How to order points anti clockwise

Question

Lets take thess points.

pt={{-4.65371,0.1},{-4.68489,0.103169},{-4.78341,0.104834},{-4.83897,0.100757},
{-4.92102,0.0949725},{-4.93456,0.100181},{-4.89166,0.122666},{-4.78298,0.129514}, 
{-4.72723,0.121442},{-4.68355,0.11023},{-4.65371,0.1},{-4.66924,0.10173}, 
{-4.93059,0.0966989},{-4.93259,0.105094},{-4.91074,0.116966},{-4.90635,0.094878}, 
{-4.66846,0.105327},{-4.92647,0.0956182},{-4.93433,0.102498},{-4.9333,0.0982262},
{-4.66257,0.10102}};

Now they are in certain order (for me is a disorder!) which can be seen if we look at the ListLinePLot

picUnorder=ListLinePlot[pt,Frame-> True,Mesh-> All,MeshStyle-> PointSize[Large]];
SeepicUnorder=ListLinePlot[pt,Frame-> True,Mesh-> All,MeshStyle-> 
PointSize[Large]]/.Line[rest_]:>{Arrowheads[Table[0.02,{i,0,1,.02}]],Arrow[rest]};
GraphicsGrid[{{picUnorder,SeepicUnorder}}]

But we need to order them like the picture below.

Does anybody has some suggestion for a algorithm to sort such 2D points in counter clockwise direction so that we can rearrange the list of points to create a geometry like the last pic just by using ListLinePlot on the rearranged points????

Using the suggestion we get something like the following.

center=Mean[pt];
pts=SortBy[pt,Function[p,{x,y}=p-center;ArcTan[x,y]]];
Show[ListPlot[pt],ListLinePlot[pts,Mesh-> All,MeshStyle->
PointSize[Large]],Frame-> True]

BR

Answer 1

I posted the following comment below your question: I don't think you'll find a general solution. This answer tries to dig a little on that.

Heike's solution seems fair, but FindShortestTour is based on the metric properties of the set, while your requirement is probably more on the topological side.

Here is a comparison on two points sets and the methods available to FindShortestTour:

pl[method_, k_] :=
  Module[{ptsorted, pt,s},
   little[x_] := {{1, 0}, {2, 1}, {1, 2}, {0, 1}}/x - (1/x) + 2;
   pt = Join[{{0, 0}, {4, 4}, {4, 0}, {0, 4}}, little[k]];
   ptsorted = Join[s = pt[[FindShortestTour[pt,Method->method][[2]]]], {s[[1]]}];
   ListPlot[ptsorted, Joined -> True, Frame -> True, 
                      PlotMarkers -> Automatic, 
                      PlotRange -> {{-1, 5}, {-1, 5}}, 
                      Axes -> False, AspectRatio -> 1, PlotLabel -> method]];
GraphicsGrid@
 Table[pl[i, j],
       {i, {"AllTours", "CCA", "Greedy", "GreedyCycle", 
            "IntegerLinearProgramming", "OrOpt", "OrZweig", "RemoveCrossings",
            "SpaceFillingCurve", "SimulatedAnnealing", "TwoOpt"}}, 
       {j, {1, 1.8}}]

     Fat Star         Slim Star

As you can see, several methods deliver the expected result on the left column, while only one does it on the right one. Moreover, the only useful method for the set on the right is completely off for the column on the left.

Answer 2

Maybe you could do something with FindShortestTour. For example

ptsorted = pt[[FindShortestTour[pt][[2]]]];
ListPlot[ptsorted, Joined -> True, Frame -> True, PlotMarkers -> Automatic]

produces something like