How to plot a point in the intersection of two functions?

Question

I'm trying to plot the intersection of a Sum function with a horizontal line and automatically highlight the intersection points. The code below shows my knowledge of Mathematica and for different values of "d" I am calculating with Solve the intersection points (ex p1,p2 and p3) and manually typing the value in the Epilog option. I tried Point[{p1,1}] but it doesn't work. I also tried it with MeshFunctions -> {g[#] - f[#] &}, Mesh -> {{0}}, MeshStyle -> PointSize[Large] but I get an error "functions must be pure functions". I would also like to be able to define the ranges of "d" and "pc" as {d, {min,max,incr}} instead of a list. Thanks in advance.

np = 2
f[pc_] := 1
q[d_, pc_] := (pc/(100*0.48)) * 
  Sum[((Pi/4)*(d - (2*n*0.48))^2), {n, 1, np}]
p[d_] := Sum[Pi*(d - (2*n - 1)*0.48), {n, 1, np}]
p1 = Solve[q[20, pc]/p[20] == f[pc], pc]
p2 = Solve[q[10, pc]/p[10] == f[pc], pc]
p3 = Solve[q[8, pc]/p[8] == f[pc], pc]
g[d_, pc_] := q[d, pc]/p[d]
Plot[{Evaluate[Table[g[d, pc], {d, {8, 10, 20}}]], f[pc]}, {pc, 0, 
  50}, PlotRange -> All, AxesLabel -> {"%", "li/lp"}, 
 FrameLabel -> {Style["pc", 12, Bold], Style["li/lp", 12, Bold]}, 
 PlotLabels -> {"d=8", "d=10", "d=20"}, PlotTheme -> "Scientific", 
 GridLines -> Automatic, PlotLabel -> "Razão comprimentos",
 Epilog -> {PointSize[0.02], Point[{10.6053, 1}], Point[{23.6134, 1}],
    Point[{31.2426, 1}]}]

Answer 1

np = 2;
f[pc_] = 1;

dmin = 8;
dmax = 24;
dincr = 4;

q[d_, pc_] = (pc/(100*12/25))*
    Sum[((Pi/4)*(d - (2*n*12/25))^2), {n, 1, np}] //
   Simplify;

p[d_] = Sum[Pi*(d - (2*n - 1)*12/25), {n, 1, np}] //
   Simplify;

g[d_, pc_] := q[d, pc]/p[d];

pt[d_] = {pc, f[pc]} /. Solve[g[d, pc] == f[pc], pc][[1]];

Plotting

Plot[{
  Evaluate[Table[g[d, pc], {d, dmin, dmax, dincr}]],
  f[pc]}, {pc, 0, 50},
 PlotRange -> All,
 FrameLabel ->
  (Style[#, 12, Bold] & /@ {"pc", "li/lp"}), 
 PlotLabels -> (StringForm["d=``", #] & /@
    Range[dmin, dmax, dincr]),
 PlotTheme -> "Scientific",
 GridLines -> Automatic,
 PlotLabel -> "Razão comprimentos",
 Epilog -> {PointSize[0.02],
   Tooltip[Point[#], N[#[[1]]]] & /@
    (pt /@ Range[dmin, dmax, dincr])}]

EDIT: To vary the level of the line

Manipulate[
 pt[d_] = {pc, fpc} /. Solve[g[d, pc] == fpc, pc][[1]];
 Plot[{
   Evaluate[Table[g[d, pc], {d, dmin, dmax, dincr}]],
   fpc}, {pc, 0, 50},
  PlotRange -> All, 
  FrameLabel -> (Style[#, 12, Bold] & /@
     {"pc", "li/lp"}),
  PlotLabels -> (StringForm["d=``", #] & /@
     Range[dmin, dmax, dincr]),
  PlotTheme -> "Scientific",
  GridLines -> Automatic,
  PlotLabel -> "Razão comprimentos",
  Epilog -> {PointSize[0.02],
    Tooltip[Point[#], N[#[[1]]]] & /@
     Select[(pt /@ Range[dmin, dmax, dincr]), #[[1]] <= 50 &]}],
 {{fpc, 1}, 0.5, 5.5, 0.1, Appearance -> "Labeled"},
 SynchronousUpdating -> False]

Answer 2

Method 1

Here we plot the intersection of three functions g[8,pc], g[10,pc], g[20,pc] respect tof[pc].

