A geometry library written in Python
Geometer is a geometry library for Python that uses projective geometry and numpy for fast geometric computation. In projective geometry every point in 2D is represented by a three-dimensional vector and every point in 3D is represented by a four-dimensional vector. This has the following advantages:
Most of the computation in the library is done via tensor diagrams (using numpy.einsum).
The source code of the package can be found on GitHub and the documentation on Read the Docs.
You can install the package directly from PyPI:
pip install geometer
from geometer import *
import numpy as np
# Meet and Join operations
p = Point(2, 4)
q = Point(3, 5)
l = Line(p, q)
m = Line(0, 1, 0)
l.meet(m)
# Point(-2, 0)
# Parallel and perpendicular lines
m = l.parallel(through=Point(1, 1))
n = l.perpendicular(through=Point(1, 1))
is_perpendicular(m, n)
# True
# Angles and distances (euclidean)
a = angle(l, Point(1, 0))
p + 2*dist(p, q)*Point(np.cos(a), np.sin(a))
# Point(4, 6)
# Transformations
t1 = translation(0, -1)
t2 = rotation(-np.pi)
t1*t2*p
# Point(-2, -5)
# Collections of points and lines
coordinates = np.random.randint(100, size=(1000, 2))
points = PointCollection([Point(x, y) for x, y in coordinates])
lines = points.join(-points)
zero = PointCollection(np.zeros((1000, 2)), homogenize=True)
lines.meet(rotation(np.pi/2)*lines) == zero
# True
# Ellipses/Quadratic forms
a = Point(-1, 0)
b = Point(0, 3)
c = Point(1, 2)
d = Point(2, 1)
e = Point(0, -1)
conic = Conic.from_points(a, b, c, d, e)
ellipse = Conic.from_foci(c, d, bound=b)
# Geometric shapes
o = Point(0, 0)
x, y = Point(1, 0), Point(0, 1)
r = Rectangle(o, x, x+y, y)
r.area
# 1
# 3-dimensional objects
p1 = Point(1, 1, 0)
p2 = Point(2, 1, 0)
p3 = Point(3, 4, 0)
l = p1.join(p2)
A = join(l, p3)
A.project(Point(3, 4, 5))
# Point(3, 4, 0)
l = Line(Point(1, 2, 3), Point(3, 4, 5))
A.meet(l)
# Point(-2, -1, 0)
p3 = Point(1, 2, 0)
p4 = Point(1, 1, 1)
c = Cuboid(p1, p2, p3, p4)
c.area
# 6
# Cross ratios
t = rotation(np.pi/16)
crossratio(q, t*q, t**2 * q, t**3 * q, p)
# 1.4408954235712448
# Higher dimensions
p1 = Point(1, 1, 4, 0)
p2 = Point(2, 1, 5, 0)
p3 = Point(3, 4, 6, 0)
p4 = Point(0, 2, 7, 0)
E = Plane(p1, p2, p3, p4)
l = Line(Point(0, 0, 0, 0), Point(1, 2, 3, 4))
E.meet(l)
# Point(0, 0, 0, 0)
Many of the algorithms and formulas implemented in the package are taken from the following books and papers: