MeshFunctions of intersections shows wrong point

Question

I am trying to plot the intersection of a function n and a line b/2, as well as showing the line b in the same plot. For this purpose, I have tried using MeshFunctions, but it not only displays the intersection of the function and the line, but also the point on b lying directly above the intersection.

n[n0_, b_, t_, r_] := (n0*b)/(n0 + (b - n0) E^(-r*t));
Manipulate[
 Plot[{n[n0, b, t, r], b, b/2}, {t, 0, 20}, 
  PlotStyle -> {{Pink, Thick}, {Purple, Medium}, {Blue, Dashed, Thin}},
  MeshFunctions -> {n[n0, b, #, r] - (b/2) &}, Mesh -> {{0.}}, 
  MeshStyle -> {Black, PointSize[0.02]}],
 {{n0, 10}, 1, 100, 1, Appearance -> "Labeled"}, 
 {{b, 100}, 0, 100, 5, Appearance -> "Labeled"}, 
 {{r, 0.4}, 0, 5, 0.1, Appearance -> "Labeled"}
 ]

How do I get around this problem, and why does it happen? From looking at similar questions (in particular Show intersection points between a curve and a line and Understand MeshFunctions with Intercepts) I understand this might not be the best method to do this. I am relatively new to Mathematica, so any help on how to plot intersections like this is appreciated!

Answer 1

Epilog works best in such cases. Just solve for the intersection and then use Epilog:

Manipulate[
 pt = {t /. Quiet@First@NSolve[n[n0, b, t, r] == b/2, t], b/2}; 
 Plot[{n[n0, b, t, r], b, b/2}, {t, 0, 20}, 
  PlotStyle -> {{Pink, Thick}, {Purple, Medium}, {Blue, Dashed, 
     Thin}}, Epilog -> {PointSize[Medium], Point[pt]}], {{n0, 10}, 1, 
  100, 1, Appearance -> "Labeled"}, {{b, 100}, 0, 100, 5, 
  Appearance -> "Labeled"}, {{r, 0.4}, 0, 5, 0.1, 
  Appearance -> "Labeled"}]

Answer 2

An alternative is to use InfiniteLine[] instead to depict your asymptote and intersecting line. This has the advantage of simplifying the expression in MeshFunctions:

Manipulate[Plot[n[n0, b, t, r], {t, 0, 20}, Mesh -> {{b/2}}, MeshFunctions -> {#2 &}, 
                MeshStyle -> Directive[Black, PointSize[0.02]], 
                PlotStyle -> Directive[Pink, Thick], 
                Prolog -> {{Directive[Purple, Thickness[Medium]],
                            InfiniteLine[{0, b}, {1, 0}]},
                           {Directive[Blue, Dashed, Thin],
                            InfiniteLine[{0, b/2}, {1, 0}]}}],
           {{n0, 10}, 1, 100, 1, Appearance -> "Labeled"},
           {{b, 100}, 0, 100, 5, Appearance -> "Labeled"},
           {{r, 0.4}, 0, 5, 0.1, Appearance -> "Labeled"}]

Answer 3

A few additional work-arounds:

1. Post-process Plot output to remove extra points: Use

Plot[...] /. Point -> Point @* First

in the first argument of Manipulate to get:

2. Instead of post-processing outside Plot use the option

 DisplayFunction -> (# /. Point -> Point@*First &)

inside Plot to get the same result.

3. Use a function for the value of the option MeshStyle and use the replacement rule above in that function:

MeshStyle -> ({Directive[Black, PointSize[0.02]], # /. Point -> Point@*First} &)

same picture

4. Remove b and b/2 from the function list in the first argument of Plot and show them using GridLines:

Manipulate[Plot[n[n0, b, t, r], {t, 0, 20}, PlotStyle -> Directive[Thick, Pink],
   MeshFunctions -> {#2 &}, Mesh -> {{b/2}}, 
  MeshStyle -> Directive[Black, PointSize[0.02]], 
  GridLines -> {None, {{b, 
      Directive[Opacity[1], Purple, Thick]}, {b/2, 
      Directive[Blue, Dashed, Thin]}}}], {{n0, 10}, 1, 100, 1, 
  Appearance -> "Labeled"}, {{b, 100}, 0, 100, 5, 
  Appearance -> "Labeled"}, {{r, 0.4}, 0, 5, 0.1, 
  Appearance -> "Labeled"}]