Projections

group proj

Projections in Geometric Algebra take on a particularly simple form. For two geometric entities \(a\) and \(b\), there are two cases to consider. First, if the grade of \(a\) is greater than the grade of \(b\), the projection of \(a\) on \(b\) is given by:

\[ \textit{proj}_b a = (a \cdot b) \wedge b \]

The inner product can be thought of as the part of \(b\) least like \(a\). Using the meet operator on this part produces the part of \(b\) most like \(a\). A simple sanity check is to consider the grades of the result. If the grade of \(b\) is less than the grade of \(a\), we end up with an entity with grade \(a - b + b = a\) as expected.

In the second case (the grade of \(a\) is less than the grade of \(b\)), the projection of \(a\) on \(b\) is given by:

\[ \textit{proj}_b a = (a \cdot b) \cdot b \]

It can be verified that as in the first case, the grade of the result is the same as the grade of \(a\). As this projection occurs in the opposite sense from what one may have seen before, additional clarification is provided below.

Summary

Members	Descriptions
public point KLN_VEC_CALL project(point a,line b) noexcept	Project a point onto a line.
public point KLN_VEC_CALL project(point a,plane b) noexcept	Project a point onto a plane.
public line KLN_VEC_CALL project(line a,plane b) noexcept	Project a line onto a plane.
public plane KLN_VEC_CALL project(plane a,point b) noexcept	Project a plane onto a point. Given a plane \(p\) and point \(P\), produces the plane through \(P\) that is parallel to \(p\).
public line KLN_VEC_CALL project(line a,point b) noexcept	Project a line onto a point. Given a line \(\ell\) and point \(P\), produces the line through \(P\) that is parallel to \(\ell\).
public plane KLN_VEC_CALL project(plane a,line b) noexcept	Project a plane onto a line. Given a plane \(p\) and line \(\ell\), produces the plane through \(\ell\) that is parallel to \(p\) if \(p \parallel \ell\).

Members

point KLN_VEC_CALL project(point a,line b) noexcept

Project a point onto a line.

point KLN_VEC_CALL project(point a,plane b) noexcept

Project a point onto a plane.

line KLN_VEC_CALL project(line a,plane b) noexcept

Project a line onto a plane.

plane KLN_VEC_CALL project(plane a,point b) noexcept

Project a plane onto a point. Given a plane \(p\) and point \(P\), produces the plane through \(P\) that is parallel to \(p\).

Intuitively, the point is represented dually in terms of a pencil of planes that converge on the point itself. When we compute \(p | P\), this selects the line perpendicular to \(p\) through \(P\). Subsequently, taking the inner product with \(P\) again selects the plane from the plane pencil of \(P\) least like that line.

line KLN_VEC_CALL project(line a,point b) noexcept

Project a line onto a point. Given a line \(\ell\) and point \(P\), produces the line through \(P\) that is parallel to \(\ell\).

plane KLN_VEC_CALL project(plane a,line b) noexcept

Project a plane onto a line. Given a plane \(p\) and line \(\ell\), produces the plane through \(\ell\) that is parallel to \(p\) if \(p \parallel \ell\).

If \(p \nparallel \ell\), the result will be the plane \(p'\) containing \(\ell\) that maximizes \(p \cdot p'\) (that is, \(p'\) is as parallel to \(p\) as possible).