ListVectorDensityPlot plots outside the data in a very wrong way

Question

I am plotting a vector field data set mydata where the background color indicates the vector orientation and its opacity for the vector magnitude. I use ColorFunction -> Function[{x, y, vx, vy, n}, Hue[ArcTan[vx, vy]/(2 π), n]] to get the rainbow-like color, in which n signifies the norm slot.

I conceived of two methods to plot it

• plot the original mydata with the option ColorFunctionScaling -> {False, False, False, False, True} that presumably normalizes the norm to be within $[0,1]$
• normalize vectors with the maximal norm in mydata to get mydatanormalized and then just plot it with the option ColorFunctionScaling -> False

mydata = ArrayReshape[#, {21, 21, 2}] &@ImportString[#, "List", "LineSeparators" -> " "] &@"-0.1053 0.1218 -0.1049 0.1505 -0.1375 0.1595 -0.1487 0.1927 -0.1451 0.2315 -0.1215 0.2768 -0.04844 0.3219 0.03322 0.3269 0.07211 0.2912 0.08432 0.2518 0.1131 0.2321 0.1724 0.2324 0.2272 0.2463 0.2196 0.2704 0.1544 0.2909 0.1152 0.2789 0.1279 0.2363 0.1452 0.1922 0.1434 0.1616 0.109 0.1527 0.09839 0.1137 -0.1213 0.136 -0.1614 0.1428 -0.1732 0.1841 -0.1728 0.2311 -0.1156 0.2995 -0.04907 0.2926 -0.1552 0.1407 -0.3534 -0.01118 -0.4592 -0.07593 -0.4765 -0.08614 -0.4852 -0.06397 -0.4895 -0.007719 -0.3949 0.06222 -0.1099 0.1005 0.2227 0.1192 0.2996 0.1942 0.1889 0.2662 0.1568 0.2377 0.1702 0.1815 0.1622 0.1482 0.122 0.1418 -0.1739 0.1208 -0.1922 0.1645 -0.1975 0.2172 -0.1333 0.291 -0.1719 0.1549 -0.4066 -0.1208 -0.3151 -0.1887 0.0631 -0.2345 0.2625 -0.3867 0.235 -0.5172 0.1542 -0.5666 0.08391 -0.545 -0.06646 -0.4208 -0.3682 -0.1554 -0.4933 0.08776 -0.06011 0.07589 0.3353 0.07367 0.2538 0.2236 0.1863 0.2226 0.192 0.159 0.1715 0.1322 -0.2041 0.1351 -0.22 0.1858 -0.173 0.2642 -0.2533 0.04761 -0.3845 -0.2177 0.1426 -0.1476 0.4323 -0.4011 0.01581 -0.6387 -0.3758 -0.4889 -0.4999 -0.2379 -0.5455 -0.1084 -0.6077 -0.1333 -0.5658 -0.3121 -0.2859 -0.5633 -0.1657 -0.4986 -0.4871 0.007491 -0.2228 0.1038 0.3369 -0.002831 0.2738 0.1983 0.2105 0.1894 0.2056 0.1288 -0.2311 0.1454 -0.218 0.2266 -0.2534 0.05052 -0.3389 -0.2386 0.3436 -0.07197 0.3474 -0.5235 -0.2275 -0.4495 -0.113 0.2096 0.2023 0.532 0.3474 0.5644 0.4112 0.5295 0.405 0.488 0.1526 0.4604 -0.4097 0.3655 -0.6207 -0.1181 -0.2755 -0.5577 -0.4798 0.004167 -0.1684 0.08243 0.3685 -0.02977 0.2602 0.1968 0.2284 0.1393 -0.253 0.1604 -0.2334 0.1419 -0.3543 -0.2607 0.271 0.01476 0.4151 -0.4706 -0.1293 -0.2393 0.129 0.5738 0.1148 