Locators, lists of variable length, all in Manipulate

Question

I'm trying to implement a complicated Manipulate[] interface. I'm not really interested in working code, but for those interested, I would like the program to

• allow you to drag around a variable number of locators,
• and have it output the motion of an object traveling from one locator to the next in straight line segments at constant velocity (just the function of time; no actual animation needed), and graph of the motion (i.e. connect the locators on the screen via ParametricPlot[]).
• One last thing: the times at which the object passes each of the locators are needed to specify a unique solution. But rather than manipulating the times themselves, I'd like to have input fields for the average velocity in each segment, and have the program infer the times.

Here is a schematic that describes how I've tried to accomplish this:

However, I have been stumped as to how to implement this. Manipulate[] doesn't seem to have enough flexibility, and Dynamic[] seems to come with its own issues. All I'd like to know is if this is a dead end, or if not, some rough outline of how I could complete this implementation. Here is how far I've gotten, with the help of a few answered stack exchange questions:

GLOBAL PARAMETERS:

Row[{SetterBar[Dynamic@\[CapitalNu], Range[1, 10, 1]],"     Sets Length of X,Y,V"}]

INTERFACE VARIABLES -> INTERNAL VARIABLES:

Internalize[{V_, {X_, Y_}}, N_] := Module[{T},
  T = Table[0, {i, 1, N + 2}];
  For[i = 2, i <= N + 2, i++,
   T[[i]] = 
     T[[i - 1]] + Norm[{X[[i - 1]], Y[[i - 1]]} - {X[[i]], Y[[i]]}]/
      V[[i]];];
  T
  ]
X := Table[
   ToExpression["x" <> ToString@i], {i, 0, \[CapitalNu] + 1}];
Y := Table[
   ToExpression["y" <> ToString@i], {i, 0, \[CapitalNu] + 1}];
V := Table[
   ToExpression["v" <> ToString@i], {i, 1, \[CapitalNu] + 1}];
T := Internalize[{V, {X, Y}}, \[CapitalNu]];

MANIPULATE INTERFACE

Procedure[T_,{X_,Y_}]:= Piecewise@
 position=Table[{{X[[i]]-(t-T[[i]])/(T[[i+1]]-T[[i]]))(X[[i]]-X[[i-1]]),Y[[i]]-(t-T[[i]])/(T[[i+1]]-T[[i]]))(Y[[i]]-Y[[i-1]])},  
   T[[i]] <= t <= T[[i + 1]]}, {i, 1, \[CapitalNu]}]

(*The rest would look like this: 
Manipulate[ParametricPlot[Procedure[Dynamic@T,{Dynamic@X,Dynamic@Y}][t],  {t,T[[1]],T[[-1]]}],
Table[{X[[i]],Y[[i]]},Locators],Table[Silders[V[[i]]]]],] *)

Does anyone know if Mathematica is capable of something like this? (I must emphasize, the Manipulate interface has to adjust automatically to changes in N; no retyping code permitted).

The key issue I haven't been able to figure out is how to apply Manipulate[] controls, including locators, to the elements of variable length lists.

While this question may seem unnecessarily long, context really helps.

Answer 1

Here's my attempt at achieving what you describes.

1) I define a display function that takes n as the number of randomly generated locators (initLoc). It also takes a list of speeds v between those locators. The rest of the function is figuring out at what time a particle will reach a particular locator (locatorTimes) given its average speed and the locator position; I think this is what you meant by "inferring the times" in your question.

Then, the points primitives to draw the red line are generated (linePrimitive) by selecting only the points that the particle had passed, along with its current position (the current position is a little tricky to determine since we're only given the time of the particle, not its position). The final result is a Manipulate of a position's path as a function of time.

Clear[display]
display[n_Integer, v_List] /; Length[v] == n - 1 := 
 DynamicModule[{initLoc, locatorTimes, linePrimitive},

  initLoc = RandomReal[{-10, 10}, {n, 2}];

  locatorTimes[pt : {{_, _} ..}, velocity_?VectorQ] := 
   Module[{ptDiff, tInterval, ptPassed, timeAtLocator},
    ptDiff = Norm[#2 - #1] & @@@ Partition[pt, 2, 1];
    tInterval = ptDiff/velocity;
    Prepend[Accumulate[tInterval], 0.]
    ];

  linePrimitive[pt : {{_, _} ..}, velocity_?VectorQ, t_] /; 
    t <= Last@locatorTimes[pt, velocity] := 
   Module[{times, lastPoint, vx, vy},
    times = locatorTimes[pt, velocity];
    lastPoint = Length@Select[times, # <= t &];
    {vx, vy} = (pt[[lastPoint + 1]] - 
        pt[[lastPoint]])/(times[[lastPoint + 1]] - times[[lastPoint]]);
    Append[pt[[1 ;; lastPoint]], 
      pt[[lastPoint]] + {vx, vy} (t - times[[lastPoint]])]~Prepend~
     velocity[[lastPoint]]
    ];

  Manipulate[
   Graphics[
    {Red, [email protected], Point[pt[[1]]], Thick, 
     Line@Rest@linePrimitive[pt, v, t]},
    PlotRange -> {{-10, 10}, {-10, 10}},
    PlotLabel -> 
     TableForm[
      Append[Flatten /@ Flatten[{pt, v}, {{2}, {1}}], 
       Flatten@linePrimitive[pt, v, t][[{-1, 1}]]],
      TableHeadings -> {("Locator " <> ToString@# & /@ 
           Range@Length@pt)~Append~"Current Position", {"x", "y", 
         "v"}}]
    ],
   {{pt, initLoc}, Locator}, {t, 0., 
    Floor[Last@locatorTimes[pt, v], 0.0001]}
   ]
  ]

display[6, {1, 2, 3, 2, 1}]

---

2) I then nest this display function within larger Manipulate's. The outermost Manipulate determines the number of locators needed. It then pass that number down to the next inner Manipulate for it to generate n - 1 velocity input fields.

The values that the user inputs in these input fields will be passed down, along with the number of locators, to display for it to generate a position vs time interactive slider that you saw above. The user can also change the velocities at once by changing constv in the outermost Manipulate.

Manipulate[
 DynamicModule[{v},
  v = Symbol["v" <> StringJoin[ToString /@ {#1, #2}]] & @@@ 
    Partition[Range@n, 2, 1];
  Manipulate[Evaluate@display[n, v], 
   Evaluate[Sequence @@ ({#, constv} & /@ v)]]
  ]
 , {n, Range[2, 5], SetterBar}, {constv, 1}]

---

If you have any questions please feel free to ask. School starts tomorrow for me so I had to code this in a hurry so I can go to bed soon. I will fix this code tomorrow if there's any issue. One of the issues is the slow dragging of my locators. Hence, if anyone has suggestions for code refactoring, I'd really appreciate it.