Kaba
Posts:
289
Registered:
5/23/11


Re: Generalizing Orthogonal Projection
Posted:
Jan 22, 2013 3:21 AM


22.1.2013 7:47, hbertaz@gmail.com wrote: > If L is a 1D subspace of R^3, and q=(x,y,z) is a point not on L. Can we > define the orthogonal projection of q into L, or do we need to have a plane > P (as subspace) , to define an ortho projection of q?
Let p, d, q in R^n, where p is a point on the line L, d is the direction vector of L, and q is the point that is to be projected onto L. Parametrize L by
f : R > R^n: f(t) = p + td.
We want to find a t' such that
dot(f(t')  q, d) = 0,
where dot stands for an inner product. Using bilinearity,
dot(p + t'd  q, d) = 0 <=> t' dot(d, d) + dot(p  q, d) = 0 <=> t' = dot(q  p, d) / dot(d, d).
Therefore the orthogonal projection of q onto L is given by
f(t') = p + t'd.
 http://kaba.hilvi.org

