @@@ coordinates]

(The mashup along the negative $x$ axis is a Mathematica problem with contouring ArcTan; it's not the result of this particular method.)

In a similar way we can reproduce the Arnold figures using the very calculations they are intended to illustrate. Here is (most) of one of them:

basis = {{1, 1}, {-1, 1}}; (* New basis *)
dual = Inverse[basis];     (* New coordinate functions *)
green = ConstantArray[Green, 8]~Join~{Black}~Join~ConstantArray[Green, 8];
coordinates = {
   {(dual.{#1, #2} // First) &, green, Range[-8, 8]},
   {(dual.{#1, #2} // Last) &, green, Range[-8, 8]},
   {#1 &, LightGray, Range[-5, 5]},
   {#2 &, LightGray, Range[-5, 5]}
   };
Show[ContourPlot[#1[x, y], {x, -5, 5}, {y, -5, 5}, Contours -> #3, 
    ContourStyle -> #2, ContourShading -> None] & @@@ coordinates]

This is not limited to two coordinate systems: coordinates can be a list of any length, allowing display of any number of coordinate functions simultaneously. And it is direct: you do not have to work out the inverse coordinate functions (which would be required with a solution based on Plot or ParametricPlot, for instance).

Answer 2

You can accomplish things like this by defining your own functions to generate descriptions for Graphics here is a simple function that generates a grid over two arbitrary basis vectors:

 genGrid[v1_, v2_, l_: 10] := {
 Line /@ Table[v1 n1 + v2 n2, {n1, -l, l}, {n2, {-l, l}}],
 Line /@ Table[v1 n1 + v2 n2, {n2, -l, l}, {n1, {-l, l}}]
 }

You can then use it along with other graphics options to style your resulting figure:

 Graphics[{
 {Black, genGrid[{1, 0}, {0, 1}]},
 {Gray, genGrid[{1, 1}, {2, 1}, 14]}
 }, PlotRange -> 3, Frame -> True, PlotRangeClipping -> True]

To add things such as arrows and anotations you can use Arrow and Text.

Answer 3

A possible starting point:

With[{θ = π/3}, 
 DeleteCases[
             ParametricPlot[{{x, y}, RotationTransform[-θ][Sqrt[2] {x, y}]},
                            {x, -5, 5}, {y, -5, 5}, PlotRange -> {{-5, 5}, {-5, 5}}],
             _Polygon, ∞]]

Answer 4

Here's a start using just simple plotting functions.

With[{m = 5},
    Plot[
        {Table[i x + j, {j, -2 m, 2 m}, {i, {-1, 1}}], Sequence[-x, x]}, {x, -m, m}, 
        AspectRatio -> 1, PlotRange -> {-m, m}, PlotStyle -> {Green, {Thick, Black}}, 
        AxesStyle -> Thick, Ticks -> None, GridLines -> ({Range[-m, m], Range[-m, m]}), 
        GridLinesStyle -> Directive@LightGray
    ]
]

You can get fancier and gain more control using graphics primitives.