See simson for details and notation.
A typical line L through F has the form kx/f+ly/g+mz)/h = 0, with k+l+m = 0.
This meets K at F twice, and at one further point. To incorporate the condition
k+l+m = 0, we replace k by -l-m. At the end, we observe that (l+m) occurs in
the formulae, so can be replaced by -k. This leads to the map
[x] -> [f(tz/h-sy/g)/x]
This maps the line at infinity to K.
We can replace the line at infinity by any other line (or any unicursal curve).
We could take the tripolar of R/F. Then the formula is [x] -> [r(y-z)/x].
This allows construction of cubics without conic tools.
We can also use a conic as domain. A point lies on a conic if its isotome - or
other isoconjugation - lies on a line.
If we choose the conic as in simson, we get the simson transform. But this
algebraic approach does not reveal the geometric significance.