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TWEET Finding right triangles in random points a guest Apr 5th, 2017 135 Never
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1. n = 12; (* How many points *)
2. pts = Table[RandomReal[], {n}, {2}];
3. Show[
4.  Graphics[{Red, PointSize[0.01], Point[pts]}],
5.  ImageSize -> 8*72, PlotRange -> {{0, 1}, {0, 1}}, Frame -> True,
6.  FrameTicks -> False
7.  ]
8.
9. \[Theta]min = 89.0; (* Range of angles to be considered "right" (in \
10. degrees) *)
11. \[Theta]max = 91.0;
12. RightAngles = {};
13. Do[(
14.   ptA = pts[[i]];
15.   ptB = pts[[j]];
16.   ptC = pts[[k]];
17.
18.   a = Norm[ptB - ptC];
19.   b = Norm[ptA - ptC];
20.   c = Norm[ptA - ptB];
21.
22.   AngleA = 180/\[Pi] ArcCos[(b^2 + c^2 - a^2)/(2 b c)];
23.   If[\[Theta]min <=
24.     AngleA <= \[Theta]max, (AppendTo[RightAngles, {ptA, ptB, ptC}];
25.     Continue[];)];
26.   AngleB = 180/\[Pi] ArcCos[(a^2 + c^2 - b^2)/(2 a c)];
27.   If[\[Theta]min <=
28.     AngleB <= \[Theta]max, (AppendTo[RightAngles, {ptA, ptB, ptC}];
29.     Continue[];)];
30.   AngleC = 180/\[Pi] ArcCos[(a^2 + b^2 - c^2)/(2 a b)];
31.   If[\[Theta]min <= AngleC <= \[Theta]max,
32.    AppendTo[RightAngles, {ptA, ptB, ptC}]];
33.   ), {i, 1, n - 2}, {j, i + 1, n - 1}, {k, j + 1, n}]
34.
35. RightAngles (* Print the list of triples of vertices that form \
36. "right-ish" triangles *)
37. RightAngles // Length (* And how many there are *)
38.
39. (* Show the "right" triangles overlaid over the random points *)
40. Show[
41.  Graphics[{Red, PointSize[0.02], Point[pts]}],
42.  Table[Graphics[{Thick, Hue[i/Length[RightAngles]],
43.     Line[{RightAngles[[i, 1]], RightAngles[[i, 2]],
44.       RightAngles[[i, 3]], RightAngles[[i, 1]]}]}], {i, 1,
45.    Length[RightAngles]}],
46.  ImageSize -> 8*72, Frame -> True, FrameTicks -> False,
47.  PlotRange -> {{0, 1}, {0, 1}}
48.  ]
49.
50. (* Create the list of unique side lengths in this set of triangles *)
51.
52.
53. lengths = {};
54. Do[(
55.    AppendTo[lengths, Norm[RightAngles[[i, 1]] - RightAngles[[i, 2]]]];
56.    AppendTo[lengths, Norm[RightAngles[[i, 1]] - RightAngles[[i, 3]]]];
57.    AppendTo[lengths, Norm[RightAngles[[i, 2]] - RightAngles[[i, 3]]]];
58.    ), {i, 1, Length[RightAngles]}];
59. lengths = Union[lengths, lengths];
60.
61. (* Here is a set of "special" numbers to search for in ratios of \
62. these lengths, along with their reciprocals *)
63. nums = {\[Pi], E, EulerGamma, GoldenRatio, Sqrt, Sqrt, Sqrt,
64.    Sqrt, Sqrt, Sqrt, Sqrt};
65.
66. (* Make a list of of side length ratios that "match" the above set of \
67. special numbers *)
68. f = 0.01; (* What fractional deviation from the "special" numbers is \
69. acceptable *)
70. Specials = {};
71. Do[(
72.    r = lengths[[i]]/lengths[[j]];
73.    Do[(
74.      x = nums[[k]];
75.      If[Abs[(r - x)/x] <= f,
76.       AppendTo[
77.        Specials, {lengths[[i]], lengths[[j]], r, x, Abs[(r - x)/x]}]];
78.      If[Abs[(r - (1/x))/(1/x)] <= f,
79.       AppendTo[
80.        Specials, {lengths[[i]], lengths[[j]], r, 1/x,
81.         Abs[(r - (1/x))/(1/x)]}]];
82.      ), {k, 1, Length[nums]}]
83.    ), {i, 1, Length[lengths]}, {j, 1, Length[lengths]}];
84.
85. Specials (* Print the list of "special" side length relationships and \
86. how accurate they are *)
87. Specials // Length
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