# Chapter 29: Planes

## Overview

A plane is defined by a point $p_1$ and a normal $\hat n$. The plane consists of all points for which $(p-p_1)$ is perpendicular to $\hat n$.

Or, using the dot product:

Recall our equation for a ray:

Plugging in for $p$ gives us:

## Derivation

Solve for $t$:

But now… $p_0$ is the origin of our ray but we need a $p1$

Povray gives us planes in this format:

plane {
<A, B, C>, D
}


Where <A, B, C> is the normal of the plane and D is the “distance” of the plane, or how far from the origin the plane is along the normal. Therefore, $\hat n * D$ is a point on the plane.

The dot product of a normalized vector with itself is 1, so we really have:

This gives us $t$ in terms of known quantities.

## Other Plane Equation

It’s important to note that there is another commonly used equation for a plane.

It looks like we can use this equation with the four parameters given by the Povray plane specification. However, for “simplicity”, Povray actually gives us -D for this equation. i.e. in Povray, the intended equation is:

You can also use this equation to find a $p_1$ for the plane.