geometry.py

Mon, 22 Jan 2018 17:47:06 +0200

author
Santeri Piippo
date
Mon, 22 Jan 2018 17:47:06 +0200
changeset 20
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parent 15
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child 22
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permissions
-rw-r--r--

rename test

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1 def degree_rep(angle):
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2 from math import degrees
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3 return '∠ %.2f°' % degrees(angle)
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4
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5 def position_vector(vertex):
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6 assert isinstance(vertex, Vertex)
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7 return Vector(*vertex.coordinates)
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8
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9 def angle_magnitude_key(angle):
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10 '''
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11 Returns how great an angle is.
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12 '''
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13 from math import pi as π
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14 normalized_angle = abs(angle) % (2 * π)
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15 if normalized_angle > π:
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16 normalized_angle = (2 * π) - normalized_angle
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17 return normalized_angle
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18
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19 class Vertex:
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20 def __init__(self, x, y, z):
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21 if not all(is_real(coordinate) for coordinate in (x, y, z)):
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22 raise ValueError(str.format('Bad vertex coordinates: {!r}',
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23 (x, y, z),
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24 ))
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25 self.x, self.y, self.z = x, y, z
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26 def __repr__(self):
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27 return str.format('Vertex({!r}, {!r}, {!r})', self.x, self.y, self.z)
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28 def distance_to(self, other):
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29 # can't use hypot because of 3 arguments
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30 from math import sqrt
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31 return sqrt(
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32 (self.x - other.x) ** 2 +
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33 (self.y - other.y) ** 2 +
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34 (self.z - other.z) ** 2
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35 )
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36 @property
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37 def coordinates(self):
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38 return self.x, self.y, self.z
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39 def __add__(self, other):
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40 assert not (type(self) == type(other) == Vertex)
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41 if type(self) == Vector and type(other) == Vector:
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42 result_type = Vector
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43 else:
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44 result_type = Vertex
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45 return result_type(self.x + other.x, self.y + other.y, self.z + other.z)
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46 def __radd__(self, other):
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47 return self + other
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48 def __neg__(self):
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49 return type(self)(-self.x, -self.y, -self.z)
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50 def __sub__(self, other):
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51 result = self + (-position_vector(other))
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52 if isinstance(other, Vertex):
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53 return Vector(*result.coordinates)
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54 else:
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55 return result
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56 def __mul__(self, scalar):
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57 assert is_real(scalar)
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58 return type(self)(self.x * scalar, self.y * scalar, self.z * scalar)
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59 def __rmul__(self, other):
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60 return self * other
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61 def __truediv__(self, scalar):
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62 assert is_real(scalar)
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63 return type(self)(self.x / scalar, self.y / scalar, self.z / scalar)
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64 def __floordiv__(self, scalar):
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65 assert is_real(scalar)
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66 return type(self)(self.x // scalar, self.y // scalar, self.z // scalar)
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67 def __matmul__(self, transformation_matrix):
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68 return transform(self, transformation_matrix)
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69 def __eq__(self, other):
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70 return all(
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71 abs(a - b) < 1e-8
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72 for a, b in zip(self.coordinates, other.coordinates)
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73 )
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74 def __ne__(self, other):
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75 return not self.__eq__(other)
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76 def __lt__(self, other):
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77 return self.coordinates < other.coordinates
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78 def __hash__(self):
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79 return hash(self.coordinates)
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80
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81 class VertexOp:
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82 def __init__(self, callable):
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83 self.callable = callable
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84 def __rmul__(self, coordinate):
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85 return self.callable(coordinate)
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86
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87 class LineSegment:
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88 def __init__(self, v1, v2):
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89 self.v1, self.v2 = v1, v2
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90 def __repr__(self):
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91 return str.format('LineSegment({!r}, {!r})', self.v1, self.v2)
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92 @property
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93 def length(self):
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94 return self.v1.distance_to(self.v2)
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95 @property
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96 def vertices(self):
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97 return self.v1, self.v2
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98 def __len__(self):
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99 return 2
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100 def __getitem__(self, index):
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101 return self.vertices[index]
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102
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103 class IndexRing:
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104 def __init__(self, container):
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105 self.container = container
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106 def __getitem__(self, index):
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107 return self.container[index % len(self.container)]
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108
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109 class Vector(Vertex):
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110 def __init__(self, x, y, z):
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111 super().__init__(x, y, z)
