Grassmann Algebra

ExploringScrewAlgebra.nb 6
The complement referred to a general point
Let X � ��Α + Β and P � � + Ν. We can refer X to the point P by adding and subtracting Ν�Α
to and from the above expression for X.
Or, equivalently:
X � � � Α�Ν� Α�Β�Ν� Α��� �Ν��Α��Β�Ν� Α�
X � P � Α�Βp Βp �Β�Ν� Α
���������
By manipulating the complement P � Α,
we can write it in the alternative form �
�����
Α����
P.
��������� ���������
P � Α ��Α�P ��Α
����� �����
� P ��Α
�����
���� P
Further, from formula 5.41, we have that:
�Α
����� ���� P � �� � Α � � ���� P
����� ����
Βp � � � Βp
So that the relationship between X and its complement X
����� , can finally be written:
X � P � Α�Βp � X
����� ���� � � � Βp � �� � Α
�
� ���� P
Remember, this formula is valid for the hybrid metric [5.33] in an n-plane of arbitrary
dimension. We explore its application to 3-planes in the section below.
7.4 The Screw
The definition of a screw
A screw is the canonical form of a 2-entity in a three-plane, and may always be written in the
form:
where:
Α is the vector of the screw
P�Α is the central axis of the screw
s
Α
�
is the pitch of the screw
is the bivector of the screw
S � P � Α�s Α �
Remember that Α � is the (free) complement of the vector Α in the three-plane, and hence is a
simple bivector.
2001 4 5
7.4
7.5