I am trying to create a blue point (please see image below) within a small radius of a red point such that the blue point lies outside of the black circle.

enter image description here

The code thatIi have used to generate the figure is as follows:

With[{radius = 0.1},
    initpoint = RandomPoint[Circle[]];
    ℛ = RegionDifference[Circle[initpoint, radius], Disk[]];
    p = RandomPoint[ℛ];
         {Black, Circle[], 
          Purple, Circle[initpoint, radius], 
          Blue, Point @ (p = RandomPoint[ℛ]), 
          Red, Point @ initpoint}]], 

However, I am not interested in generating images. I am only interested in finding the position of the blue point given a red point and the radius. The code I use is the same as before except that i do not show graphics.

With[{radius = 0.001},
   initpoint = RandomPoint[Circle[]];
   ℛ = RegionDifference[Circle[initpoint, radius], Disk[]];
   {100}]] // AbsoluteTiming

(* {9.02254, {{-0.6495, -0.761656}, {-0.122595, -0.99263}, <<96>>,{-0.142981,
0.990732}, {0.86886, 0.49706}}} *)

The above shows that it is taking 9 seconds to generate 100 configurations of blue given red. The process is extremely slow. Is there a way to speed the code up or is there a better algorithm?

Would be grateful for help.

  • $\begingroup$ Why not discretize the region first? $\endgroup$ Dec 29, 2016 at 18:43
  • $\begingroup$ @J.M. if i discretize them first i get an error saying the first argument should be a parameter free region. I have no idea what it means. $\endgroup$
    – Ali Hashmi
    Dec 29, 2016 at 18:51
  • 2
    $\begingroup$ Region computations are often slow. Best to avoid them when possible, as it is for this problem. $\endgroup$
    – m_goldberg
    Dec 29, 2016 at 23:51

2 Answers 2


Here is a better algorithm.

First a faster function for finding the blue point given the red point and the radius of the circle on which the blue point lies.

selectPt[cntr_, r_] :=
  Module[{pt = {0, 0}},
    While[Norm[pt] < 1, pt = RandomPoint[Circle[cntr, r]]]; 

This constrains the blue point be further from the origin than the radius of the black circe; i.e., outside of the black circle.

With[{r = 0.2},
        cntr = RandomPoint[Circle[]];
           Black, Circle[],
           Purple, Circle[cntr, r],
           Blue, Point @ selectPt[cntr, r],
           Red, Point @ cntr}], {2}, {2}]]]]



  Table[cntr = RandomPoint[Circle[]]; selectPt[cntr .0011], {100}]]; 
// AbsoluteTiming

{0.055384, Null}

  • 3
    $\begingroup$ But you still use RandomPoint which is way slower than just calculating {Cos[#], Sin[#]} &@RandomReal[2*Pi]. The RandomPoint solution is ~ 30 times slower for 10000 points. $\endgroup$
    – BlacKow
    Dec 29, 2016 at 23:54
  • 2
    $\begingroup$ @BlacKow. Because it makes my code easier to understand, and it may be fast enough for the OP's purposes. I believe simplicity is preferable to speed as long as it fast enough. $\endgroup$
    – m_goldberg
    Dec 30, 2016 at 0:06
  • 1
    $\begingroup$ Actually now I'm puzzled why the RandomPoint@Circle[] is so slow? Trace doesn't give anything useful ... $\endgroup$
    – BlacKow
    Dec 30, 2016 at 0:32
  • $\begingroup$ @m_goldberg thanks so much. I would only propose a small edit if you like. Maybe the GraphicsGrid can be implemented on the end result i.e. outside of With $\endgroup$
    – Ali Hashmi
    Dec 30, 2016 at 12:50
  • $\begingroup$ @BlacKow thanks, i upvoted your answer $\endgroup$
    – Ali Hashmi
    Dec 30, 2016 at 13:05

Why do you need regions at all?

rBlack = 1.0;
rPurple = 0.1;    
randPoint[rB_, rP_][x_] := Module[{},
   rB {Cos[#], Sin[#]} &@RandomReal[2*Pi] + rP {Cos[#], Sin[#]} &@RandomReal[2*Pi]
Table[NestWhile[randPoint[rBlack, rPurple], {0, 0}, 
  Norm[#] < rBlack &], {100}]

So it takes a random point on black circle and adds (as vector) a random point on the purple circle. Then NextWhile repeats it until the point is actually beyond the black circle. Here is code that does the drawing - I modified the randPoint because I need the center of purple circle in addition to blue point coordinates.

randPointDraw[rB_, rP_][x_] := Module[{initPoint},
   initPoint = rB {Cos[#], Sin[#]} &@RandomReal[2*Pi];
   {initPoint, initPoint + rP {Cos[#], Sin[#]} &@RandomReal[2*Pi]}

GraphicsGrid@Partition[#, 3] &@ Table[Show[
 Graphics[{Black, Circle[{0, 0}, rBlack], Purple, Circle[#[[1]], rPurple], Blue, 
 Point[#[[2]]]}]] &@ NestWhile[randPointDraw[rBlack, rPurple], {0, 0}, 
    Norm[#[[2]]] < rBlack &], {9}]

enter image description here


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