How to rotate text in user coordinate system?

Question

Applying Rotate to text rotates it relative to the Cartesian page coordinates. I want it rotated relative to the current user coordinate system. As a simple illustration (of a more general problem), consider the following manipulation. When the AspectRatio is 1, all is good; but otherwise not. (I've included a plot for an apect ratio of 1/2, to illustrate.) How can I keep the text on the line?

Manipulate[With[{f = x |-> (x + 0.4)^2, x0 = 1, x1 = 2},
  Module[{y0 = f[x0], y1 = f[x1]},
   Graphics[{
     Line[{{x0, y0}, {x1, y1}}],
     Text[
      Rotate["test", ArcTan[1, (y1 - y0)/(x1 - x0)]], {(x0 + x1)/2, (y0 + y1)/2}, {0, -2}]
     },
    PlotRange -> {{0, 6}, {0, 6}},
    Axes -> True, AspectRatio -> a,
    AxesOrigin -> {0, 0}]]],
 {{a, 1}, 1/2, 1}]

More generally, it seems that Mathematica has much to learn from TikZ about making annotation easy for users.

Answer 1

Well, if @kglr doesn't know a direct way to make Text[]/Inset[] work, I guess there isn't one. First, the 4th argument to Text[] makes the use of Rotate[] unnecessary. Second, the 4th argument doesn't work any better than Rotate[]. Third, the angle depends on both the PlotRange and the AspectRatio, so the solution in the comments and @anon's answer work only if the PlotRange has equal horizontal and vertical lengths.

For decades I did this as a two step process (or forced the graphics to work with AspectRatio -> Automatic). But @b3m2a1's function getBoxSize[] allows us to combine the steps into a single function. The poorly named withAspectRatio[] calculates the correction factor (also incorrectly called $aspectRatio) and stores it in a global variable, which may appear in the Graphics[..] code; the graphics will be returned with the correction factor replacing the global variable. The user may use a different variable name and pass this as the second argument to withAspectRatio[].

(* @b2m2a1, https://mathematica.stackexchange.com/a/166772/46569 *)
getBoxSize[c_Cell] := {#[[1]], Total@#[[2 ;;]]} &@
   First@FrontEndExecute@GetBoundingBoxSizePacket[c];
getBoxSize[c_CellObject] := getBoxSize[NotebookRead[c]];
getBoxSize[b_BoxObject] := 
  getBoxSize[
   Cell[BoxData@NotebookRead[b], "Output", PageWidth -> Infinity, 
    ShowCellBracket -> False, CellMargins -> {{0, 0}, {0, 0}}]];
getBoxSize[e_] := 
  getBoxSize[
   Cell[BoxData@ToBoxes[e], "Output", PageWidth -> Infinity, 
    ShowCellBracket -> False, CellMargins -> {{0, 0}, {0, 0}}]];

withAspectRatio // ClearAll;
withAspectRatio // Attributes = {HoldRest};
withAspectRatio[g_, a_Symbol : $aspectRatio] := Block[{a},
   a = 1;
   With[{boxes = 
      Show[g, Frame -> None, Ticks -> None, ImagePadding -> 0] // 
       MakeBoxes[#, StandardForm] &},
    a = First@
      Ratios[getBoxSize@Cell[BoxData@boxes, "Output"]/
        Flatten[Differences /@ PlotRange@g]];
    g
    ]];

Manipulate[
 withAspectRatio[
  Plot[{Exp[x], Exp[a] (x - a + 1)}
   , {x, a - 1, a + 1}
   , Epilog -> {
      Text[
       "my tangent line is parallel to this",
       {a, Exp[a]}, {0, 2}, {1, $aspectRatio * Exp[a]}]}
   , AspectRatio -> 2.3 + a]
  ]
 , {{a, 0}, -2, 1}]

Also for decades, I became used to this and assumed that the 4th argument was supposed to behave this way. This question made the scales fall, and I don't see why this behavior is at all convenient. Furthermore I think the community should consider it a bug and bug WRI to fix it.

The OP's code refactored:

Manipulate[
 With[{f = x |-> (x + 0.4)^2, x0 = 1, x1 = 2},
  Module[{y0 = f[x0], y1 = f[x1]},
   withAspectRatio[
    Graphics[{Line[{{x0, y0}, {x1, y1}}],
      Text[
       "test test test", {(x0 + x1)/2, (y0 + y1)/2}, {0, -2}, {x1 - 
         x0, $aspectRatio (y1 - y0)}]}
     , PlotRange -> {{0, 6}, {0, 6}}, Axes -> True, AspectRatio -> a, 
     AxesOrigin -> {0, 0}]
    ]
   ]]
 , {{a, 1}, 1/2, 2}]

Answer 2

Manipulate[With[{f = Function[x, (x + 0.4)^2], x0 = 1, x1 = 2}, 
   Module[{y0 = f[x0], y1 = f[x1]}, 
    Graphics[{Line[{{x0, y0}, {x1, y1}}], 
      Text[Rotate["test", ArcTan[x1 - x0, a*(y1 - y0)]], 
       {(x0 + x1)/2, (y0 + y1)/2}, {0, -2*a}]}, 
     PlotRange -> {{0, 6}, {0, 6}}, Axes -> True, 
     AspectRatio -> a, AxesOrigin -> {0, 0}]]], 
  {{a, 1}, 1/2, 1}]

The two argument form of ArcTan handles getting the rotation angle into the correct quadrant.

The AspectRatio effectively scales the y coordinate. Therefore, another correction is needed in the ArcTan function evocation. For similar reasons I also scaled the offset parameter.

Of course, I may have missed something.

Answer 3

Manipulate[
 With[{f = x |-> (x + 0.4)^2, x0 = 1, x1 = 2}, 
  Module[{y0 = f[x0], y1 = f[x1]}, 
   Graphics[{Arrowheads[{{Automatic, .5, 
        Graphics[{Text["test", {0, 0}, {0, -2}]}]}}], 
     Arrow[{{x0, y0}, {x1, y1}}]}, PlotRange -> {{0, 6}, {0, 6}}, 
    Axes -> True, AspectRatio -> a, AxesOrigin -> {0, 0}]]], 
{{a, 1}, 1/2, 1}]