Composition of projections


Geometric algebra permits a formula for the composition of two arbitrary projections. This applet illustrates the formula in 2D, where it reads for projections Pa and: Pb onto the lines a and b:

2 Pa(Pb(x)) = Pa(x) + Pb(x) - Pab(x)

Here Pab is the projection onto a versor. The result is due to Tim Bouma.



I have not put any labels to clutter the figure, but follow the red and green arrows to get the two composite projections. The blue parallellogram is spanned from Pa(x)/2 and Pb(x)/2, and the final connections to the results use half the versor projections.

You can drag x and a or b to get a feeling for the construction. Make sure you do the special cases where a and b make an angle of 90 degrees, and of 0 degrees.


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