Coneheads instead of Arrowheads

Question

I was reading a paper by Holten and van Wijk called "A User Study on Visualizing Directed Edges in Graphs". They show a number of graphical alternatives to using arrowheads:

I find the final one (f), that replaces the arrowheads with coneheads, particularly appealing, and would like to try using them. It appears there is no built in conehead functionality, and I was wondering if there is an easy way to do this.

For arrowheads, it is easy:

Graphics[Arrow[{{1, 0}, {2, 1}}]]

My first thought was to replace the line with a cone:

Graphics3D[Cone[{{1, 0, 0}, {2, 1, 0}}, 0.05], Boxed -> False]

This works to some extent, but is a 3D command instead of 2D, and it was not obvious how to modify it to work with Graphics instead of Graphics3D. My second though was to use a Polygon:

conehead[{{a1_, b1_}, {a2_, b2_}}, r_] := 
          Graphics[Polygon[{{a1, b1}, {a2, b2 - r}, {a2, b2 + r}}]];
conehead[{{0, 0}, {1, 0.2}}, 0.02]

This also works to some extent, but has several problems: the end of the cone isn't at the right angle, it doesn't work for many a's and b's, and there is no shading from transparent to dark.

So my question is: is there a straightforward way to replace arrowheads with coneheads?

Answer 1

How about:

conehead[r_][{p1_, ___, p2_}, ___] := With[
  {n = Normalize[{{0, 1}, {-1, 0}}.(p2 - p1)]},
  Polygon[{p1 - r n, p1 + r n, p2}]
  ]
Graphics[conehead[0.02][{{0, 0}, {1, 0.5}}]]

Used in a graph:

RandomGraph[
 {20, 40},
 EdgeShapeFunction -> conehead[0.02],
 EdgeStyle -> Directive[Black, Opacity@0.5]
 ]

Answer 2

StreamStyle glyph "Pointer" can be made to look the same as the desired shape, so we can use

f[w_: .05][pts_] := Graphics`Glyphs`GlyphData["Pointer", GlyphWidth -> w, 
  GlyphControlFunction -> (1 - #&), PlotPoints -> 300][Line @ pts] 

to generate the desired Graphics primitive:

Graphics[{Red,f[][{{1, 0}, {3, 1}}]}]

The function glyph below is slightly more general than f above in that other built-in glyph functions (such as "Drop", "Dart", etc) can be used as the first argument if desired.

ClearAll[glyph]
glyph[g_: "Pointer", w_: .1] := Graphics`Glyphs`GlyphData[g, GlyphWidth -> w][
    Line[(g /. {"Drop" -> Reverse, _ :> Identity})@#]] &;  

Examples:

SeedRandom[1]
rg = RandomGraph[{20, 30}];
SetProperty[rg, {EdgeShapeFunction -> glyph[], VertexSize -> .3, 
  EdgeStyle -> Directive[Opacity[.5], Black], ImageSize -> 500}]

The same idea can be used to have curved versions using the above function in combination with GraphElementData["CurvedArc"]:

ClearAll[glyph2]
glyph2[g_: "Pointer", w_: .1, curv_: .5] := Module[{bf = BezierFunction @@ 
   (GraphElementData[{"CurvedArc", "Curvature" -> curv}][(g /. 
      {"Drop" -> Reverse, _ :> Identity})@#, ##2])}, 
  {Graphics`Glyphs`GlyphData[g, GlyphWidth -> w][Line[bf /@ Subdivide[10]]]}] &; 

SetProperty[rg, {EdgeShapeFunction -> glyph2[], VertexSize -> .3, 
  EdgeStyle -> Directive[Opacity[.5], Black], ImageSize -> 500}] 

Further examples:

SeedRandom[1] 
Graphics[{Opacity[.5], RandomColor[], glyph["Pointer", RandomReal[{1, 10}]]@#} & /@ 
  RandomInteger[100, {50, 2, 2}]] 

Graphics[{Opacity[.5], RandomColor[], glyph["Drop", RandomReal[{1, 10}]]@#} & /@ 
  RandomInteger[100, {50, 2, 2}]]  

SeedRandom[123]
Graphics[{Opacity[.5], RandomColor[], glyph2["Pointer", RandomReal[{1, 10}]]@#} & /@
   RandomInteger[100, {20, 3, 2}]]