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Comment by djhworld | Hacker News Reader

What I'm failing to understand is how do matrices relate to 3D graphics. What is a matrix representing?

I think I get the idea of a vertex, as that can be used to represent a point on a shape, like a 3d model.

Matrices are a way of accumulating a bunch of separate transformations you might want to perform on a point (in the lingo, points are vectors because each coordinate of the point is an element of the vector).

Suppose figuring out where a world point (in a game map, for example) should show up on screen (in pixels) looked like a bit like this function call sequence (this is simplified):

    screenPoint = perspective(move(rotate(move(worldPoint))))

Think of a camera moving around the world; in virtual reality, this is exactly equivalent to the camera staying put and the world moving around instead. That's more or less how 3D graphics works. You have a virtual box that logically lives behind your screen, with the same x and y coordinates as your screen resolution, and z coordinates handled by a Z buffer that's used to determine what overlaps what. A big chunk of the matrix work is about moving, rotating and squishing the whole virtual world until the bit you want to look at fits into the virtual box that lives behind the screen.

Each function (perspective, move (aka translate), rotate, scale, etc.) can be expressed in the form of multiplication by a matrix. See [1], for example. That turns it into something like this:

    screenPoint = perspectiveMatrix * moveMatrix * rotateMatrix * moveMatrix * worldPoint

But because matrix multiplication is associative, we can calculate a single matrix to do all the work:

    transformMatrix = perspectiveMatrix * moveMatrix * rotateMatrix * moveMatrix
    screenPoint = transformMatrix * worldPoint

And this is much more efficient because there's fewer calculations that need to be performed per point.

[1] https://en.wikipedia.org/wiki/Rotation_matrix

Wonderful explanation, thanks

A transformation from world coordinates (x, y, z) to screen coordinates (pixels).

A matrix corresponds to a linear mapping of some vector space to another. In case of graphics you usually start with local coordinates of your object. These have to be mapped to world coordinates, camera coordinates, and finally a perspective projection to get the screen coordinates. All these mappings can be expressed as matrices and applying mapping A after B is the same as applying the matrix product A*B to your initial coordinate vector. Precomputing the entire transformation matrix in this way causes quite a speedup for long transformation pipelines.

Check out this article! http://www.senocular.com/flash/tutorials/transformmatrix/

It helped me understand them quickly, with easy to understand, practical examples.

Don't mind the "Flash 8" context, there really isn't anything flash-specific in there.

That's a helpful site, especially the interactive examples. Thanks for the link

Without getting too mathy - matrices let you express useful transformations (eg. translation, rotation, scale) and multiplying two matrices combines those two translations - so you can have matrix tansform = rotationA * translation * rotationB

and now multiplying a vector (~ a 3d point) by that transform matrix is the same as doing each individual transformation (in the reverse order they appear in matrix multiplication) on a point to get the point after transform

It represents the axes of a space. Suppose you have a 3d space. It has 3 axes (traditionally referred to as X, Y and Z), usually all pointing in orthogonal directions, in some other space. The matrix holds the directions of these axes, one per column.

Here's an example 3-d matrix:

    [ a d g ]
    [ b e h ]
    [ c f i ]

(a,b,c) is the vector representing the direction of the X axis; (d,e,f), Y; (g,h,i), Z.

Matrices can be thought of as the coefficients to linear systems of equations. Things that linear systems of equations can do to their input variables are rotations, translations, scaling etc. Perspective distortion is another linear transform so it can be represented by a matrix.

Yes. Imagine you have a cube plotted in XYZ coordinates. The eight vertex points are (0,0,0), (0,0,10), (0,10,0), (0,10,10), (10,0,0), (10,0,10), (10,10,0) and (10,10,10).

Now let us say you want to rotate that cube 30 degrees on the X axis. You take the last vertex and make it a matrix, adding 1 at the end like you will do with the others, i.e. (10 10 10 1). Then you multiply it by the X rotation matrix ( https://open.gl/transformations ) using the appropriate angle. You now have the new correct position of that point. Now do the same for the other vertexes on the cube. The cube is still the same size and shape but its position has shifted.

You can do other easy transformations with matrix math. You can scale the cube with a scaling matrix - doubling it, halving it, or whatnot. Just multiply each vertex to the same scaling matrix. The new coordinates have the same size.

You can have a translation matrix to, where each vertex is multiplied by the same translation matrix, so that the cube is translated x, y, z coordinates from its origin.

You can also chain these matrix multiplications, so that the cube size is doubled, then rotated 60 degrees on the Y axis, then moved 20 X, -30 Y, 15 Z via translation. So that is multiplying three matrices in a row, then multiplying them against each vertex.

My intention with this first post was to show how to work with the mathematical constructs.

I have already started on the next article, which will provide the context for what these mathematical constructs provide.

To briefly answer your question, for 3d graphics the Matrix provides an efficient mechanism to translate, scale, rotate, shear and convert between different coordinate systems.

Here's a set of articles I wrote that might help?

http://webglfundamentals.org/webgl/lessons/webgl-2d-matrices...

That's 2d but if you follow them forward it will go to 3d

A 4x4 matrix can represent a position and a rotation, where the 3x3 matrix (called the transform) is the rotation and the 4th row (calls the transpose) is the location in space.

The neat thing is that when you multiply matrices together you update the original matrix. e.g. If you want to rotate something 90degrees to the right, you make a transform matrix with a zero transpose and multiply it with the original matrix. The resulting matrix will be an object in the same place rotated by 90 degrees.

Most modeling applications since about 2000 though use quaternions and a point to represent rotation and location. That's because it uses less memory and is easier/faster to compute.

Matrices and Linear Transformations by Cullen (Dover Books on Mathematics) will answer all your questions and more. It's written in plain English, doesn't require a mathematical background, and you really only need the first three chapters.

Yeah, while being informative the article is not necessarily inspiring. I just gives a rundown of linear algebra without talking about how it relates to the graphics.

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