Rendering polygon as graph with directed edges in Mathematica

Question

In Mathematica I'm trying to render a polygon as a set of vertices and directed edges.

What I have so far:

Graphics[Polygon[{{1, 10}, {2, 4}, {10, 5}, {20, 10}}]]

I see Mathematica has the Graph and PathGraph commands - both of which have a DirectedEdges option - but it seems like I have no control over the position of the vertices with these commands.

If I could customize the fill and edge/stroke of the Polygon command, that would be acceptable - but I'm not seeing how to do it. It looks like this command is specifically meant to draw filled polygons.

I'd also be OK with a custom Mathematica routine to draw what I want using a loop and multiple Line commands within a Graphics command - but I can't see how to draw the lines as arrows (I'm sure I could make the routine do this with three Lines per edge, but I really just think I'm missing something here.)

Answer 1

Graph has the option VertexCoordinates which allows you to specify the coordinates of the vertices, so you could do something like

crds = {{1, 10}, {2, 4}, {10, 5}, {20, 10}};
vertices = Range[Length[crds]];
edges = Thread[vertices \[DirectedEdge] RotateLeft[vertices]];

Graph[vertices, edges, VertexCoordinates -> crds,
 EdgeShapeFunction -> GraphElementData[{"Arrow", "ArrowSize" -> .1}]]

You can also use Graphics primitives, for example

edges1 = Thread[{crds, RotateLeft[crds]}];

Graphics[{Red, Arrow /@ edges1}]

Answer 2

n = 5;
Graphics@Arrow@Table[{Sin[2 Pi i/n], Cos[2 Pi i/n]}, {i, 1, n + 1}]

Edit

GraphicsGrid[Partition[
  Table[Graphics@
    Arrow@Partition[Table[{Sin[2 Pi i/j], Cos[2 Pi i/j]}, {i, 1, j + 1}], 2, 1], 
   {j,3, 12}], 3], Frame -> All]

Answer 3

You had this already

pol = Graphics[Polygon[{{1, 10}, {2, 4}, {10, 5}, {20, 10}}]];

You can turn it into something similar to what you want with

pol /. Polygon[i_] :> Thread@Arrow@Partition[i, 2, 1, 1]