0.5778 -0.09508 0.505 -0.116 0.5551 -0.004217 0.5867 0.1707 0.5495 0.423 0.4366 0.5743 0.3617 0.0175 0.5135 -0.618 0.1333 -0.3555 -0.5048 -0.5089 0.1697 0.08054 -0.06534 0.358 0.048 0.2506 0.1613 -0.2582 0.1718 -0.308 -0.1391 -0.05642 -0.07989 0.5825 -0.1667 0.008296 -0.3498 0.08211 0.6251 -0.1607 0.4409 -0.4303 0.4275 -0.3143 0.5939 -0.1327 0.6513 0.03406 0.6559 0.1883 0.6363 0.2908 0.5872 0.4234 0.4293 0.6435 0.2429 0.03715 0.5649 -0.579 -0.01969 -0.4309 -0.2641 -0.3364 0.195 0.3361 -0.16 0.2943 0.1476 -0.2673 0.1044 -0.3049 -0.2994 0.3626 0.1816 0.3727 -0.472 0.001924 0.4332 -0.1696 0.4304 -0.5591 0.2757 -0.4592 0.4918 -0.3519 0.5183 -0.2659 0.5616 -0.09348 0.6183 0.1314 0.5989 0.361 0.513 0.4824 0.4538 0.5533 0.2773 0.6248 0.2195 -0.2896 0.5875 -0.4982 -0.3974 -0.4863 0.2603 0.1407 -0.1899 0.3439 0.05582 -0.2786 -0.0324 -0.1941 -0.2044 0.5641 0.1394 0.1785 -0.2592 -0.05808 0.6511 -0.5027 0.08997 -0.6128 0.3006 -0.5299 0.3401 -0.5251 0.3802 -0.3475 0.5655 -0.01929 0.6477 0.2287 0.6177 0.3663 0.5083 0.5362 0.3325 0.5904 0.272 0.6829 0.03091 0.2214 0.5812 -0.5499 -0.08476 -0.4961 0.1056 -0.06765 -0.06919 0.359 -0.07163 -0.2761 -0.1554 -0.08473 -0.03167 0.6094 0.01855 0.07798 0.1012 -0.1947 0.4449 -0.6302 -0.03477 -0.6369 0.1938 -0.6219 0.13 -0.6111 0.2534 -0.303 0.4264 0.09962 0.3719 0.4239 0.3184 0.5455 0.3559 0.5647 0.234 0.648 0.131 0.6571 -0.03595 0.4996 0.3753 -0.4662 0.2026 -0.51 -0.00885 -0.1832 0.04382 0.3393 -0.171 -0.2615 -0.2213 -0.04926 0.07117 0.6149 0.00897 0.024 0.2857 -0.2641 0.2578 -0.6493 -0.1399 -0.6557 0.02893 -0.6399 -0.08074 -0.6387 -0.0007441 -0.3608 0.09465 0.007346 -0.0009028 0.3769 -0.08724 0.6345 0.03261 0.6116 0.06239 0.6586 -0.03337 0.6447 -0.1409 0.5834 0.2681 -0.3904 0.2822 -0.5222 0.009543 -0.2105 0.06824 0.3155 -0.2184 -0.2472 -0.2323 -0.0873 0.05752 0.5964 0.1293 0.04292 0.2784 -0.2628 0.2395 -0.5928 -0.2914 -0.6531 -0.1266 -0.5859 -0.2606 -0.5652 -0.3251 -0.4119 -0.3033 -0.08226 -0.3658 0.3187 -0.4094 0.6001 -0.2264 0.5953 -0.1647 0.6324 -0.1912 0.6217 -0.2794 0.5702 0.3252 -0.4316 0.1705 -0.4969 0.1579 -0.1811 -0.02403 0.3108 -0.2095 -0.245 -0.1963 -0.1483 -0.08369 0.4783 0.3497 0.2037 0.1327 -0.2708 0.4133 -0.4239 -0.4292 -0.6278 -0.2823 -0.5278 -0.3669 -0.4059 -0.5047 -0.2513 -0.5934 0.01406 -0.6254 0.3336 -0.5451 0.4879 -0.3919 0.5248 -0.3732 0.5617 -0.3413 0.6158 -0.3397 0.4355 0.488 -0.5753 -0.0458 -0.4104 0.3839 -0.1143 -0.2249 0.3265 -0.1465 -0.2622 -0.133 -0.1644 -0.2877 0.1814 0.5119 0.4972 0.03027 -0.2991 0.5835 -0.2053 -0.2953 -0.4827 -0.4978 -0.5046 -0.4433 -0.3522 -0.5418 -0.1633 -0.6184 0.04957 -0.6333 0.2286 -0.5891 0.3474 -0.5458 0.438 -0.4952 0.4755 -0.5012 0.6741 -0.06038 0.01007 0.4247 -0.6218 -0.0444 -0.306 0.4313 -0.0005853 -0.4014 0.3409 -0.07037 -0.2828 -0.0945 -0.1467 -0.3714 -0.1334 0.2989 0.537 0.2901 0.00376 0.3698 -0.293 0.2969 -0.1096 -0.5334 -0.3542 -0.6048 -0.3511 -0.5872 -0.1966 -0.6331 -0.02812 -0.6585 0.125 -0.6499 0.2403 -0.6243 0.3452 -0.608 0.6147 -0.312 0.4136 0.3586 -0.5743 0.03703 -0.3848 0.3885 -0.2612 0.01957 0.1751 -0.3491 0.321 -0.05854 -0.2716 -0.1112 -0.19 -0.258 -0.1576 -0.21 0.01572 0.5461 0.5296 0.2337 -0.2536 0.4973 -0.246 0.1792 0.08305 -0.4828 -0.007954 -0.6817 -0.07472 -0.702 -0.005609 -0.707 0.1391 -0.6944 0.3642 -0.6015 0.6275 -0.253 0.3941 0.2408 -0.4934 0.1225 -0.4678 0.2694 -0.2555 0.389 -0.117 -0.4085 0.316 -0.1561 0.2688 -0.114 -0.234 -0.1335 -0.2529 -0.1575 -0.09523 -0.3728 -0.2317 0.03112 0.07002 0.5562 0.4624 0.3282 -0.2308 0.4643 -0.4138 0.355 -0.009134 -0.03503 0.2844 -0.2779 0.4195 -0.3189 0.4808 -0.1992 0.368 0.038 -0.1098 0.1978 -0.6147 0.1328 -0.3873 0.3639 -0.2082 0.4786 -0.2768 -0.2929 0.2105 -0.2902 0.2958 -0.121 0.2281 -0.1462 -0.2065 -0.1259 -0.2296 -0.1659 -0.2005 -0.2228 -0.05204 -0.385 -0.2771 0.04309 -0.1131 0.5339 0.3871 0.4707 0.1428 0.4324 -0.3477 0.4414 -0.533 0.3754 -0.5441 0.3051 -0.5711 0.2708 -0.5883 0.2652 -0.407 0.3585 -0.1193 0.5688 -0.2342 0.377 -0.331 -0.2804 0.1355 -0.3338 0.3 -0.1523 0.2203 -0.1758 0.2069 -0.1301 -0.1755 -0.1277 -0.1917 -0.1584 -0.2112 -0.1947 -0.1623 -0.2553 -0.01455 -0.3841 -0.209 -0.159 -0.3922 0.292 -0.1193 0.5134 0.1762 0.5547 0.2226 0.5624 0.1554 0.5717 0.09772 0.5905 0.01863 0.5886 -0.1891 0.4332 -0.4181 0.01418 -0.2568 -0.3562 0.1765 -0.3024 0.2786 -0.1863 0.2019 -0.202 0.1944 -0.1618 0.1725 -0.1201 -0.1219 -0.1427 -0.1668 -0.1434 -0.1696 -0.1812 -0.1793 -0.2191 -0.154 -0.2562 -0.00981 -0.3511 0.00569 -0.3594 -0.2114 -0.2023 -0.4123 -0.01746 -0.4789 0.08845 -0.4855 0.09717 -0.4873 0.009834 -0.4405 -0.1557 -0.2497 -0.3138 0.06111 -0.3378 0.2535 -0.2536 0.221 -0.2233 0.1696 -0.226 0.1755 -0.1816 0.1603 -0.1414 0.1192 -0.138 -0.09769 -0.1143 -0.109 -0.1534 -0.1482 -0.1577 -0.1453 -0.1907 -0.137 -0.232 -0.139 -0.2565 -0.07987 -0.2917 0.03069 -0.3354 0.1005 -0.3555 0.11 -0.3516 0.1107 -0.3425 0.1407 -0.3325 0.1894 -0.3114 0.2091 -0.2807 0.1743 -0.2626 0.1346 -0.2582 0.1387 -0.2337 0.1524 -0.1891 0.1351 -0.1592 0.1026 -0.1522 0.1058 -0.1212"; 