For example,if we want to find the intersetion of g[8,pc] and y=f[x] when we plot g[8,pc], we can set the MeshFunction of g[8,pc] to y-f[x],that is MeshFunctions -> {#2 - f[#1] &}, Mesh -> {{0}}

np = 2;
f[pc_] := 1;
q[d_, pc_] := (pc/(100*0.48))*
   Sum[((Pi/4)*(d - (2*n*0.48))^2), {n, 1, np}];
p[d_] := Sum[Pi*(d - (2*n - 1)*0.48), {n, 1, np}];
g[d_, pc_] := q[d, pc]/p[d];
plot = Plot[Evaluate[Table[g[d, pc], {d, {8, 10, 20}}]], {pc, 0, 50}, 
  MeshFunctions -> {#2 - f[#1] &}, Mesh -> {{0}}, 
  MeshStyle -> {PointSize[Large], Automatic}, PlotRange -> All, 
  AxesLabel -> {"%", "li/lp"}, 
  FrameLabel -> {Style["pc", 12, Bold], Style["li/lp", 12, Bold]}, 
  PlotLabels -> {"d=8", "d=10", "d=20"}, PlotTheme -> "Scientific", 
  GridLines -> Automatic, PlotLabel -> "Razão comprimentos"]

Show[plot, Plot[f[pc], {pc, 0, 50}]]

Method 2

On the other hand, we can also set three pure functions Table[Function[pc, g[d, pc] - f[pc] // Evaluate], {d, {8, 10, 20}}]

when we plot f[pc] to get the three intersections.

np = 2;
f[pc_] := 1;
q[d_, pc_] := (pc/(100*0.48))*
   Sum[((Pi/4)*(d - (2*n*0.48))^2), {n, 1, np}];
p[d_] := Sum[Pi*(d - (2*n - 1)*0.48), {n, 1, np}];
g[d_, pc_] := q[d, pc]/p[d];

Plot[f[pc], {pc, 0, 50}, 
 MeshFunctions -> 
  Table[Function[pc, g[d, pc] - f[pc] // Evaluate], {d, {8, 10, 20}}],
  Mesh -> {{0}}, MeshStyle -> Directive[PointSize[Large], Red]]

Animation

Clear[np, f, q, p, g, plot];
np = 2;
f[pc_] := 1;
q[d_, pc_] := (pc/(100*0.48))*
   Sum[((Pi/4)*(d - (2*n*0.48))^2), {n, 1, np}];
p[d_] := Sum[Pi*(d - (2*n - 1)*0.48), {n, 1, np}];
g[d_, pc_] := q[d, pc]/p[d];
plot[c_] := 
  Plot[Evaluate[Table[g[d, pc], {d, {8, 10, 20}}]], {pc, 0, 50}, 
   MeshFunctions -> {#2 - f[#1] &}, Mesh -> {{c}}, 
   MeshStyle -> {PointSize[Large], Automatic}, PlotRange -> All, 
   AxesLabel -> {"%", "li/lp"}, 
   FrameLabel -> {Style["pc", 12, Bold], Style["li/lp", 12, Bold]}, 
   PlotLabels -> {"d=8", "d=10", "d=20"}, PlotTheme -> "Scientific", 
   GridLines -> Automatic, PlotLabel -> "Razão comprimentos"];
Manipulate[Show[plot[c], Plot[f[pc] + c, {pc, 0, 50}]], {c, -1, 2}, 
 ControlPlacement -> Bottom]