Interactive angles

Question

The following is nearly what I am after, but the angles don't behave quite as I'd like:

r[a_, b_] := RegionNearest[Circle[{0, 0}, 1], a[[b]]];
Manipulate[Graphics[{Circle[{0, 0}, 1],
   Line /@ {{Join[{{0, 0}}, a, {{0, 0}}]}},
   Circle[{0, 0}, 1/4, {
     If[a[[3, 2]] < 0, -#, #] &@VectorAngle[{1, 0}, a[[3]]], 
     If[a[[1, 2]] < 0, -#, #] &@VectorAngle[{1, 0}, a[[1]]]}],
   Circle[a[[2]], 1/4, {
     If[a[[3, 2]] - 1 < 0, -#, #] &@ VectorAngle[{1, 0}, {0, -1} + a[[3]]], 
     If[a[[1, 2]] - 1 < 0, -#, #] &@ VectorAngle[{1, 0}, {0, -1} + a[[1]]]}]
   }, PlotRange -> 1.1],
  {{a, CirclePoints[3]}, Locator,
    TrackingFunction -> (Switch[CurrentValue["CurrentLocatorPaneThumb"],
    1, a[[1]] = r[#, 1],
    2, a[[2]] = r[#, 2],
    3, a[[3]] = r[#, 3]
  ] &), Appearance -> None}]

Is there a more straightforward way to go about this, or is VectorAngle the simplest?

Answer 1

Does this approximate what you want? It is not the most efficient. Since you're always referencing to {1,0} with VectorAngle, why not just use ArcTan. I used a couple of helper functions from messing with circular arcs for Indra's Pearls.

r[a_, b_] := RegionNearest[Circle[{0, 0}, 1], a[[b]]];

ang[p_, q_] := 
   Block[{a, b}, 
         If[(a = N[ArcTan @@ p]) < 0, a = a + 2 Pi];
         If[(b = N[ArcTan @@ q]) < 0, b = b + 2 Pi];
         If[b < a, {a, b + 2 Pi}, {a, b}]]

ang1[p_, q_] := 
   Block[{a, b}, 
         If[(a = N[ArcTan @@ p]) < 0, a = a + 2 Pi];
         If[(b = N[ArcTan @@ q]) < 0, b = b + 2 Pi];
         {a, b} = Sort[{a, b}];
         If[b - a > Pi, {b, a + 2 Pi}, {a, b}]]

Manipulate[
   Graphics[{Thick,
      Circle[{0, 0}, 1], 
      Line /@ {{Join[{{0, 0}}, a, {{0, 0}}]}}, Blue, 
      Circle[{0, 0}, 1/4, ang[a[[1]], a[[3]]]], Red, 
      Circle[a[[2]], 1/8, ang1[a[[1]] - a[[2]], a[[3]] - a[[2]]]]}, 
      PlotRange -> 1.1], {{a, CirclePoints[3]}, Locator, 
      TrackingFunction -> (Switch[CurrentValue["CurrentLocatorPaneThumb"],
         1, a[[1]] = r[#, 1],
         2, a[[2]] = r[#, 2],
         3, a[[3]] = r[#, 3]] &), Appearance -> None}]