Homogeneous coordinates

This continues the series on perspective projection, but again, the subject of homogeneous coordinates is extemely useful on its own.  Homogeneous coordinates are sort of notorious for being non-intuitive, or maybe it’s just hard to see what the point of them is.  I’ve always sort-of kind-of understood basically what the rules were, but to be honest, I don’t think I ever really had that aha! moment until just a few hours ago.

I’m going to assume that you understand how plain old coordinates work.  You’ve got x, y, and z values that describe a position in a cartesian coordinate system.  Usually this is represented by a 3-dimensional vector.  Simple enough.

To get so-called homogeneous coordinates, you add a fourth component, conventionally called w.  So now you have x, y, z, and w, but they still represent a point in 3 dimensions, not 4.  Then what’s the w for?  Isn’t it superfluous?  Before we answer that, let’s look at a few examples.

Consider this 3d point:

<10, 10, 10>

The simplest way to make these into homogeneous coordinates is to just add w=1:

<10, 10, 10, 1>

There, now they’re homogeneous.  Big deal!  Well, what happens if we change w?

<10, 10, 10, 5>

Uh oh.  They’re still homogeneous coordinates, but now they’re misleading.  Here’s the thing:

<10, 10, 10, 5> != <10, 10, 10>.  They are not the same point in space!  In fact:

<10, 10, 10, 5> = <2, 2, 2>!

The rule is, <x, y, z, w> is the same point as <x, y, z> only when w = 1.  If w != 1, you can normalize the whole thing by dividing every component by w (obviously, doing this will give you w = 1).

Now we can sort of start to see one part of what w is good for.  It gives us a way to package a scaling factor into a vector.  Notice that because of the rules, this scaling factor is upside down – the rest of the vector scales inversely with w.  When w gets bigger, the rest of the vector gets smaller, and when w gets smaller, the rest of the vector gets bigger.

Ok, so we can pack an inverse scaling factor in with our coordinates.  Why is that useful?  To understand that, you have to look at how these 4-component vectors interact with the 4×4 transformation matrices used everywhere in computer graphics.  I’ll get into that in the next post, where I’ll examine the perspective projection matrix.

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Vector – matrix multiplication: What matrices do to vectors

This is the first post in a series on perspective projection, but it’s useful on its own, too.

In order to understand how a projection matrix works, first you have to understand what a matrix actually does to a vector when you multiply them.  It can seem a little mysterious if you don’t look too closely, but once you look, it’s almost intuitive.

First, a refresher on dot products.  Remember that you get the dot product of two vectors by adding the component-wise products, like so:

You can look at vector/matrix multiplication as taking the dot products of the vector with each row of the matrix.  Each dot product becomes a component of the result vector.  Here’s how I like to think about this:  All of the components of the original vector can potentially contribute to each component of the result vector.  Each row of the matrix determines the contributions for one component of the result vector.  So, in the result vector, x is determined by row 1 in the matrix, y is determined by row 2, and so on.  Let’s see how this works for the simplest case: the identity matrix.  After this, it should make sense why the identity matrix has ones on a diagonal.

Each component of the original vector can contribute additively to any component in the result vector, and the weightings of the contributions are determined by the corresponding row in the matrix.  So, P’x is determined by row 1.  You can see that in the identity matrix, row 1 has a one in the x position, and zeros in all other positions.  So Px from the original vector is preserved intact by the one in that row, while Py, Pz, and Pw are suppressed by the zeros.  In the second row, there is a one in the y position, and the rest are zeros.  Since row 2 determines the contribution weightings for P’y, the contribution from Py is preserved, while Px, Pz, and Pw get zeroed.

Hopefully that makes sense after going over it once or twice.  The trick is to see each row in the matrix as selecting the way that the original vector’s components combine to produce each component of the result.  The tools used to do this are multiplication (to scale or cancel a source component) and addition (to combine those components into a single result component).

Of course, most matrices are more interesting than the identity matrix.  Try applying this to some of the transformation matrices.  Notice how the rows of a scaling matrix only affect one vector component at a time, just like the identity matrix, since x scaled is always a function of just x.  In contrast, notice how the rows of the rotation matrices affect multiple components at once, since x rotated can be a function of both x and y, or x and z, depending on the axis of rotation.  Once you absorb this, you’ll be able to see intuitively what a given matrix is doing to a vector, and make up your own matrices to do interesting things to vectors.