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:
- There are points at infinity that can be treated just like normal points.
- Projective transformations are described by matrices but they can also represent translations and general affine transformations.
- Two lines have a unique point of intersection if they lie in the same plane. Parallel lines have a point of intersection at infinity.
- Points of intersection, planes or lines through given points can be calculated using simple cross products or tensor diagrams.
- Special complex points at infinity and cross ratios can be used to calculate angles and to construct perpendicular geometric structures.
- Collections of points and lines can be represented by tensors. Their connecting lines and intersections can be calculated using fast matrix multiplications.
Most of the computation in the library is done via tensor diagrams (using numpy.einsum).
Geometer was originally built as a learning exercise and is based on two graduate courses taught at the Technical University Munich. After investing a lot of time in the project, it is now reasonably well tested and the API should be stable.
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 geometerfrom 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:
- Jürgen Richter-Gebert, Perspectives on Projective Geometry
- Jürgen Richter-Gebert and Thorsten Orendt, Geometriekalküle
- Olivier Faugeras, Three-Dimensional Computer Vision
- Jim Blinn, Lines in Space: The 4D Cross Product
- Jim Blinn, Lines in Space: The Line Formulation
- Jim Blinn, Lines in Space: The Two Matrices
- Jim Blinn, Lines in Space: Back to the Diagrams
- Jim Blinn, Lines in Space: A Tale of Two Lines
- Jim Blinn, Lines in Space: Our Friend the Hyperbolic Paraboloid
- Jim Blinn, Lines in Space: The Algebra of Tinkertoys
- Jim Blinn, Lines in Space: Line(s) through Four Lines