Equations of Planes

A plane is simply a flat sheet that goes on forever, like a tabletop with no edges, and you can describe it by one point on it and which way it faces. Planes are the flat pieces that 3D models, walls and solar panels are made of. In $\vec n\cdot(\vec r - \vec r_0) = 0$, $\vec n$ is the direction the sheet faces (straight out of it), $\vec r - \vec r_0$ is the arrow from a known point on the sheet to some other point, and "$= 0$" means those two are at right angles. If the arrow lies flat in the sheet, the point is on the plane. That's it; the formula is just that sentence written in symbols.

A line needed a point and a direction along it. A plane has too many directions along it to list, so we flip the idea around and use the one direction that's perpendicular to it. Think of a tabletop and a pencil standing straight up on it. Every arrow lying flat on the table, from any point to any other, is perpendicular to the pencil.

That turns into an equation right away. Fix a point $\vec r_0$ on the plane. Another point $\vec r$ is on the plane exactly when $\vec r - \vec r_0$ lies flat, which means $\vec n\cdot(\vec r - \vec r_0) = 0$. That's the perpendicularity test from the dot product lesson, and writing it out in components gives $a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$.

Reading the normal. Multiply that out and you get $ax + by + cz = d$, so the normal is sitting right there in the coefficients: $\langle a, b, c\rangle$. Changing $d$ slides the plane along $\vec n$ without tilting it.

Plane through three points $P$, $Q$, $R$: we need a vector perpendicular to the plane, which means perpendicular to both edges $\overrightarrow{PQ}$ and $\overrightarrow{PR}$. That's exactly what the cross product gives, so $\vec n = \overrightarrow{PQ}\times\overrightarrow{PR}$.

Relationships between planes all come from their normals, since the normal is what sets a plane's tilt. Parallel normals mean parallel planes, perpendicular normals mean perpendicular planes, and the angle between two planes is the angle between their normals.

A line meets a plane where the line's point satisfies the plane's equation. Substitute the parametric equations into the plane's equation, solve for $t$, and plug that $t$ back into the line.

3D models are made of triangles, each lying in a plane; collision detection checks which side of each plane a point is on with the sign of $\vec n\cdot(\vec r - \vec r_0)$.