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Norm of Lines in 3D PGA

The 3DPGA cheat sheet states the norm of a euclidean line to be \lVert \ell \rVert = \sqrt{\ell^2} = \sqrt{d^2+e^2+f^2}.

When I use \ell = a{\bf e_{01}} + b{\bf e_{02}} + c{\bf e_{03}} + d{\bf e_{12}} + e{\bf e_{31}} + f{\bf e_{23}} and compute \ell^2 I get \ell^2 = -d^2-e^2-f^2 + 2(af + be + cd){\bf e_{0123}} which as the Section 8.1.3 of the SIGGRAPH lecture note mentions is a dual number. Even though I haven’t read it anywhere (?) I assume that all lines are simple bivectors (since they can be seen as the intersection of two planes and the product of two planes always results in a simple bivector), and similarly the Plücker quadric relation equals the pseudoscalar part being zero. Nevertheless, the norm then should be \sqrt{-\ell^2} or \sqrt{|\ell^2|}, right?

Bonus question: If not all bivectors are lines, what do non-simple bivectors represent then in 3D PGA?

Hi @phew,

Yes, the norm of a line should be \sqrt{l \tilde l} or \sqrt{\lvert l^2 \rvert}. (and yes, all lines are simple bivectors). The simple bivectors encode all translations and rotations.

On the bonus question, all euclidean non-simple bivectors are screw axi. (see secion 8.1.3 and 8.1.4 of the course notes).

Cheers,

S.

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