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112 def __repr__(self):
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113 return str.format('Vector({!r}, {!r}, {!r})', self.x, self.y, self.z)
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114 def length(self):
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115 from math import sqrt
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116 return sqrt(self.x ** 2 + self.y ** 2 + self.z ** 2)
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117 def normalized(self):
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118 length = self.length()
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119 if length > 0:
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120 return Vector(self.x / length, self.y / length, self.z / length)
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121 else:
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122 return Vector(0, 0, 0)
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123
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124 def dot_product(v1, v2):
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125 ''' Returns the dot product v1 · v2. '''
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126 return v1.x * v2.x + v1.y * v2.y + v1.z * v2.z
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127
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128 def cross_product(v1, v2):
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129 ''' Returns the cross product v1 × v2. '''
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130 return Vector(
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131 x = v1.y * v2.z - v1.z * v2.y,
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132 y = v1.z * v2.x - v1.x * v2.z,
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133 z = v1.x * v2.y - v1.y * v2.x,
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134 )
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135
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136 def unit_normal(a, b, c):
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137 x = Matrix3x3([
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138 1, a.y, a.z,
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139 1, b.y, b.z,
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140 1, c.y, c.z,
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141 ]).determinant()
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142 y = Matrix3x3([
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143 a.x, 1, a.z,
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144 b.x, 1, b.z,
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145 c.x, 1, c.z,
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146 ]).determinant()
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147 z = Matrix3x3([
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148 a.x, a.y, 1,
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149 b.x, b.y, 1,
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150 c.x, c.y, 1,
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151 ]).determinant()
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152 return Vector(x, y, z).normalized()
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153
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154 class NoIntersection(Exception):
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155 pass
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156
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157 def pairs(iterable, *, count = 2):
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158 '''
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159 Iterates the given iterable and returns tuples containing `count`
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160 sequential values in the iterable.
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161
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162 Formally, for a vector A with indices 0…n and pair count k, this
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163 function yields:
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164 (A₀, A₁, …, Aₖ),
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165 (A₁, A₂, …, Aₖ₊₁),
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166 …,
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167 (Aₙ₋₂, Aₙ₋₁, …, Aₖ₋₂),
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168 (Aₙ₋₁, A₀, …, Aₖ₋₁).
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169 '''
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170 iterable_count = len(iterable)
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171 for index in range(iterable_count):
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172 yield tuple(
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173 iterable[(index + offset) % iterable_count]
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174 for offset in range(count)
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175 )
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176
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177 class Polygon:
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178 def __init__(self, vertices):
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179 self.vertices = vertices
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180 def __repr__(self):
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181 return str.format('Polygon({!r})', self.vertices)
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182 def area(self):
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183 total = Vector(0, 0, 0)
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184 for v1, v2 in pairs(self.vertices):
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185 total += cross_product(v1, v2)
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186 return dot_product(total, unit_normal(self.vertices[0], self.vertices[1], self.vertices[2]))
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187 def centroid(self):
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188 ...
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189 raise NotImplementedError
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190 def __getitem__(self, index):
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191 return self.vertices[index % len(self.vertices)]
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192 def __len__(self):
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193 return len(self.vertices)
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194 @property
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195 def perimeter_lines(self):
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196 for v1, v2 in pairs(self.vertices):
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197 yield LineSegment(v1, v2)
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198 @property
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199 def angles(self):
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200 from math import acos
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201 for v1, v2, v3 in pairs(self.vertices, count = 3):
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202 vec1 = (position_vector(v3) - position_vector(v2)).normalized()
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203 vec2 = (position_vector(v1) - position_vector(v2)).normalized()
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204 cosine = dot_product(vec1, vec2) / vec1.length() / vec2.length()
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205 yield acos(cosine)
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206 @property
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207 def smallest_angle(self):
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208 return min(
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209 angle_magnitude_key(angle)
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210 for angle in self.angles)
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211 @property
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212 def hairline_ratio(self):
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213 lengths = [line.length for line in self.perimeter_lines]
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214 return max(lengths) / min(lengths)
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215
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216 def is_real(number):
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217 return isinstance(number, int) or isinstance(number, float)
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218
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219 def matrix_indices():
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220 '''
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221 Yields index pairs for matrix iteration
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222 '''
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223 from itertools import product
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224 yield from product(range(3), range(3))
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225
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226 class Matrix3x3:
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227 '''
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228 A 3×3 matrix forming the top-left portion of a full 4×4 transformation
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229 matrix.
