Two-way locator & slider controls

Question

I was wondering whether it was possible to have a two way control system, whereby one could click and drag both the locator (which would in turn move the sliders), AND click & drag the sliders (which in turn would move the locator). Here is an image of what I would like to achieve:

The code below controls the locator only:

f[x_] := x^2;
Manipulate[n, {n, 1, 20, 1}]
Manipulate[With[{x0 = Clip[p[[1]], {0., 10}]}, 
Plot[f[x], {x, 0, 10}, ImagePadding -> 20, LabelStyle -> (FontFamily -> "Ariel"),
Epilog -> {PointSize[Large], Red,
Tooltip[Point[#], #] &@{x0, f[x0]}}]], {{p, {0, 0}}, Locator, Appearance -> None},
AppearanceElements -> None]

for which I must credit Michael E2 and Timothy Wofford :-)

Answer 1

Manipulate does not remember the previous value of its dynamic variables. So we save their values in another variable. Then whenever the Manipulate begins another cycle, we compare the new values to the saved values.

To lock the sliders together we need a function y=f[x] and its inverse x=g[y]. You have to supply both of these relationships; Mathematica will not give you an inverse function.

(* Relationships constraining sliders *)
f[x_] := x^2;
g[y_] := Sqrt[y];

(* External variables for remembering previous values *)
xSave = 0;
ySave = 0;
pSave = {0, 0};

Manipulate[
 (* compare values and handle changes appropriately *)
 If[xx != xSave, xSave = xx; p[[1]] = xx; yy = f[xx]];
 If[yy != ySave, ySave = yy; xx = g[yy]];
 If[p != pSave, pSave = p; xx = p[[1]]; yy = f[xx]];

 (* The manipulated object *)
With[{x0 = Clip[p[[1]], {0., 10}]}, 
  Plot[f[x], {x, 0, 10}, ImagePadding -> 20, 
   LabelStyle -> (FontFamily -> "Ariel"), 
   Epilog -> {PointSize[Large], Red, 
    Tooltip[Point[#], #] &@{x0, f[x0]}}]],

 (* The controls *)
 {xx, 0, 10},
 {yy, 0, 100},
 {{p, {0, 0}}, Locator, Appearance -> None}, 
 AppearanceElements -> None]