and the codes

mydatanormalized = With[{max = Max@Map[Norm[#] &, mydata, {2}]}, mydata/max];

MatrixPlot[Map[Norm[#] &, mydata, {2}], PlotLegends -> Automatic]

ListVectorDensityPlot[mydata, DataRange -> {{-1.2, 1.2}, {-1.2, 1.2}}, 
 PlotRange -> Full, PlotRangeClipping -> False,
 VectorColorFunction -> None, 
 VectorScaling -> "Linear",
 VectorRange -> All, VectorStyle -> Black, 
 ColorFunctionScaling -> {False, False, False, False, True}, 
 ColorFunction -> 
  Function[{x, y, vx, vy, n}, 
   Hue[ArcTan[vx, vy]/(2 π), n]], 
 MaxRecursion -> 2]

ListVectorDensityPlot[mydatanormalized, DataRange -> {{-1.2, 1.2}, {-1.2, 1.2}}, 
 PlotRange -> Full, PlotRangeClipping -> False,
 VectorColorFunction -> None, 
 VectorScaling -> "Linear",
 VectorRange -> All, VectorStyle -> Black, 
 ColorFunctionScaling -> False, 
 ColorFunction -> 
  Function[{x, y, vx, vy, n}, 
   Hue[ArcTan[vx, vy]/(2 π), n]], 
 MaxRecursion -> 2]

However, there is a big difference as shown below. In both plots, spurious color regions (exaggeratedly opaque, I think) show up at the boundary. They are actually outside the data coordinate range, $[-1.2,1.2]$. And such distortion looks to suppress the true data in the first plot. However, the original data is nothing special at all in the norm near the boundary (one can easily see by MatrixPlot the norm as shown.)

One can restrict the plot range to the original data coordinates

ListVectorDensityPlot[mydatanormalized, DataRange -> {{-1.2, 1.2}, {-1.2, 1.2}}, 
 PlotRange -> {{-1.2, 1.2}, {-1.2, 1.2}}(*,PlotRangeClipping\[Rule]False*), 
 VectorColorFunction -> None, VectorScaling -> "Linear", VectorRange -> All, 
 VectorStyle -> Black, ColorFunctionScaling -> False, 
 ColorFunction -> Function[{x, y, vx, vy, n}, Hue[ArcTan[vx, vy]/(2 π), n]], 
 MaxRecursion -> 2]

for Method 2 and it gets rid of the spurious region (not shown here), although this is not helpful for Method 1. Here, one has to comment out PlotRangeClipping -> False. However, one needs PlotRangeClipping -> False for good reasons, otherwise

1. one cannot place labels outside the plotting box (I care about this the most)
2. arrows at the boundary are cut off, like only half of an arrow is shown

So we still need some remedy/workaround here. How should one correctly plot the data without these weird distortions? (I came up with an idea but weirdly got stuck. Solving that one would do as well.)

Only plot norm: Method 1: Method 2:

Answer 1

I think for both methods, a better solution would be to use RegionFunction to restrict the plotting region:

ListVectorDensityPlot[mydata, DataRange -> {{-1.2, 1.2}, {-1.2, 1.2}}, 
 PlotRange -> Full, 
 PlotRangeClipping -> False, 
 RegionFunction -> (-1.2 <= #1 <= 1.2 && -1.2 <= #2 <= 1.2 &),
  VectorColorFunction -> None, 
  VectorScaling -> "Linear",
  VectorRange -> All, VectorStyle -> Black, 
  ColorFunctionScaling -> {False, False, False, False, True}, 
  ColorFunction -> 
    Function[{x, y, vx, vy, n}, 
      Hue[ArcTan[vx, vy]/(2 π), n]], 
  MaxRecursion -> 2]

produces

and

ListVectorDensityPlot[mydatanormalized, DataRange -> {{-1.2, 1.2}, {-1.2, 1.2}},
 PlotRange -> Full, 
 PlotRangeClipping -> False, 
 RegionFunction -> (-1.2 <= #1 <= 1.2 && -1.2 <= #2 <= 1.2 &),
  VectorColorFunction -> None, 
  VectorScaling -> "Linear",
  VectorRange -> All, VectorStyle -> Black, 
  ColorFunctionScaling -> False, 
  ColorFunction -> 
    Function[{x, y, vx, vy, n}, 
      Hue[ArcTan[vx, vy]/(2 π), n]], 
  MaxRecursion -> 2]

produces

In either case, you can remove the thin border around the plotted region using RegionBoundaryStyle -> None.

Answer 2

It is difficult to understand how ListVectorDensityPlot[] works in this case. But for my opinion for some presentation it could be better to use something less rainbow, for instance,

ListVectorDensityPlot[mydatanormalized, 
 DataRange -> {{-1.2, 1.2}, {-1.2, 1.2}}, 
  PlotRange -> Full, PlotRangeClipping -> False,
  VectorColorFunction -> None, 
  VectorScaling -> "Linear",
  VectorRange -> All, VectorStyle -> Black, 
  ColorFunctionScaling -> False, 
  ColorFunction -> 
    Function[{x, y, vx, vy, n}, 
      Hue[ArcCos[vy/Sqrt[vx^2 + vy^2]]/(2 \[Pi]), .5 + n/2]], 
  MaxRecursion -> 2]