Locator following other locator in DynamicModule

Question

I'm trying to create a dynamic environment where I specify, using two locators, the initial condition (position (x,y) and velocity angle (alpha)) of a 2D particle.

To simplify, I'm assuming the initial position is restricted to a circle, and the velocity angle is always in respect to an angle in relation to the normal vector, perpendicular to the tangent vector.

This means that in the end, I would like to have 3 parameters: x,y initial position of the positional locator, and an initial angle, defined as the angle between the vector (positional locator - angle locator) and the normal vector.

The main problem resides on the coupling between the positional locator, and the angle locator: when I move the positional locator, I would like to have the angle locator moved in such a way that the initial angle is preserved.

To manipulate those, I'm using two Locators: one for position (initialPoint), one for initial angle (pt2)

DynamicModule[{initialPoint = {0, 0}, pt2 = {0, 0}},
 Style[Graphics[{
    Dashed,
    Circle[{0, 0}, 1],
    Locator[
     Dynamic[pt2, (pt2 = 
         initialPoint + 0.5 Normalize[# - initialPoint]) &]], 
    Locator[Dynamic[initialPoint, (initialPoint = Normalize[#]) &]]}, 
   PlotRange -> {{-3.3, 3.3}, {-3.3, 3.3}}, AspectRatio -> 1], 
  Selectable -> False]]

however, when I dynamically pick the initialPoint, the second locator does not follow it. What can I do it in order to keep pt2 always such that the initial angle is preserved?

Answer 1

If by "normal" you mean normal to the circle, then you were almost there. You need to update the angle point when initialPoint is changed.

DynamicModule[{initialPoint = {1, 0}, anglePoint = {0.5, 0}}, 
 Style[Graphics[{Dashed, Circle[{0, 0}, 1], 
    Locator[Dynamic[
      anglePoint, (anglePoint = 
         initialPoint + 0.5 Normalize[# - initialPoint]) &]], 
    Locator[Dynamic[initialPoint, 
      Module[{angle = ArcTan @@ (anglePoint - initialPoint) - ArcTan @@ initialPoint},
        initialPoint = Normalize[#];
        angle += ArcTan @@ initialPoint;
        anglePoint = initialPoint + 0.5 {Cos[angle], Sin[angle]}] &]]}, 
   PlotRange -> {{-3.3, 3.3}, {-3.3, 3.3}}, AspectRatio -> 1], 
  Selectable -> False]]

Answer 2

I"m not sure exactly what motion you are looking for, but here is a way to make the second locator follow the first around. As in your code, I locked the first locator to the unit circle and the second locator to a smaller orbit about the first.

pt1Old = {0, 0}; pt2Old = {0, 0};
Manipulate[If[pt1Old != pt1, pt1Old = pt1; pt1 = Normalize[pt1];
              pt2 = pt1 + Normalize[pt2 - pt1]/2; pt2Old = pt2;];
           If[pt2Old != pt2, pt2 = pt1 + Normalize[pt2 - pt1]/2; pt2Old = pt2;];
           Graphics[{Dashed, Circle[{0, 0}, 1]}, 
              PlotRange -> {{-2.3, 2.3}, {-2.3, 2.3}}], 
           {{pt1, {1, 0}}, Locator}, {{pt2, {0, 1}}, Locator}]

The way this works is to maintain a set of "old" locations for the locators and to only update the positions when they are dragged around. That's what the two Ifs are for. Note that the second locator can be changed independent of the first, but that whenever the first is moved, the second follows. Accordingly, when you want to change the motion of the second (to whatever angle you wish) you will need to update it in both places.