Choosing $n$ equidistant points on a circle with given radius and center

Question

I would like to have a function circle which takes two inputs: a tuple {x,y} and a real number r and outputs the cooridinates of points on the circumference of a circle that is centered at $(x,y)$ which has radius $r$.

Importantly, I want the points to be equidistant on the circle.

I tried to implement the procedure here described as Circle Point Picking. But I don't get equidistant points.

Kindly help me. Here is the code I tried:

num = 20; r = 50;
circle = Module[{},
  random = RandomReal[{-2, 2}, {num + 10, 2}];
  random = Take[DeleteCases[random, #1^2 + #2^2 >= r^2 &], num];
  Table[r {(x[[1]]^2 - x[[2]]^2)/(x[[1]]^2 + x[[2]]^2), 
     2 x[[1]] x[[2]]/(x[[1]]^2 + x[[2]]^2)}, {x, random}]
  ]

The above code will output num=20 points on a circle that are not equidistant on the circumference. Here is the output represented as a ListPlot:

ListPlot[circle]

Irrelevant as such for the question, but if anyone is curious:

The context for getting equidistant points is that I want to embed a graph with some vertices located on the points. If you instead know how to do this, kindly tell me. I cannot find a use for CircularEmbedding because I only want a circular embedding on some vertices of the graph, not all vertices. The only way I see is to explicitly give the vertex cooridinates as points on a circle.

Answer 1

func[cntr_, rad_, n_, ang0_] := 
 Graphics[{Circle[cntr, rad], {Red, PointSize@0.02, 
    Point[Table[rad{Cos [ang0 + j],Sin[ang0 + j]}, {j, 0, 
       2 Pi - 2 Pi/n, 2 Pi/n}]]}}]

Here cntr is centre of circle, rad is radius, n is the number of points/segments, ang0 is just specifying where to start.

Manipulate[
 func[{0, 0}, 1, num, a], {{num, 3}, Range[3, 20]}, {a, 0, 2 Pi}]

You could randomize your starting position. Apologies, if have misunderstood the intent.

Answer 2

Just for fun:

SetAttributes[parts, Listable];
parts[z_] := {Re[z], Im[z]};
randomCirclePoints[n_, center_, radius_] := Block[{z},
  parts[
   z /. NSolve[(Exp[I RandomReal[{0, 2 Pi}]] (z - {1, I}.center))^n ==
       radius^n, z]
   ]
  ]

Graphics[
 {Red, Point@randomCirclePoints[33, {2.5, 1}, 4]},
 Frame -> True]

Answer 3

Here is a simple one, p sets how many points you want, q the radius and point the position of the center

 mycircle[p_, q_, point_:{0,0}] := Table[{q Cos[2 Pi k /p], q Sin[2 Pi k /p]} + point, {k, p}]


 Graphics[{Yellow, Point /@ mycircle[12, 1], Black, 
           Point /@ mycircle[12, 1, {1, 1}], Green, 
           Point /@ mycircle[12, 1, {2, 0}], Red, 
           Point /@ mycircle[12, 0.2, {3, 1}], Blue, 
           Point /@ mycircle[12, 1, {-1, 1}]}]

The randomness then it is just a random phase to the points

 mycirclerd[p_, q_, point_: {0, 0}] := 
     Module[{phase = RandomReal[{0, 2 Pi}]}, 
     Table[{q Cos[2 Pi k /p + phase], q Sin[2 Pi k /p + phase]} + point, {k, p}]]

Answer 4

Here is a quick and dirty (read approximate) way to achieve this using Version 10 capabilities:

circ[ctr_, r_, n_] := MeshCoordinates @ DiscretizeRegion[Circle[ctr, r], 
                                        MaxCellMeasure -> {"Length" -> 2 Pi r /(n - 1.5)}]

Here ctr is the center, r, the radius and n number of points.

Visualize (for $n = 5$):

Graphics[{Red, PointSize[0.02], Point@circ[{0, 0}, 3, 5], Blue, Circle[{0, 0}, 3]}]

Note: You'll have to tweak that number 1.5 in the denominator of "Length" definition in the MaxCellMeasure option to get the right number of points. Ideally it should just be $2\pi r/n$, but that doesn't always give the desired $n$. At least, it does produce equidistant points on the circle.

Answer 5

One compact way to get $n$ points on a circle would be

CirclePoints[center_,radius_,n_] := (center+radius #&) /@ 
                                    Transpose @ Through[{Re,Im}[Exp[2\[Pi] I Range[n]/n]]]

Answer 6

Another simple routine:

myCircle[cent_?VectorQ, r_?Positive, n_Integer] := 
         AffineTransform[{DiagonalMatrix[{r, r}], cent}][CirclePoints[n]]

Using lalmei's example:

Graphics[{Yellow, Point[myCircle[{0, 0}, 1, 12]],
          Black, Point[myCircle[{1, 1}, 1, 12]],
          Green, Point[myCircle[{2, 0}, 1, 12]],
          Red, Point[myCircle[{3, 1}, 1/5, 12]],
          Blue, Point[myCircle[{-1, 1}, 1, 12]]}]

Adding a phase is easy, too:

myCircle[cent_?VectorQ, r_?Positive, ph_?NumericQ, n_Integer] := 
         AffineTransform[{DiagonalMatrix[{r, r}].RotationMatrix[ph], cent}] @ 
         CirclePoints[n]

(CirclePoints[] is new in version 10.1.)

---

Pickett mentions that CirclePoints[] actually supports a more extensive syntax, one that I missed in the first reading of the docs: CirclePoints[cent, {r, ph}, n].

Thus, one can redo the first example as

Graphics[{Yellow, Point[CirclePoints[{0, 0}, 1, 12]],
          Black, Point[CirclePoints[{1, 1}, 1, 12]],
          Green, Point[CirclePoints[{2, 0}, 1, 12]],
          Red, Point[CirclePoints[{3, 1}, 1/5, 12]],
          Blue, Point[CirclePoints[{-1, 1}, 1, 12]]}]

Answer 7

As you didn't specify that any randomness was required, the natural solution to your problem for $n$ points and circle of radius $r$ is to find the $n$ complex $n$th roots of $r$, then translate them by the complex number $z = (x, y)$.

To implement that, a "natural" thing is to use David Park's Presentations add-on (http://home.comcast.net/~djmpark/DrawGraphicsPage.html), which allows you to work directly with complex numbers instead of having to use trig functions:

<< Presentations`

circlePoints[ctr_, r_, n_] := 
   Draw2D[{
          ComplexCirclePoint[#, 4, Black, Orange] & /@ (ctr + r Exp[Range[n] 2 Pi I/n]),
          Legacy@DarkGreen, ComplexCircle[ctr, r]
          },
   Axes -> True, Background -> Legacy@Linen, ImageSize -> 200]

For example:

circlePoints[0.5 + 0.75 I, 3, 12]

Should you wish randomness, you could include a random rotation (implemented as a complex multiplication or, equivalently, an addition of phase in the argument in the exponential).