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In slide 84 of "Dual Quaternions Demystified", how are the expressions equal?

Examining the equation for the GP between two points on slide 84 (time code 33:32) of @enkimute’s presentation:

How is x_1\mathbf{e}_{01}+y_1\mathbf{e}_{20}-x_2\mathbf{e}_{01}-y_2\mathbf{e}_{20}-1 = 1+(x_2-x_1)\mathbf{e}_{01} + (y_2-y_1)\mathbf{e}_{20} a valid manipulation? For one, I can’t for the life of me see how the -1 in the scalar part on the LHS could become a 1 in the RHS.

Hi @jogloran,

I swapped the outcome R to -R, these represent the same isometry but the form with 1 + .. is more easily recognizable as being a translation. (should have used \cong instead of =).

I’ll see if I can add it as text/overlay on the vid, thanks for noticing!



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