How to draw a Line on a Plot, with a small scale, using Prolog or Epilog

Question

If I want to draw a line on a Plot I can use Prolog or Epilog. However, this goes wrong if the scale of the plot is small. Here is a minimum working example.

Plot[Sin[x], {x, 0, 3 π}, Frame -> True, 
 Prolog -> {Red, Line[{{4, -1}, {4, 1}}]}]
Plot[10^-13 Sin[x], {x, 0, 3 π}, Frame -> True, 
 Prolog -> {Red, Line[{{4, -1}, {4, 1}}]}]

No vertical line is drawn in the second case. This is the problem. I guess that it is due to the small scale and the line is not drawn to this resolution. If I change the scale of the vertical line it works

Plot[10^-13 Sin[x], {x, 0, 3 π}, Frame -> True, 
 Prolog -> {Red, Line[{{4, -10^-13}, {4, 10^-13}}]}]

However this requires me to find the scale of the plot first and put this into the line. I am working with the strains from the Finite Element Method within a Dynamic and I can't predict the scale in advance. I thought of this

Show[Plot[10^-13 Sin[x], {x, 0, 3 π}, Frame -> True],
 Graphics[{Red, Line[{{4, -1}, {4, 1}}]}]
 ]

However this does not work either. Is it possible to use Scaled or Offset? I failed when I tried. Thanks for suggestions.

Answer 1

(Too long for a comment.)

Yes, you can use Scaled[], as I did in this answer. To wit,

{Plot[Sin[x], {x, 0, 3 π}, Frame -> True, 
      Prolog -> {Red, Line[{Scaled[{0, 1}, {4, 0}], Scaled[{0, -1}, {4, 0}]}]}], 
 Plot[1*^-13 Sin[x], {x, 0, 3 π}, Frame -> True, 
      Prolog -> {Red, Line[{Scaled[{0, 1}, {4, 0}], Scaled[{0, -1}, {4, 0}]}]}]}
// GraphicsRow

Or, you can use InfiniteLine[]:

Plot[1*^-13 Sin[x], {x, 0, 3 π}, Frame -> True, 
     Prolog -> {Red, InfiniteLine[{4, 0}, {0, 1}]}]

which should yield the same picture as the one on the right.