Understanding projective change of coordinates.

Question

An affine change of coordinates on $\mathbb A^n$ ,the affine $n$-space is defined to be a polynomial map $T:\mathbb A^n\to \mathbb A^n$ which is given by linear homogeneous polynomials and is also bijective.In short,an affine coordinate change can be visualized as a linear bijection $T$ composed with a translation map.i.e. we are just changing the axes linearly and are allowed to shift the origin.It is quite intuitive.But I think it is more difficult to understand projective change of coordinates.It is defined by the induced map $\mathbb P^n\to \mathbb P^n$ obtained from $\mathbb A^{n+1}\setminus \{0\}\to \mathbb A^{n+1}\setminus \{0\}$,noting that it sends lines to lines.But if we think of a projective real plane $\mathbb P^2$,then how should I visualize a projective change of coordinates.Is there some way which makes things clear.Some help will be appreciated.

Answer 1

First, repeating what you said, projective $n$-space is the set of all lines in $\mathbb{R}^{n+1}$, and projective transformations are the linear transformations of $\mathbb{R}^{n+1}$. So if you are able to visualize linear transformations of a vector space, then you might be able to visualize what a linear tranformation does to the lines. Since any linear transformation can be decomposed into $LDR$, where $L$ and $R$ are rotations and $D$ is a diagonal matrix, I wold say that visualizing what $D$ does is the most difficult part.

Another way I like to visualize projective transformations is in terms of affine space. Affine space consists of the lines passing through the hperplane $A = \{x^{n+1} = 1\}$. The missing part of projective space are the lines in $H = \{x^{n+1} = 0\}$, often called the hyperplane at infinity. Projective transformations decompose into ones that map the hyperplane to itself, which are simply the affine transformations of $A$, and those that swap the hyperplane at infinity with a hyperplane in $A$. This is still in some sense a linear transformation of $A$ composed a translation.