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

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

September 29, 2008 Leave a comment Go to 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.

REFERENCES:
Homogeneous coordinates. (2008, September 29). In Wikipedia, The Free Encyclopedia. Retrieved 04:33, September 29, 2008, from http://en.wikipedia.org/w/index.php?title=Homogeneous_coordinates&oldid=241693659

About these ads
  1. _Felix
    December 2, 2008 at 9:11 pm | #1

    OK, but why do matrices in openGL have sixteen values, rather than twelve? What are the other four?

  2. Andrew Harvey
    December 3, 2008 at 1:15 am | #2

    (I think, though am not sure) Because if you have a matrix A times a vector B of dimension 4, then the matrix A will need to have 4 columns for the multiplication to make sense, and it will have 4 rows (probably) because the resulting vector needs 4 rows. I’m not to sure but some perspective transformations utilise this.

  3. Rupert
    June 12, 2009 at 3:30 pm | #3

    If you are interested in learning more about this, you should take MATH5785 Geometry next time it runs

  4. Andrew Harvey
    June 12, 2009 at 11:08 pm | #4

    Yeh, that course looks great. I hope I can find space to do it.

  5. Anonymous
    August 24, 2010 at 1:38 pm | #5

    Thanks for the explain. The 2D translation using matrix math really illustrates it.

  6. Anonymous
    December 3, 2011 at 11:06 pm | #6

    louda explanation…

  7. Anonymous
    December 3, 2011 at 11:07 pm | #7

    sorry lauda explanation…

  1. No trackbacks yet.

I don't read comments anymore due to an increase in spam comments. If you want to get in touch please send me an email (see tianjara.net for details).

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Follow

Get every new post delivered to your Inbox.

%d bloggers like this: