Rotors and Rotations in the Euclidean Plane

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Let i denote the unit bivector for the plane. As you were asked to prove in the previous section, i2 = −1, so i has the properties of the unit imaginary in the complex number system (although it has extra properties, too, since, for example, you also proved that i anticommutes with vectors in its plane). However, by relating it multiplicatively to vectors, GA endows i with two new geometric interpretations:

Operator interpretation: Multiplication by rotates vectors in its plane through a right angle. We can assign an orientation to i so that left multiplication by i rotates vectors in its plane through a CCW (counterclockwise) right angle. Thus, any vector a is rotated into a vector b given by the equation ia –ai b, with the action of depicted by the directed arc in the diagram (Alternatively, we could just as well assign the opposite orientation to i, so that right multiplication by i still rotates a into b in the diagram.):

a b

Object interpretation: Bivector i represents a unit plane segment, as expressed by the following depictions and equations (for a and b perpendicular unit vectors):

a b a b = ab i R = ba i L

If we'd like i to behave as in the previous paragraph (Operator interpretation), choose i = iL, so that:

          ia (counterclockwise rotation) = iLa = baab = aab = aiR = - ai

Note: most GA authors think of the green bivector iL as having a CW (clockwise) orientation and the pink bivector iR as having a CCW orientation. That is independent of how we wish to view their behavior as right angle rotators for vectors in their plane. Since bivectors anticommute with vectors in their plane, multiplying vectors in the plane on the right or the left will produce opposite rotations. As we see here, multiplying a vector in the plane on the right by a CCW bivector will result in a CCW rotation (and hence most authors favor this, feeling comforted by the operation resulting in the same orientation as the bivector used for the operation). However, here we chose to multiply on the left by a CW bivector to produce a CCW rotation. We did it this way (this time) to point out the equally reasonable possibility and because you are probably more accustomed to seeing operators "attack" from the left. 

The operator interpretation of i generalizes to the concept of rotor Uθ , the entity produced by the product ba of unit vectors with relative direction θ.

Rotor Uθ = ba is depicted as a directed arc on the unit circle. Reverse Uθ = ab.

a b θ U θ U θ

Defining sine and cosine functions from products of unit vectors: 

          ab2 1,               i2 = −1               ba  cos θ               ba  sin θ

i cos θ sin θ a b θ The blue bivector sin θ i represents

Rotor: Uθ     ba bba cos θsin θ  eiθ

Rotor equivalence of directed arcs is like vector equivalence of directed line segments:

U U U a a a

Product of coplanar rotors is equivalent to addition of angles:    UθUφ = Uθ+φ UφUθ  or    eiθeiφ =  ei(θ+φ) :

U θ U φ U θ U φ θ+ U φ


Rotor-vector product = vector:    Uθ v  =  eiθ v  =  u :

U θ U θ v u v

Thus rotor algebra represents the algebra of 2d rotations!

The concept of rotor generalizes to the concept of complex number interpreted as a directed arc:

0 1 λ U z z 0 1 λ U

          z = λ= λeiθ ba (for some two vectors in the plane, of as yet unspecified lengths, separated by θ

= complex conjugation:    
z = λU = λeiθ ab

          zz = λ2 (ba)(aba2b2 = |z|2 

          Modulus:    |z| = λ = |a| |b|

Relation to standard complex number notation:    z = Re(z) + i Im(z) = ba 

          with         Re(z) ½ (z + z) = ba                    i Im(z) ½ (z z) = ba

(This representation of complex numbers in a real GA is a special case of spinors for 3d.)

Note the new GA Roots of unity:   

\[\large{\begin{array}{l}{\rm{Two}}\,\,{\rm{roots}}\,\,{\rm{of}}\,\,{\rm{ - 1:}}\,\,\,\,\,\,\sqrt { - 1} \,\,\, = \,\,\, \pm {\bf{i}},\\{\rm{Many}}\,\,{\rm{roots}}\,\,{\rm{of}}\,\,{\rm{ + 1:}}\,\,\,\,\,\,\sqrt { + 1} \,\,\, = \,\,\,{\bf{\hat a}},\,\,\,1.\,\end{array}}\]

Previous sectionDefining and Interpreting the Geometric Product.

Next sectionVector Identities and Plane Trigonometry with GA.

© David Hestenes 2005, 2014