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230 '''
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231 class Row:
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232 def __init__(self, matrix_ref, i1, i2, i3):
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233 self.matrix_ref = matrix_ref
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234 self.matrix_indices = i1, i2, i3
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235 def __getitem__(self, row_index):
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236 return self.matrix_ref[self.matrix_indices[row_index]]
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237 def __setitem__(self, row_index, value):
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238 self.matrix_ref[self.matrix_indices[row_index]] = value
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239 def __init__(self, values = None):
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240 if values is not None:
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241 assert(all(is_real(x) for x in values))
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242 assert len(values) == 9
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243 self.values = values
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244 else:
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245 self.values = [1, 0, 0, 0, 1, 0, 0, 0, 1]
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246 def __repr__(self):
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247 if self != Matrix3x3():
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248 return str.format('Matrix3x3({!r})', self.values)
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249 else:
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250 return 'Matrix3x3()'
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251 def __getitem__(self, index):
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252 if isinstance(index, tuple):
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253 assert all(k in [0, 1, 2] for k in index)
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254 return self.values[index[0] * 3 + index[1]]
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255 else:
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256 return Matrix3x3.Row(self, index * 3, index * 3 + 1, index * 3 + 2)
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257 def determinant(self):
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258 return sum([
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259 +(self[0, 0] * self[1, 1] * self[2, 2]),
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260 +(self[0, 1] * self[1, 2] * self[2, 0]),
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261 +(self[0, 2] * self[1, 0] * self[2, 1]),
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262 -(self[0, 2] * self[1, 1] * self[2, 0]),
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263 -(self[0, 1] * self[1, 0] * self[2, 2]),
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264 -(self[0, 0] * self[1, 2] * self[2, 1]),
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265 ])
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266 @property
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267 def scaling_vector(self):
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268 '''
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269 Extracts scaling factors from this transformation matrix.
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270 '''
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271 from math import sqrt
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272 return Vector(
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273 x = sqrt(self[0, 0] ** 2 + self[1, 0] ** 2 + self[2, 0] ** 2),
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274 y = sqrt(self[0, 1] ** 2 + self[1, 1] ** 2 + self[2, 1] ** 2),
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275 z = sqrt(self[0, 2] ** 2 + self[1, 2] ** 2 + self[2, 2] ** 2),
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276 )
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277 @property
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278 def rotation_component(self):
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279 '''
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280 Extracts rotation from this matrix.
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281 '''
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282 return self.split()[0]
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283 def split(self):
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284 '''
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285 Extracts the rotation matrix and scaling vector.
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286 '''
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287 vec = self.scaling_vector
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288 return Matrix3x3([
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289 self[i, j] / vec.coordinates[j]
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290 for i, j in matrix_indices()
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291 ]), vec
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292 @property
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293 def contains_scaling(self):
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294 '''
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295 Returns whether this matrix contains scaling factors.
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296 '''
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297 vec = self.scaling_vector
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298 return abs((vec.x * vec.y * vec.z) - 1) >= 1e-10
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299 @property
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300 def contains_rotation(self):
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301 '''
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302 Returns whether this matrix contains rotation factors.
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303 '''
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304 return self.rotation_component != Matrix3x3()
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305 def __eq__(self, other):
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306 '''
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307 Returns whether two matrices are equivalent.
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308 '''
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309 return all(
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310 abs(self[(i, j)] - other[(i, j)]) < 1e-10
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311 for i, j in matrix_indices()
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312 )
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313 def __matmul__(self, other):
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314 '''
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315 Computes the matrix multiplication self × other.
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316 '''
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317 return Matrix3x3([
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318 sum(self[i, k] * other[k, j] for k in range(3))
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319 for i, j in matrix_indices()
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320 ])
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321 def __add__(self, other):
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322 return Matrix3x3([
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323 a + b
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324 for a, b in zip(self.values, other.values)
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325 ])
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326 def __mul__(self, scalar):
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327 return Matrix3x3([
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328 x * scalar
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329 for x in self.values
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330 ])
0
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331
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332 def complete_matrix(matrix, anchor):
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333 '''
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334 Combines a 3×3 matrix and an anchor vertex into a full 4×4
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335 transformation matrix.
