Homogenous coordinates

Idea: take any 3D vector and add a fourth component called the $w$-component. This transforms the vector from Cartesian coordinates to homogeneous coordinates. To convert back to Cartesian, we divide the first three components by $w$.

$$⟨x,y,z,w⟩\leftrightarrow ⟨\frac{x}{w},\frac{y}{w},\frac{z}{w}⟩$$

The most common value of $w$ is 1. Dividing $x,y,z$ by 1 gives us $⟨x,y,z⟩$ back, so to convert Cartesian to homogeneous we simply set $w=1$.

Why? Big scary math stuff that I don’t care too much to understand. The math doesn’t really come up anyway. The following are some applications/examples that provide some intuition about what it means in day to day usage.

2D example

A 2D point $⟨x,y⟩$ has an equivalent homogenous point $⟨x,y,w⟩$. Unlike 3D, however, we can visualize this. In particular, every possible 2D point $⟨x,y⟩$ exists on the 2D plane that spans $w=1$, which represents all homogenous points $⟨x,y,1⟩$.

For each Cartesian 2D point $⟨x,y⟩$, there are an infinite number of homogenous points $⟨kx,ky,k⟩$, where $k\ne 0$.

Enables representing translation transformations

A translation of $(x,y,z)$ is represented by the translation matrix

$$\left[\begin{array}{cccc}1& 0& 0& x\\ 0& 1& 0& y\\ 0& 0& 1& z\\ 0& 0& 0& 1\end{array}\right]$$

This is obviously a good thing, and furthermore allows us to compose them with the scale and rotation transformation matrices. See Homogeneous transformation matrices.

Differentiates between “points” and “directions”

Consider a translation of $(1,2,3)$. To apply it to a point, we do:

$$\left[\begin{array}{cccc}1& 0& 0& 1\\ 0& 1& 0& 2\\ 0& 0& 1& 3\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right]=\left[\begin{array}{c}1\\ 2\\ 3\\ 1\end{array}\right]$$

This translates the origin point to $(1,2,3)$. And if we look at the math, it’s the $w$-component value of 1 that is doing all the heavy lifting. Contrast this to a vector that represents a direction, let’s say $⟨4,5,6⟩$. Let’s try “translating” this direction:

$$\left[\begin{array}{cccc}1& 0& 0& 1\\ 0& 1& 0& 2\\ 0& 0& 1& 3\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}4\\ 5\\ 6\\ 0\end{array}\right]=\left[\begin{array}{c}4\\ 5\\ 6\\ 0\end{array}\right]$$

It doesn’t do anything because mathematically, the $w$-component being zero, in fact, zeroes everything out. But also, it doesn’t make sense to translate a direction in the first place, because it just doesn’t have that notion.

So another way of seeing the $w$-component is its ability to tell us whether the vec4 we’re working with is a 3D point or just a direction. For example, this is why the $w$-component of a surface normal vector is zero, because it’s a direction.

Allows us to represent infinity

Setting $w=0$ means that we cannot divide by $w$ to get back to Cartesian coordinates. This tells us something important: Cartesian coordinates literally can’t represent the value captured by the homogenous point $⟨x,y,z,0⟩$.

This makes sense in the context of thinking them as directions. A direction is simply a point at infinity. And as seen above, the math also shows this. It doesn’t make sense to “translate” a point at infinity, just like how you can’t translate a direction.

Furthermore, it explains why we need an additional fourth component. Standard Cartesian points simply don’t contain enough information to encapsulate infinity. Homogenous does.

Also see

• More on Matrices - 3D Math Primer for Graphics and Game Development: good overview from the online version of the book.
• Homogenous coordinates - Wikipedia: especially the “Introduction” section that talks about homogenous points at infinity.