Geometry

Introduction

The geometry module for SymPy allows one to create two-dimensional geometrical entities, such as lines and circles, and query for information about these entities. This could include asking the area of an ellipse, checking for collinearity of a set of points, or finding the intersection between two lines. The primary use case of the module involves entities with numerical values, but it is possible to also use symbolic representations.

Available Entities

The following entities are currently available in the geometry module:

Most of the work one will do will be through the properties and methods of these entities, but several global methods exist:

  • intersection(entity1, entity2)

  • are_similar(entity1, entity2)

  • convex_hull(points)

For a full API listing and an explanation of the methods and their return values please see the list of classes at the end of this document.

Example Usage

The following Python session gives one an idea of how to work with some of the geometry module.

>>> from sympy import *
>>> from sympy.geometry import *
>>> x = Point(0, 0)
>>> y = Point(1, 1)
>>> z = Point(2, 2)
>>> zp = Point(1, 0)
>>> Point.is_collinear(x, y, z)
True
>>> Point.is_collinear(x, y, zp)
False
>>> t = Triangle(zp, y, x)
>>> t.area
1/2
>>> t.medians[x]
Segment2D(Point2D(0, 0), Point2D(1, 1/2))
>>> m = t.medians
>>> intersection(m[x], m[y], m[zp])
[Point2D(2/3, 1/3)]
>>> c = Circle(x, 5)
>>> l = Line(Point(5, -5), Point(5, 5))
>>> c.is_tangent(l) # is l tangent to c?
True
>>> l = Line(x, y)
>>> c.is_tangent(l) # is l tangent to c?
False
>>> intersection(c, l)
[Point2D(-5*sqrt(2)/2, -5*sqrt(2)/2), Point2D(5*sqrt(2)/2, 5*sqrt(2)/2)]

Intersection of medians

>>> from sympy import symbols
>>> from sympy.geometry import Point, Triangle, intersection

>>> a, b = symbols("a,b", positive=True)

>>> x = Point(0, 0)
>>> y = Point(a, 0)
>>> z = Point(2*a, b)
>>> t = Triangle(x, y, z)

>>> t.area
a*b/2

>>> t.medians[x]
Segment2D(Point2D(0, 0), Point2D(3*a/2, b/2))

>>> intersection(t.medians[x], t.medians[y], t.medians[z])
[Point2D(a, b/3)]

An in-depth example: Pappus’ Hexagon Theorem

From Wikipedia ([WikiPappus]):

Given one set of collinear points \(A\), \(B\), \(C\), and another set of collinear points \(a\), \(b\), \(c\), then the intersection points \(X\), \(Y\), \(Z\) of line pairs \(Ab\) and \(aB\), \(Ac\) and \(aC\), \(Bc\) and \(bC\) are collinear.

>>> from sympy import *
>>> from sympy.geometry import *
>>>
>>> l1 = Line(Point(0, 0), Point(5, 6))
>>> l2 = Line(Point(0, 0), Point(2, -2))
>>>
>>> def subs_point(l, val):
...    """Take an arbitrary point and make it a fixed point."""
...    t = Symbol('t', real=True)
...    ap = l.arbitrary_point()
...    return Point(ap.x.subs(t, val), ap.y.subs(t, val))
...
>>> p11 = subs_point(l1, 5)
>>> p12 = subs_point(l1, 6)
>>> p13 = subs_point(l1, 11)
>>>
>>> p21 = subs_point(l2, -1)
>>> p22 = subs_point(l2, 2)
>>> p23 = subs_point(l2, 13)
>>>
>>> ll1 = Line(p11, p22)
>>> ll2 = Line(p11, p23)
>>> ll3 = Line(p12, p21)
>>> ll4 = Line(p12, p23)
>>> ll5 = Line(p13, p21)
>>> ll6 = Line(p13, p22)
>>>
>>> pp1 = intersection(ll1, ll3)[0]
>>> pp2 = intersection(ll2, ll5)[0]
>>> pp3 = intersection(ll4, ll6)[0]
>>>
>>> Point.is_collinear(pp1, pp2, pp3)
True

References

[WikiPappus]

“Pappus’s Hexagon Theorem” Wikipedia, the Free Encyclopedia. Web. 26 Apr. 2013. <https://en.wikipedia.org/wiki/Pappus’s_hexagon_theorem>

Miscellaneous Notes

  • The area property of Polygon and Triangle may return a positive or negative value, depending on whether or not the points are oriented counter-clockwise or clockwise, respectively. If you always want a positive value be sure to use the abs function.

  • Although Polygon can refer to any type of polygon, the code has been written for simple polygons. Hence, expect potential problems if dealing with complex polygons (overlapping sides).

  • Since SymPy is still in its infancy some things may not simplify properly and hence some things that should return True (e.g., Point.is_collinear) may not actually do so. Similarly, attempting to find the intersection of entities that do intersect may result in an empty result.

Future Work

Truth Setting Expressions

When one deals with symbolic entities, it often happens that an assertion cannot be guaranteed. For example, consider the following code:

>>> from sympy import *
>>> from sympy.geometry import *
>>> x,y,z = map(Symbol, 'xyz')
>>> p1,p2,p3 = Point(x, y), Point(y, z), Point(2*x*y, y)
>>> Point.is_collinear(p1, p2, p3)
False

Even though the result is currently False, this is not always true. If the quantity \(z - y - 2*y*z + 2*y**2 == 0\) then the points will be collinear. It would be really nice to inform the user of this because such a quantity may be useful to a user for further calculation and, at the very least, being nice to know. This could be potentially done by returning an object (e.g., GeometryResult) that the user could use. This actually would not involve an extensive amount of work.

Three Dimensions and Beyond

Currently a limited subset of the geometry module has been extended to three dimensions, but it certainly would be a good addition to extend more. This would probably involve a fair amount of work since many of the algorithms used are specific to two dimensions.

Geometry Visualization

The plotting module is capable of plotting geometric entities. See Plotting Geometric Entities in the plotting module entry.

Submodules