Archive for September, 2008

(x,y,z,w) in OpenGL/Direct3D (Homogeneous Coordinates)

September 29, 2008 7 comments

I always wondered why 3D points in OpenGL, Direct3D and in general computer graphics were always represented as (x,y,z,w) (i.e. why do we use four dimensions to represent a 3D point, what’s the w for?). This representation of coordinates with the extra dimension is know as homogeneous coordinates. Now after finally getting formally taught linear algebra I know the answer, and its rather simple, but I’ll start from the basics.

Points can be represented as vectors, eg. (1,1,1). Now a common thing we want to do in computer graphics is to move this point (translation). So we can do this by simply adding two vectors together,

\begin{pmatrix}x'\\y'\\z'\end{pmatrix} = \begin{pmatrix}x\\y\\z\end{pmatrix} + \begin{pmatrix}a\\b\\c\end{pmatrix} = \begin{pmatrix}x + a\\y + b\\z + c\end{pmatrix}.

If we wanted do some kind of linear transformation such as rotate about the origin, scale about the origin, etc, then we could just multiply a certain matrix with the point vector to obtain the image of the vector under that transformation. For example,

\begin{pmatrix}x'\\ y'\\ z' \end{pmatrix} = \begin{pmatrix}\cos \theta &-\sin \theta &0\\ \sin \theta &\cos \theta &0\\ 0&0&1\end{pmatrix} \begin{pmatrix}x\\ y\\ z\end{pmatrix}

will rotate the vector (x,y,z) by angle theta about the z axis.

However as you may have seen you cannot do a 3D translation on a 3D point by just multiplying a 3 by 3 matrix by the vector. To fix this problem and allow all affine transformations (linear transformation followed by a translation) to be done by matrix multiplication we introduce an extra dimension to the point (denoted w in this blog). Now we can perform the translation,

\mathbb{R}^2 : (x,y) \to (x+a, y+b)

by a matrix multiplication,

\begin{pmatrix}1 & 0 & a\\ 0 & 1 & b\\ 0 & 0 & 1\end{pmatrix} \begin{pmatrix}x\\ y\\ 1\end{pmatrix} = \begin{pmatrix}x + a\\ y + b\\ 1 \end{pmatrix}.

We need this extra dimension for the multiplication to make sense, and it allows us to represent all affine transformations as matrix multiplication.

Homogeneous coordinates. (2008, September 29). In Wikipedia, The Free Encyclopedia. Retrieved 04:33, September 29, 2008, from