04-06.14 More geometric algebra

Note: the April project presentations have been shifted to run from Monday, April 7 to Wednesday, April 9.  (No one had signed up for Thursday and there was interest in Monday.)

If you’re wondering what I’m expecting in the project, take a look (again) at EvaluationFormMVWS13 based on last year’s projects.

On Tuesday February 4 Mr. Gunn continued to develop the geometric algebra he began the previous week.  In particular he focused on sandwich operators in the geometric algebra of $\mathbb{R}^3$.  He reviewed the sandwiches with a 1-vector as bread, and showed they are rotations of 180 degrees around the vector.  (These are called half-turns in English or Umwendungen in German).  He then looked into the product of two unit 1-vectors $\mathbf{u}$ and $\mathbf{v}$, which gives rise to a sandwich, rotating around the common perpendicular of $\mathbf{u}$ and $\mathbf{v}$, through twice the angle separating the vectors. Such an element is called a rotor, and has the form \[\cos{\frac{\alpha}{2}} + \sin{\frac{\alpha}{2}} \mathbf{U} \] where $\mathbf{U} := \dfrac{\mathbf{u} \wedge \mathbf{v}}{\sin{\frac{\alpha}{2}}}$ is the normalized form of the plane spanned by the 1-vectors.  There is again an exponential form for such rotors: \[\cos{\frac{\alpha}{2}} + \sin{\frac{\alpha}{2}} \mathbf{U} =e^{\frac{\alpha}{2}\mathbf{U}}\]

Mr. Gunn then gave an overview of the steps remaining to obtain the geometric algebra of the Euclidean plane $\mathbf{E}^2$ from the geometric algebra of the euclidean vector space $\mathbb{R}^3$:

  1. Replace the standard exterior algebra based on the join operator $\vee$ with the dual exterior algebra based on the meet operator $\wedge$.
  2. Projectivize the vector space structure of the exterior algebra to obtain an exterior algebra describing the subspace structure of projective space $P(\mathbb{R}^3)$ instead of the vector space $\mathbb{R}^3$.
  3. Replace the standard inner product $+++$ with the “slightly” degenerate inner product $++0$.

Notationally: instead of the Clifford algebra $Cl(\mathbb{R}_{3,0,0})$ we need the Clifford algebra $P(Cl(\mathbb{R}^*_{2,0,1}))$.  Here, the triple $(3,0,0)$ represents an inner product with signature +++ while $(2,1,0)$ represents one with signature $++-$ and $(2,0,1)$ represents signature $++0$.  The latter is important for the euclidean plane.  To motivate this fact Mr. Gunn turned to a  a speedy review of the geometry of lines in the plane.

Review of lines in the plane. A line whose equation is $ax+by+c=0$ is represented by the vector $\mathbf{v} := [a,b,c]$ or any positive multiple.  The oppositely oriented line is obtained by $-\mathbf{v}$ or any negative multiple. The normal vector to the line is the vector $(-b, a)$. I can normalize  line equations so that $a^2+b^2=1$ (except the  triple $(0,0,c)$ which represents the ideal line of the euclidean plane). For two such normalized euclidean lines, it’s then easy to see that $\mathbf{u} \cdot \mathbf{v} := a_u a_v + b_u b_v$ is the cosine of the angle between the two lines. Notice that this inner product does not involve the $c-$ coordinate.  It’s signature is $(2,0,1)$.  The important point is that the angle between two lines is totally determined by this inner product, even though it is “degenerate”.

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