Tion matrix of reflection through an arbitrary plane with the same deduction. Three dimensional point transformation is one of the well known computer graphics.
![Arbitrary Arbitrary](/uploads/1/2/5/4/125482849/357069358.jpg)
![Reflection about arbitrary line in graphics examples Reflection about arbitrary line in graphics examples](/uploads/1/2/5/4/125482849/500036153.jpg)
I don't suppose that there are many methods other than the two you suggested, but here's what you could do which is closer to your first one. The following requires no knowledge of vectors or matrices.Given a line $M:y=mx+k$ and a point $A(a,b)$ on the curve $f(x)$, the line perpendicular to $M$ through $A$ is $P:(a-x)/m+b$. The point of intersection of $M$ and $P$ is $I(m(b+k)+a/m^2+1,m^2(b+k)+ma/m^2+1+k)$.Suppose that after the reflection of $(a,b)$, the new point is $B(c,d)$, or $B(c,(a-c)/m+b)$. Then this requires $AI=IB$, which is a quadratic in terms of $c$. Solving this and choosing the correct root gives $c=g(a,b)$, and hence $d$.
The algebra may become quite fiddly though.