Chapter 29: Planes


A plane is defined by a point and a normal . The plane consists of all points for which is perpendicular to .

Or, using the dot product:

Recall our equation for a ray:

Plugging in for gives us:


Solve for :

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

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, is a point on the plane.

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

This gives us 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 for the plane.