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336 '''
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337 return [
9
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338 matrix[0, 0], matrix[0, 1], matrix[0, 2], anchor.x,
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339 matrix[1, 0], matrix[1, 1], matrix[1, 2], anchor.y,
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340 matrix[2, 0], matrix[2, 1], matrix[2, 2], anchor.z,
0
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341 0, 0, 0, 1,
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342 ]
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343
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344 def transform(vertex, transformation_matrix):
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345 '''
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346 Transforms a vertex by a 4×4 transformation matrix.
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347 '''
9
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348 return Vertex(
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349 x = transformation_matrix[0] * vertex.x \
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350 + transformation_matrix[1] * vertex.y \
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351 + transformation_matrix[2] * vertex.z \
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352 + transformation_matrix[3],
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353 y = transformation_matrix[4] * vertex.x \
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354 + transformation_matrix[5] * vertex.y \
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355 + transformation_matrix[6] * vertex.z \
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356 + transformation_matrix[7],
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357 z = transformation_matrix[8] * vertex.x \
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358 + transformation_matrix[9] * vertex.y \
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359 + transformation_matrix[10] * vertex.z \
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360 + transformation_matrix[11],
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361 )
14
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362
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363 def transform_to_xy(polygon):
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364 '''
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365 Transforms the provided polygon into the XY plane. The polygon is
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366 assumed to be planar.
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367
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368 Implementation is based on this StackOverflow answer:
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369 https://math.stackexchange.com/a/476311
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370 '''
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371 v1, v2, v3 = polygon.vertices[:3]
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372 a, b = v3 - v2, v1 - v2
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373 normal = cross_product(a, b).normalized()
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374 v_matrix = Matrix3x3([
15
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375 0, 0, -normal.x,
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376 0, 0, -normal.y,
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377 normal.x, normal.y, 0,
14
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378 ])
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379 try:
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380 rotation_matrix = Matrix3x3()
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381 rotation_matrix += v_matrix
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382 rotation_matrix += (v_matrix @ v_matrix) * (1 / (1 + normal.z))
14
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383 except ZeroDivisionError:
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384 rotation_matrix = Matrix3x3()
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385 full_matrix = complete_matrix(rotation_matrix, Vertex(0, 0, 0))
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386 return Polygon([
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387 transform(vertex = vertex, transformation_matrix = full_matrix)
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388 for vertex in polygon.vertices
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389 ])
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390
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391 def vector_angle(vec_1, vec_2, normalized = False):
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392 from math import acos, pi as π
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393 cosine = dot_product(vec_1, vec_2)
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394 try:
d383f319f35b transform_to_xy implemented and concave test online
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395 cosine /= vec_1.length() * vec_2.length()
d383f319f35b transform_to_xy implemented and concave test online
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396 except ZeroDivisionError:
d383f319f35b transform_to_xy implemented and concave test online
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397 return 0
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398 angle = acos(cosine)
d383f319f35b transform_to_xy implemented and concave test online
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399 if normalized and angle > π / 2:
d383f319f35b transform_to_xy implemented and concave test online
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400 angle = π - angle
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401 return angle
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402
d383f319f35b transform_to_xy implemented and concave test online
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403 def split_quadrilateral(polygon):
d383f319f35b transform_to_xy implemented and concave test online
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404 assert(len(polygon.vertices) == 4)
d383f319f35b transform_to_xy implemented and concave test online
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405 vertices = IndexRing(polygon.vertices)
d383f319f35b transform_to_xy implemented and concave test online
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406 for i in (0, 1):
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407 a, b = vertices[0 + i], vertices[1 + i]
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408 c, d = vertices[2 + i], vertices[3 + i]
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409 yield Polygon([a, b, c]), Polygon([b, c, d])
d383f319f35b transform_to_xy implemented and concave test online
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410
d383f319f35b transform_to_xy implemented and concave test online
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411 def triangle_plane_normal(polygon):
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412 assert(len(polygon.vertices) == 3)
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413 a, b, c = polygon.vertices[:3]
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414 return cross_product(b - a, b - c)

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