quoracle/quorums/geometry.py

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from typing import Any, Callable, List, NamedTuple, Optional, Tuple
import math
import unittest
class Point(NamedTuple):
x: float
y: float
class Segment:
def __init__(self, l: Point, r: Point) -> None:
assert l != r
assert l.x < r.x
self.l = l
self.r = r
def __str__(self) -> str:
return f'{tuple(self.l)} -> {tuple(self.r)}'
def __repr__(self) -> str:
return f'Segment({self.l}, {self.r})'
def __eq__(self, other) -> bool:
if isinstance(other, Segment):
return (self.l, self.r) == (other.l, other.r)
else:
return False
def __hash__(self) -> int:
return hash((self.l, self.r))
def __call__(self, x: float) -> float:
assert self.l.x <= x <= self.r.x
return self.slope() * (x - self.l.x) + self.l.y
def approximately_equal(self, other: 'Segment') -> float:
return (math.isclose(self.l.y, other.l.y, rel_tol=1e-5) and
math.isclose(self.r.y, other.r.y, rel_tol=1e-5))
def compatible(self, other: 'Segment') -> float:
return self.l.x == other.l.x and self.r.x == other.r.x
def slope(self) -> float:
return (self.r.y - self.l.y) / (self.r.x - self.l.x)
def above(self, other: 'Segment') -> bool:
assert self.compatible(other)
return self != other and self.l.y >= other.l.y and self.r.y >= other.r.y
def above_eq(self, other: 'Segment') -> bool:
assert self.compatible(other)
return self == other or self.above(other)
def intersects(self, other: 'Segment') -> bool:
assert self.compatible(other)
if self == other:
return True
elif self.l.y == other.l.y or self.r.y == other.r.y:
return True
elif self.above(other) or other.above(self):
return False
else:
return True
def intersection(self, other: 'Segment') -> Optional[Point]:
assert self.compatible(other)
if self == other or not self.intersects(other):
return None
x = ((other.l.y - self.l.y) /
(self.r.y - other.r.y + other.l.y - self.l.y))
return Point(x, self(x))
def max_of_segments(segments: List[Segment]) -> List[Tuple[float, float]]:
assert len(segments) > 0
assert len({segment.l.x for segment in segments}) == 1
assert len({segment.r.x for segment in segments}) == 1
# We compute the x-coordinate of every intersection point. We sort the
# x-coordinates and for every x, we compute the highest line at that point.
xs: List[float] = [0.0, 1.0]
for (i, s1) in enumerate(segments):
for (j, s2) in enumerate(segments[i + 1:], i + 1):
p = s1.intersection(s2)
if p is not None:
xs.append(p.x)
xs.sort()
return [(x, max(segments, key=lambda s: s(x))(x)) for x in xs]
class TestGeometry(unittest.TestCase):
def test_eq(self):
l = Point(0, 1)
r = Point(1, 1)
m = Point(0.5, 0.5)
self.assertEqual(Segment(l, r), Segment(l, r))
self.assertNotEqual(Segment(l, r), Segment(l, m))
def test_compatible(self):
s1 = Segment(Point(0, 1), Point(1, 2))
s2 = Segment(Point(0, 2), Point(1, 1))
s3 = Segment(Point(0.5, 2), Point(1, 1))
self.assertTrue(s1.compatible(s2))
self.assertTrue(s2.compatible(s1))
self.assertFalse(s1.compatible(s3))
self.assertFalse(s3.compatible(s1))
self.assertFalse(s2.compatible(s3))
self.assertFalse(s3.compatible(s2))
def test_call(self):
segment = Segment(Point(0, 0), Point(1, 1))
for x in [0.0, 0.25, 0.5, 0.75, 1.0]:
self.assertEqual(segment(x), x)
segment = Segment(Point(0, 0), Point(1, 2))
for x in [0.0, 0.25, 0.5, 0.75, 1.0]:
self.assertEqual(segment(x), 2*x)
segment = Segment(Point(1, 2), Point(3, 6))
for x in [1.0, 1.25, 1.5, 1.75, 2.0, 2.25, 2.5, 2.75, 3.0]:
self.assertEqual(segment(x), 2*x)
segment = Segment(Point(0, 1), Point(1, 0))
for x in [0.0, 0.25, 0.5, 0.75, 1.0]:
self.assertEqual(segment(x), 1 - x)
def test_slope(self):
self.assertEqual(Segment(Point(0, 0), Point(1, 1)).slope(), 1.0)
self.assertEqual(Segment(Point(0, 1), Point(1, 2)).slope(), 1.0)
self.assertEqual(Segment(Point(1, 1), Point(2, 2)).slope(), 1.0)
