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How to perfectly align combined 2D and 3D graphics?

Posted 9 years ago
1 Reply
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I need a loose replication of this plot from Wikipedia. My best attempt so far seems to have some unwanted padding around the vector plot, on the bottom layer. Am I missing anything to properly align the two coordinate systems? Is there an alternative implementation?

Define a potential field and its gradient:
f = Function[{x, y}, -(Cos^2 + Cos^2)^2];
g = Function[{x, y}, Evaluate@D[f[x, y], {{x, y}}]];
Make 2D plot for the texture 
tx = VectorDensityPlot[g[x, y], {x, -Pi/2, Pi/2}, {y, -Pi/2, Pi/2},
          Frame -> False, PlotRangePadding -> None,   ColorFunction -> "DarkRainbow", VectorPoints -> 30]
Combine 3D filed plot with 2D vector plot of its gradient via texture
Show[Plot3D[f[x, y], {x, -Pi/2, Pi/2}, {y, -Pi/2, Pi/2},
          ColorFunction -> "DarkRainbow", PlotStyle -> Opacity[.8], MeshStyle -> Opacity[.3]],
                  Polygon[{{-Pi/2, -Pi/2, -4}, {Pi/2, -Pi/2, -4}, {Pi/2, Pi/2, -4}, {-Pi/2, Pi/2, -4}},
                  VertexTextureCoordinates -> {{0, 0}, {1, 0}, {1, 1}, {0, 1}}]},   Lighting -> "Neutral"]]

Posted 9 years ago
Adding PlotRange -> All to VectorPlot gets rid of the padding. I'm not quite sure why PlotRange -> Full adds the padding. I would expect the padding for the arrowheads to be controlled by ImagePadding so that 
Graphics[{Arrowheads[.2], Arrow[{{0, 0}, {1, 0}, {1, 1}}]}]
wouldn't clip the arrowhead. 

How about a different approach, not using Texture. We can extract graphics primitives from VectorPlot and put them literally in the same coordinate system used by Plot3D. This guarantees perfect alignment.
Show[Plot3D[f[x, y], {x, -Pi/2, Pi/2}, {y, -Pi/2, Pi/2},   
         PlotStyle -> FaceForm[],   MeshStyle -> Directive[Red, Thick, Opacity[.5]],BoundaryStyle->None], 
    Graphics3D[{Arrowheads[Small], Blue, Arrow@PadRight[#, {Length@#, 2, 3}, -4.] &@
         Cases[VectorPlot[g[x, y], {x, -Pi/2, Pi/2}, {y, -Pi/2, Pi/2}, VectorPoints -> 20],
                Arrow[l_] :> l, -1]}], Boxed -> False]
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