Round brackets as Cartesian coordinates and square brackets as homogenous coordinates.

I picked this up idea from Needham's *Visual Complex Analysis*, and I'm not sure how commonly it's used elsewhere. In the book, the convention was to use round-bracketed matrices
$$
\begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix}
$$
for representing linear transformations, and square-bracketed matrices
$$
\begin{bmatrix}
a & b \\
c & d \\
\end{bmatrix}
$$
for representing Möbius transformations. More generally, we can let round brackets be for all "ordinary", "Cartesian", "vector"-ish things, and square brackets be for all "projective" or "homogeneous" things. It's useful to be able to tell them apart quickly, as the algebra looks the same but the meaning is very different.

Outside of complex analysis, there's an even plainer context where this convention would help: computer graphics (a.k.a. the actual "real-life" application of projective geometry). In computer graphics, there are two common ways to think of a point, say, in 2-dimensional space. One way is a pair $(x,y)$, the way people normally think of coordinates. The other way is to add an extra coordinate: $x$ and $y$ along with an extra $1$. This is a practical representation, since it works naturally with affine and projective transformations, which we represent in homogeneous coordinates too. The use of both systems leads to a problem. Without any notational convention, is $(3,-4,1)$ supposed to mean a 2D point or a 3D point? Is
$$
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 1 \\
0 & 0 & 1 \\
\end{bmatrix}
\begin{bmatrix}
-2 \\
3 \\
1 \\
\end{bmatrix}
$$
supposed to be a translation applied to a 2D point, or a shear mapping applied to a 3D point? You need context and you can't tell at a glance. If we used round/square brackets with our designated interpretations, though, we'd know right away that $(3,-4,1)$ means a 3D point and the square-bracketed matrix above means a 2D translation.

Unfortunately, they don't actually make any notational distinction at all in computer graphics. They're just going to stay confused ;) - but you don't have to. You can use the round and square to distinguish ordinary and projective coordinates. It deserves to be more standard.

(There's another convention out there to make the distinction with commas and colons, i.e. $(a,b,c)$ is Cartesian and $(a \mathbin{:} b \mathbin{:} c)$ is homogeneous. It works well for some purposes, but not if you need the notation to generalize to matrices.)

17more comments