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In [1]:

    
import SymPy: symbols, sympy, Sym
using GAlgebra



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galgebra.printer.Format()

The following follows definitions in Homogeneous Coordinates for Computational Geometry by Hestenes and Rockwood.



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CGA3D = galgebra.ga.Ga("e_1 e_2 e_3 e e_{0}", g="1 0 0 0 0,0 1 0 0 0,0 0 1 0 0,0 0 0 0 -1,0 0 0 -1 0")









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PyObject <galgebra.ga.Ga object at 0x119eba160>



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galgebra.ga.Ga.dual_mode_value = "Iinv+"









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"Iinv+"



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sympy.Matrix(CGA3D.g)









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\[\left[ \begin{array}{rrrrr}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&0&-1\\0&0&0&-1&0\end{array}\right]\]



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e1,e2,e3,e,e0 = CGA3D.mv()
e1+e2+e3+e+e0









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\begin{align*} \boldsymbol{e}_{1} + \boldsymbol{e}_{2} + \boldsymbol{e}_{3} + \boldsymbol{e} + \boldsymbol{e}_{{0}}\end{align*}



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A = CGA3D.mv("A", "mv")









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\begin{align*}A  + A^{1} \boldsymbol{e}_{1} + A^{2} \boldsymbol{e}_{2} + A^{3} \boldsymbol{e}_{3} + A^{} \boldsymbol{e} + A^{{0}} \boldsymbol{e}_{{0}} + A^{12} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2} + A^{13} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3} + A^{1} \boldsymbol{e}_{1}\wedge \boldsymbol{e} + A^{1{0}} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{{0}} + A^{23} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3} + A^{2} \boldsymbol{e}_{2}\wedge \boldsymbol{e} + A^{2{0}} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{{0}} + A^{3} \boldsymbol{e}_{3}\wedge \boldsymbol{e} + A^{3{0}} \boldsymbol{e}_{3}\wedge \boldsymbol{e}_{{0}} + A^{{0}} \boldsymbol{e}\wedge \boldsymbol{e}_{{0}} + A^{123} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3} + A^{12} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e} + A^{12{0}} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{{0}} + A^{13} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e} + A^{13{0}} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}_{{0}} + A^{1{0}} \boldsymbol{e}_{1}\wedge \boldsymbol{e}\wedge \boldsymbol{e}_{{0}} + A^{23} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e} + A^{23{0}} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}_{{0}} + A^{2{0}} \boldsymbol{e}_{2}\wedge \boldsymbol{e}\wedge \boldsymbol{e}_{{0}} + A^{3{0}} \boldsymbol{e}_{3}\wedge \boldsymbol{e}\wedge \boldsymbol{e}_{{0}} + A^{123} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e} + A^{123{0}} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}_{{0}} + A^{12{0}} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}\wedge \boldsymbol{e}_{{0}} + A^{13{0}} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}\wedge \boldsymbol{e}_{{0}} + A^{23{0}} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}\wedge \boldsymbol{e}_{{0}} + A^{123{0}} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}\wedge \boldsymbol{e}_{{0}}\end{align*}



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I = CGA3D.I()









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\begin{align*} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}\wedge \boldsymbol{e}_{{0}}\end{align*}



In [9]:

    
display("text/latex", A.Fmt(2))









    





 \begin{align*}  & A  \\  &  + A^{1} \boldsymbol{e}_{1} + A^{2} \boldsymbol{e}_{2} + A^{3} \boldsymbol{e}_{3} + A^{} \boldsymbol{e} + A^{{0}} \boldsymbol{e}_{{0}} \\  &  + A^{12} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2} + A^{13} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3} + A^{1} \boldsymbol{e}_{1}\wedge \boldsymbol{e} + A^{1{0}} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{{0}} + A^{23} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3} + A^{2} \boldsymbol{e}_{2}\wedge \boldsymbol{e} + A^{2{0}} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{{0}} + A^{3} \boldsymbol{e}_{3}\wedge \boldsymbol{e} + A^{3{0}} \boldsymbol{e}_{3}\wedge \boldsymbol{e}_{{0}} + A^{{0}} \boldsymbol{e}\wedge \boldsymbol{e}_{{0}} \\  &  + A^{123} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3} + A^{12} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e} + A^{12{0}} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{{0}} + A^{13} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e} + A^{13{0}} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}_{{0}} + A^{1{0}} \boldsymbol{e}_{1}\wedge \boldsymbol{e}\wedge \boldsymbol{e}_{{0}} + A^{23} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e} + A^{23{0}} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}_{{0}} + A^{2{0}} \boldsymbol{e}_{2}\wedge \boldsymbol{e}\wedge \boldsymbol{e}_{{0}} + A^{3{0}} \boldsymbol{e}_{3}\wedge \boldsymbol{e}\wedge \boldsymbol{e}_{{0}} \\  &  + A^{123} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e} + A^{123{0}} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}_{{0}} + A^{12{0}} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}\wedge \boldsymbol{e}_{{0}} + A^{13{0}} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}\wedge \boldsymbol{e}_{{0}} + A^{23{0}} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}\wedge \boldsymbol{e}_{{0}} \\  &  + A^{123{0}} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}\wedge \boldsymbol{e}_{{0}}  \end{align*}



In [10]:

    
display("text/latex", (A').Fmt(2))









    





 \begin{align*}  & A^{123{0}}  \\  &  + A^{23{0}} \boldsymbol{e}_{1} - A^{13{0}} \boldsymbol{e}_{2} + A^{12{0}} \boldsymbol{e}_{3} - A^{123} \boldsymbol{e} + A^{123{0}} \boldsymbol{e}_{{0}} \\  &  - A^{3{0}} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2} + A^{2{0}} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3} - A^{23} \boldsymbol{e}_{1}\wedge \boldsymbol{e} + A^{23{0}} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{{0}} - A^{1{0}} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3} + A^{13} \boldsymbol{e}_{2}\wedge \boldsymbol{e} - A^{13{0}} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{{0}} - A^{12} \boldsymbol{e}_{3}\wedge \boldsymbol{e} + A^{12{0}} \boldsymbol{e}_{3}\wedge \boldsymbol{e}_{{0}} + A^{123} \boldsymbol{e}\wedge \boldsymbol{e}_{{0}} \\  &  - A^{{0}} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3} + A^{3} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e} - A^{3{0}} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{{0}} - A^{2} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e} + A^{2{0}} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}_{{0}} + A^{23} \boldsymbol{e}_{1}\wedge \boldsymbol{e}\wedge \boldsymbol{e}_{{0}} + A^{1} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e} - A^{1{0}} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}_{{0}} - A^{13} \boldsymbol{e}_{2}\wedge \boldsymbol{e}\wedge \boldsymbol{e}_{{0}} + A^{12} \boldsymbol{e}_{3}\wedge \boldsymbol{e}\wedge \boldsymbol{e}_{{0}} \\  &  + A^{} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e} - A^{{0}} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}_{{0}} - A^{3} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}\wedge \boldsymbol{e}_{{0}} + A^{2} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}\wedge \boldsymbol{e}_{{0}} - A^{1} \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}\wedge \boldsymbol{e}_{{0}} \\  &  - A \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}\wedge \boldsymbol{e}_{{0}}  \end{align*}



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I









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\begin{align*} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}\wedge \boldsymbol{e}_{{0}}\end{align*}



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(I)⁻¹









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\begin{align*}- \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}\wedge \boldsymbol{e}_{{0}}\end{align*}



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I == - (I)⁻¹









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true



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dual(A) == A * inv(I)









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true



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A' == A * (I)⁻¹









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true



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(I)⁻¹ == (I)ǂ == I^2 * I









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true



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e'









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\begin{align*} \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}\end{align*}



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e0'









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\begin{align*}- \boldsymbol{e}_{1}\wedge \boldsymbol{e}_{2}\wedge \boldsymbol{e}_{3}\wedge \boldsymbol{e}_{{0}}\end{align*}



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