Projection preserves Ratios I have a projection $P$ onto a vector $v$,  $P = \cfrac{vv^T}{||v||^2}$. For intuitive reasons it seems, that ratios of distances are preserved under such projections. 
1: How is this property called exactly (I only found similarity transforms in this context, but the projection being singular is somewhat different).
2: Is there a proof, that $P$ preserves ratios for $d$ dimensions? I think by having a clear definition this point would follow from some kind of linearity argument.
 A: *

*This is called similarity. A singular linear map (i.e. a linear map with non-trivial $\ker$) cannot preserve the ratios of distances (unless it's the zero map). Indeed, let $T\ne 0$ such that $\ker T\ne\{0\}$. Pick $w\in\ker T$ a non-zero vector and $s$ a vector such that $Ts\ne 0$. Then, $$\lVert s+nw\rVert\ge n\lVert w\rVert-\lVert s\rVert$$ while $T(s+nw)=Ts$. But it stands clear that, for sufficiently large $n$, $$\dfrac{\lVert s+nw\rVert}{\lVert s\rVert}\ge\dfrac{n\lVert w\rVert-\lVert s\rVert }{\lVert s\rVert}>1\\\dfrac{\lVert T(s+nw)\rVert }{\lVert Ts\rVert}=1$$

*By the observation above, if $v^Tv=1$, that map does not preserve ratios of distances.
Added: The different property $P(\alpha v)=\alpha Pv$ is sometimes called "$1$-homogenity", though it is most commonly coupled with the additive condition $P(x+y)=P(x)+P(y)$. Together, they are the conditions of linearity, which obviously holds for $P$.
A: One should make a fundamental commentary to the "NO PRESERVATION" as explained by @G. Sassatelli (without any polemics; there is no contradiction) : any affine tranformation $f$ (and in particular a linear transformation, and still more specifically, a projection) DO preserve ratios of distances for ALIGNED points $A,B,C$, because this property is amenable to a preservation of barycenters:
$$C=\lambda A + (1-\lambda)B \ \ \text{implies} \ \ f(C)=\lambda f(A) + (1-\lambda)f(B)$$
by linearity of $f$. 
See e.g, http://mathworld.wolfram.com/AffineTransformation.html
(for example, if $\lambda=1/2$, $C$ is the midpoint of [AC], i.e. $CA/CB=1$ and its image is the midpoint of images, thus $C'A'/C'B'=1$ as well).
This (classical) invariant linked to 3 aligned points is extended into the (also classical) invariant called the cross-ratio for 4 aligned points in the framework of projective transformations.
