As Applied in This Library

All of the concepts in Quick Intro to Geometric Algebra are applied in this library.

Blades

• multivectors.x, y, z, and w represent the first four basis vectors. Bases past 4D use \(\hat e_n\) syntax: multivectors.e5 or multivectors.e_5 represent the 5th basis vector.

• Basis names can be swizzled on the module:

  >>> from multivectors import x, y, z, xyz
  >>> x * y * z == xyz
  True
  

• multivectors._ is a \(0\)-blade: a scalar, but with MultiVector type. It is differentiated from normal scalars when repr()-d by surrounding parentheses:

  >>> from multivectors import _
  >>> 2 * _
  (2.0)
  

• The rules of arithmetic with blades as described above apply:

  >>> from multivectors import x, y, z
  >>> x * 2 + x * 3
  (5.0 * x)
  >>> x * 5 * y
  (5.0 * x*y)
  >>> (x * y) * z == x * (y * z)
  True
  

MultiVector

• A multivector is a sum of zero or more blades. All scalars and blades are multivectors, but not all multivectors are scalars or blades.

• Basis names can be swizzled on class instances to get the coefficient of that basis:

  >>> from multivectors import x, y
  >>> (x + 2*y).x
  1.0
  

• Basis indices can also be used:

  >>> from multivectors import xy, yz
  >>> (xy + 2*yz) % (0, 1)
  1.0
  

• Choosing by grade is supported:

  >>> from multivectors import x, y, xy, yz
  >>> V = 1 + 2*x + 3*y + 4*xy + 5*yz
  >>> V[0]
  (1.0)
  >>> V[1]
  (2.0 * x + 3.0 * y)
  >>> V[2]
  (4.0 * x*y + 5.0 * y*z)
  

• Getting the grade of a multivector will only work if if the multivector is a blade. Otherwise, you get None:

  >>> from multivectors import w, xy, yz
  >>> w.grade
  1
  >>> xy.grade
  2
  >>> yz.grade
  2
  >>> (w + xy).grade
  >>> # returns None
  >>> (xy + yz).grade
  >>> # even multivectors consisting of blades with the same grade don't work
  >>> # the multivector must be an actual blade, with only one basis sequence
  

• The rules of arithmetic with multivectors as described above apply:

  >>> from multivectors import x, y, z
  >>> x * (y + z)
  (1.0 * x*y + 1.0 * x*z)
  >>> (y + z) * (2 * x)
  (-2.0 * x*y + -2.0 * x*z)
  >>> (1*x + 2*y) * (3*x + 4*y)
  (11.0 + -2.0 * x*y)
  

• The extra products apply too:

  >>> from multivectors import x, y, z
  >>> 1*3 + 2*4
  11
  >>> (1*x + 2*y) @ (3*x + 4*y)
  (11.0)
  >>> (1*5 - 2*4, 1*6 - 3*4, 2*6 - 3*5)
  (-3, -6, -3)
  >>> (1*x + 2*y + 3*z) ^ (4*x + 5*y + 6*z)
  (-3.0 * x*y + -6.0 * x*z + -3.0 * y*z)
  

• A convenience method is provided to rotate multivectors:

  >>> from math import radians
  >>> from multivectors import x, y, z, xz
  >>> round((3*x + 2*y + 4*z).rotate(radians(90), xz), 2)
  (-4.0 * x + 2.0 * y + 3.0 * z)