By the wikipedia definitions found here and here, and especially by the definition implicit in this MSE post, it seems two images are homographic if they are renderings of the same set of points in the same world plane.
In the context of n-body orbital central configurations the notion of homography is also thrown around quite a bit, but seems to be much more restrictive than the "official" definition from projective geometry. In that context, two configurations of points are said to be "homographic" if they are scaled, rotated copies of each other. Formally, the configuration is said to homographic if at any time $t$ its coordinates $q(t)$ satisfy $$q(t)=\lambda(t)Q(t)q(t_0)$$ where $\lambda(t)$ is a time-dependent scalar, $Q(t)$ is a time-dependent rotation matrix, and $t_0$ is a specific point in time. In other words, the configuration of points is said to be homographic if at any specific time in its orbit it is a scaled, rotated copy of itself at any other time in the orbit.
So, isn't this a quite restricted version of the definition of homography offered in the projective geometry links at the top of this post? It seems to me the "official" definition of homography allows for revolving the oculus around the configuration (i.e. the world plane), which leads to foreshortening, meeting of parallel lines, and perhaps other complicated effects which do not result from mere rotation and scaling.
(One researcher Moeckel has suggested that a perhaps more accurate way to refer to scale- and rotation-invariant n-body configurations is to just call them self-similar. Sounds like a good idea to me.)
Can anyone think of any interesting ramifications of applying the broader definition of homography to the field of central configurations?
This other post is relevant.
