Evenly spaced points on boundary of polygon

I have a polygon and I would like to generate $n$ evenly spaced points along the boundary.

5 Answers
5

You can use piecewise linear interpolation:

Let's generate a random polygon:

R = ConvexHullMesh[RandomPoint[Disk, 100]]
polygon = Append[#, #[[1]]] &@MeshCoordinates[R][[MeshCells[R, 2][[1, 1]]]];


Creating a piecewise-linear interpolation which is parameterized by arc length.

t = Prepend[Accumulate[Norm /@ Differences[polygon]], 0.];
γ = Interpolation[Transpose[t, polygon],
InterpolationOrder -> 1,
PeriodicInterpolation -> True
];


Sampling it uniformly:

s = Subdivide[γ[[1, 1, 1]], γ[[1, 1, 2]], 100];
newpts = γ[s];


And plotting the generated points:

Show[
R,
ListPlot[newpts]
]


If you just want to generate a graphic, then you could use an Arrow with a custom Arrowheads directive. For example, here is a Polygon object:

Arrow

Arrowheads

Polygon

poly = Polygon[
7.361093855790543, 0.2777667989114292, 8.37955832193947, 4.837661706721057,
8.064449639390588, 8.016729782194783, 3.518023168010062, 8.74578450698846,
0.8343834304870779, 2.2170380309534554, 4.807772222882807, 0.09815872747936005,
7.361093855790543, 0.2777667989114292
];

Graphics[RegionBoundary @ poly]


Show 10 evenly spaced points:

Graphics[
Arrowheads @ Append[
ConstantArray[
.1, Automatic, Graphics[PointSize[Large], Point[0,0]],
10
],
0
],
Arrow @@ RegionBoundary[poly]
]


You can use BSplineFunction and ParametricPlot with ArcLength as option value for the option MeshFunctions:

BSplineFunction

ParametricPlot

ArcLength

MeshFunctions

SeedRandom[77]
pts = RandomReal[-1, 1, 7, 2];
coords = pts[[FindShortestTour[pts][[2]]]];
n = 10;

bsF = BSplineFunction[coords, SplineDegree -> 1];
ParametricPlot[bsF[t], t, 0, 1, Frame -> True, Axes -> False, AspectRatio -> Automatic,
MeshFunctions -> ArcLength,
Mesh -> n,
MeshStyle -> Directive[Red, PointSize[Large]],
Epilog -> Red, PointSize[Large] , Point@bsF[0]]


We could also use mesh functions, as Vitaliy showed here:

plot = ListLinePlot[
coords,
Mesh -> 20,
MeshFunctions -> "ArcLength"
];
pts = Cases[Normal[plot], Point[pt_] :> pt, Infinity];
pts = Prepend[pts, First[coords]];

Graphics[
Line[coords],
PointSize[Large], Red, Point[pts]
]


I reused the polygon in Carl's answer.

I think that the broad scheme of doing it would be as follows:

Load the Polytopes package:

<< Polytopes`

Use the vertices command (e.g. assuming that you are working with an octagon):

a=Vertices[Octagon]

Set up a function which interpolates between two points using a parameter t.

Use the Map command to calculate the interpolated co-ordinates for any given value of t on successive values of your vertices.

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