Date: Jan 22, 2013 3:21 AM
Subject: Re: Generalizing Orthogonal Projection
22.1.2013 7:47, firstname.lastname@example.org wrote:
> If L is a 1-D 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.