Suppose that r,s,u are distinct real numbers.
Let z=(0,0,1), r = (r,1-r,r(r-1)), s = (s,1-s,s(s-1)), u = (u,1-u,u(u-1)).

The p-points [r], [s], [z] lead to a symmetry with matrix

A =   | r/(r-s) s/(r-s) 0 | | (1-r)/(r-s) (1-s)/(r-s) 0 | | r(r-1)/(r-s) s(s-1)/(r-s) (r-s) | The p-points [r], [s], [u] lead to A =   | (s-u)r/(r-s)(u-r) (u-r)s/(r-s)(s-u) (r-s)u/(s-u)(u-r) | | (s-u)(1-r)/(r-s)(u-r) (u-r)(1-s)/(r-s)(s-u) (r-s)(1-u)/(s-u)(u-r) | | (s-u)r(r-1)/(r-s)(u-r) (u-r)s(s-1)/(r-s)(s-u) (r-s)u(u-1)/(s-u)(u-r) | projective symmetries