Are the following sets of coordinates affinely equivalent?

Question

Given the coordinates $\{(0,t^{k}),(t^{k},0)\}$ for $k\in\{1,...,n\}$ and for $t>1$, and $\{(\alpha s^{k-1},0),(0,\alpha s^{k-1})\}$ for $k\in \{1,...,n\}$ and for $s>1$ and $\alpha>0$, are the two sets of coordinates affinely equivalent to one another? I know that affine equivalence must preserve the parallelity / collinearity of lines, which both sets of points seem to do with respect to the other set. Is there anything additional that I should also take into consideration?

Answer 1

No, they're not. In both cases you have points lying on exactly two straight lines; any affine transformation that maps the first set of points onto the second will necessarily preserve the intersection of the two lines (i.e., the origin). An affine transformation also preserves the ratios of distances between points lying on a straight line (Wikipedia:Affine_transformation). For the first set of points, there are distances $t$, $t^2$, etc. along each line (measured from the origin), with a common ratio $t$; for the second set, it's $\alpha$, $\alpha s$, $\alpha s^2$, etc., with a common ratio $s$. If $s\neq t$, the two sets of points can't be affinely equivalent.