Transform matrix to scale away from/towards an arbitrary plane in 3D space

Question

I'm not entirely sure if this belongs in Mathematics or GameDev. I'm trying Mathematics first, so please let me know if it's in the wrong place.

In 3D space, I have a plane A given by 3 points A1, A2, and A3, and I need a transform matrix M to scale arbitrary points so their distance to the plane changes by a factor x, along the normal of the plane.

The plane doesn't necessarily pass through the origin so homogeneous coordinates are necessary.

So far I only have convoluted solutions (e.g. rotate and translate plane to match xy plane, scale along z, then reverse rotation and translation) and feel something simpler is possible.

Answer 1

A normal to the plane can be calculated by $n = (A_2 - A_1)\times(A_3 - A_1)$. The unit normal is then $\hat n = \frac n{\|n\|}$ The distance from the plane to the origin is $d = \hat n \cdot A_1$ (or $A_2$ or $A_3$). Thus the equation for points $v$ in the plane is $$\hat n \cdot v = d$$ And the distance of some arbitrary point $w$ from the plane is given by $\hat n \cdot w - d$. More generally, the projection of $w$ onto the plane is $$\widetilde w = w - (\hat n \cdot w - d)\hat n$$

So if we want $w_x$ to a point on the same normal line from the plane passing through $w$, but at a distance of $x$ times the distance of $w$ from the plane, then it would be given by $\widetilde w + x(\hat n \cdot w - d)\hat n$, or: $$\begin{align}w_x &= w - (\hat n \cdot w - d)\hat n+ x(\hat n \cdot w - d)\hat n\\&= w -(1 - x)(\hat n \cdot w - d)\hat n\end{align}$$