Editing Geometric algebra (section)

=== Projection and rejection ===
[[File:GA plane subspace and projection.svg|right|300px|thumb|In 3D space, a bivector <math>a \land b</math> defines a 2D plane subspace (light blue, extends infinitely in indicated directions). Any vector <math>c</math> in 3D space can be decomposed into its projection <math>c_\Vert</math> onto a plane and its rejection <math>c_\perp</math> from this plane.]]

For any vector <math>a</math> and any invertible vector {{tmath|1= m }},
: <math> a = amm^{-1} = (a\cdot m + a \wedge m)m^{-1} = a_{\| m} + a_{\perp m} ,</math>
where the '''projection''' of <math>a</math> onto <math>m</math> (or the parallel part) is
: <math> a_{\| m} = (a \cdot m)m^{-1} </math>
and the '''rejection''' of <math>a</math> from <math>m</math> (or the orthogonal part) is
: <math> a_{\perp m} = a - a_{\| m} = (a\wedge m)m^{-1} .</math>

Using the concept of a {{tmath|1= k }}-blade {{tmath|1= B }} as representing a subspace of {{tmath|1= V }} and every multivector ultimately being expressed in terms of vectors, this generalizes to projection of a general multivector onto any invertible {{tmath|1= k }}-blade {{tmath|1= B }} as{{efn|This definition follows {{harvp|Dorst|Fontijne|Mann|2007}} and {{harvp|Perwass|2009}} – the left contraction used by Dorst replaces the ("fat dot") inner product that Perwass uses, consistent with Perwass's constraint that grade of {{tmath|1= A }} may not exceed that of {{tmath|1= B }}.}}
: <math> \mathcal{P}_B (A) = (A \;\rfloor\; B) \;\rfloor\; B^{-1} ,</math>
with the rejection being defined as
: <math> \mathcal{P}_B^\perp (A) = A - \mathcal{P}_B (A) .</math>

The projection and rejection generalize to null blades <math>B</math> by replacing the inverse <math>B^{-1}</math> with the pseudoinverse <math>B^{+}</math> with respect to the contractive product.{{efn|Dorst appears to merely assume <math>B^{+}</math> such that {{tmath|1= B \;\rfloor\; B^{+} = 1 }}, whereas {{harvp|Perwass|2009}} defines {{tmath|1= B^{+} = B^{\dagger}/(B \;\rfloor\; B^{\dagger}) }}, where <math>B^{\dagger}</math> is the conjugate of {{tmath|1= B }}, equivalent to the reverse of <math>B</math> up to a sign.}} The outcome of the projection coincides in both cases for non-null blades.{{sfn|ps=|Dorst|Fontijne|Mann|2007|loc=§3.6 p. 85}}{{sfn|ps=|Perwass|2009|loc=§3.2.10.2 p. 83}} For null blades {{tmath|1= B }}, the definition of the projection given here with the first contraction rather than the second being onto the pseudoinverse should be used,{{efn|That is to say, the projection must be defined as {{tmath|1= \mathcal{P}_{B}(A) = (A \;\rfloor\; B^{+}) \;\rfloor\; B }} and not as {{tmath|1= (A \;\rfloor\; B) \;\rfloor\; B^{+} }}, though the two are equivalent for non-null blades {{tmath|1= B }}.}} as only then is the result necessarily in the subspace represented by {{tmath|1= B }}.{{sfn|ps=|Dorst|Fontijne|Mann|2007|loc=§3.6 p. 85}}
The projection generalizes through linearity to general multivectors {{tmath|1= A }}.{{efn|This generalization to all {{tmath|1= A }} is apparently not considered by Perwass or Dorst.}} The projection is not linear in {{tmath|1= B }} and does not generalize to objects {{tmath|1= B }} that are not blades.