self.assertEqual(Segment(Point(1, 1), Point(2, 3)).slope(), 2.0)
self.assertEqual(Segment(Point(1, 1), Point(2, 0)).slope(), -1.0)
def test_above(self):
s1 = Segment(Point(0, 0), Point(1, 0.5))
s2 = Segment(Point(0, 0.5), Point(1, 2))
s3 = Segment(Point(0, 1.5), Point(1, 0.5))
self.assertFalse(s1.above(s1))
self.assertFalse(s1.above(s2))
self.assertFalse(s1.above(s3))
self.assertTrue(s2.above(s1))
self.assertFalse(s2.above(s2))
self.assertFalse(s2.above(s3))
self.assertTrue(s3.above(s1))
self.assertFalse(s3.above(s2))
self.assertFalse(s3.above(s3))
def test_above_eq(self):
s1 = Segment(Point(0, 0), Point(1, 0.5))
s2 = Segment(Point(0, 0.5), Point(1, 2))
s3 = Segment(Point(0, 1.5), Point(1, 0.5))
self.assertTrue(s1.above_eq(s1))
self.assertFalse(s1.above_eq(s2))
self.assertFalse(s1.above_eq(s3))
self.assertTrue(s2.above_eq(s1))
self.assertTrue(s2.above_eq(s2))
self.assertFalse(s2.above_eq(s3))
self.assertTrue(s3.above_eq(s1))
self.assertFalse(s3.above_eq(s2))
self.assertTrue(s3.above_eq(s3))
def test_intersects(self):
s1 = Segment(Point(0, 0), Point(1, 0.5))
s2 = Segment(Point(0, 0.5), Point(1, 2))
s3 = Segment(Point(0, 1.5), Point(1, 0.5))
self.assertTrue(s1.intersects(s1))
self.assertFalse(s1.intersects(s2))
self.assertTrue(s1.intersects(s3))
self.assertFalse(s2.intersects(s1))
self.assertTrue(s2.intersects(s2))
self.assertTrue(s2.intersects(s3))
self.assertTrue(s3.intersects(s1))
self.assertTrue(s3.intersects(s2))
self.assertTrue(s3.intersects(s3))
def test_intersection(self):
s1 = Segment(Point(0, 0), Point(1, 1))
s2 = Segment(Point(0, 1), Point(1, 0))
s3 = Segment(Point(0, 1), Point(1, 1))
s4 = Segment(Point(0, 0.25), Point(1, 0.25))
self.assertEqual(s1.intersection(s1), None)
self.assertEqual(s1.intersection(s2), Point(0.5, 0.5))
self.assertEqual(s1.intersection(s3), Point(1, 1))
self.assertEqual(s1.intersection(s4), Point(0.25, 0.25))
self.assertEqual(s2.intersection(s1), Point(0.5, 0.5))
self.assertEqual(s2.intersection(s2), None)
self.assertEqual(s2.intersection(s3), Point(0, 1))
self.assertEqual(s2.intersection(s4), Point(0.75, 0.25))
self.assertEqual(s3.intersection(s1), Point(1, 1))
self.assertEqual(s3.intersection(s2), Point(0, 1))
self.assertEqual(s3.intersection(s3), None)
self.assertEqual(s3.intersection(s4), None)
self.assertEqual(s4.intersection(s1), Point(0.25, 0.25))
self.assertEqual(s4.intersection(s2), Point(0.75, 0.25))
self.assertEqual(s4.intersection(s3), None)
self.assertEqual(s4.intersection(s4), None)
def test_max_one_segment(self):
s1 = Segment(Point(0, 0), Point(1, 1))
s2 = Segment(Point(0, 1), Point(1, 0))
s3 = Segment(Point(0, 1), Point(1, 1))
s4 = Segment(Point(0, 0.25), Point(1, 0.25))
s5 = Segment(Point(0, 0.75), Point(1, 0.75))
def is_subset(xs: List[Any], ys: List[Any]) -> bool:
return all(x in ys for x in xs)
for s in [s1, s2, s3, s4, s5]:
self.assertEqual(max_of_segments([s]), [s.l, s.r])
expected = [
([s1, s1], [(0, 0), (1, 1)]),
([s1, s2], [(0, 1), (0.5, 0.5), (1, 1)]),
([s1, s3], [(0, 1), (1, 1)]),
([s1, s4], [(0, 0.25), (0.25, 0.25), (1, 1)]),
([s1, s5], [(0, 0.75), (0.75, 0.75), (1, 1)]),
([s2, s2], [(0, 1), (1, 0)]),
([s2, s3], [(0, 1), (1, 1)]),
([s2, s4], [(0, 1), (0.75, 0.25), (1, 0.25)]),
([s2, s5], [(0, 1), (0.25, 0.75), (1, 0.75)]),
([s3, s3], [(0, 1), (1, 1)]),
([s3, s4], [(0, 1), (1, 1)]),
([s3, s5], [(0, 1), (1, 1)]),
([s4, s4], [(0, 0.25), (1, 0.25)]),
([s4, s5], [(0, 0.75), (1, 0.75)]),
([s5, s5], [(0, 0.75), (1, 0.75)]),
([s1, s2, s4], [(0, 1), (0.5, 0.5), (1, 1)]),
([s1, s2, s5], [(0, 1), (0.25, 0.75), (0.75, 0.75), (1, 1)]),
]
for segments, path in expected:
self.assertTrue(is_subset(path, max_of_segments(segments)))
self.assertTrue(is_subset(path, max_of_segments(segments[::-1])))
if __name__ == '__main__':
unittest.main()