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To translate a vector by 10 unit in the X direction, why do we have to use a matrix?

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We can just add 10 to the mat[0][0], and we got the same result too.

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5 Answers 5

up vote 17 down vote accepted

Yes, you can add a vector in the case of translation. The reason to use a matrix boils down to having a uniform way to handle different combined transformations.

For example, rotation is usually done using a matrix (check @MickLH comment for other ways to deal with rotations), so in order to deal with multiple transformations (rotation/translation/scaling/projection...etc) in a uniform way, you need to encode them in a matrix.

Well, more technically speaking; a transformation is mapping a point/vector to another point/vector.

p` = T(p); 

where p` is the transformed point T(p) is the transformation function.

Given that we don't use a matrix we need to do this to combine multiple transformations:

p1= T(p);

pfinal = M(p1);

Not only can a matrix combine multiple types of transformations into a single matrix (e.g. affine, linear, projective).

Using a matrix gives us the opportunity to combine chains of transformations and then batch multiply them. This saves us a ton of cycles usually by the GPU (thanks to @ChristianRau for pointing it out).

Tfinal = T * R * P; // translate*rotate*project

pfinal = Tfinal*p;

It's also good to point out that GPUs and even some CPUs are optimized for vector operations; CPUs using SIMD and GPUs being data driven parallel processors by design, so using matrices fits perfectly with hardware acceleration (actually, GPUs were designed to fit matrix/vector ops).

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yes, i know that matrix is useful for rotation. but every tutorial guide me using matrix to do such simple calculation :D –  ngoaho91 Mar 17 at 10:33
    
Saying rotation can "only" be done with a matrix is incorrect, off the top of my head Quaternions and Trigonometry would work just fine also –  MickLH Mar 17 at 10:34
    
@MickLH I knew someone would point this out, TBH I only said that for simplicity –  concept3d Mar 17 at 10:36
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And even more than that, once you have rotation and translation both as 4x4 matrices, you can just multiply them and have the combined transformation in one single matrix without the need to transform every vertex by a thousands of different transformations using different constructs. The fact that a 4x4 matrix is overkill for a single translation or a single rotation is outweight by the fact that you usually don't just transform a vertex by single translation or a single rotation. –  Christian Rau Mar 17 at 16:21
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@concept3d Yeah, I know, the answer is good. Yet the even bigger advantage gained from the uniform way of using a matrix is not only uniformity, but representation of an entire chain of transformations in a single operation. While that might have been implied, I found it unclear and important enough to mention it explicitly. But the answer was still good anyway, it wasn't a critique. –  Christian Rau Mar 17 at 17:05

If all you are ever going to do is move along a single axis and never apply any other transformation then what you are suggesting is fine.

The real power of using a matrix is that you can easily concatenate a series of complex operations together, and apply the same series of operations to multiple objects.

Most cases aren't that simple and if you rotate you object first, and want to transform along its local axes instead of the world axes you'll find you can't simply add 10 to one of the numbers and have it work out correctly.

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the reason to use a 4x4 matrix is so that the operation is a linear transformation. this is an example of homogeneous coordinates. The same thing is done in the 2d case (using a 3x3 matrix). The reason for using homogeneous coordinates is so that all 3 geometric tansformations can be done using one operation; otherwise one would need to do a 3x3 matrix multiply and a 3x3 matrix addition (for the translation). this link from cegprakash is useful.

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You should elaborate. A succinct explanation is better than only linking to wikipedia. –  Seth Battin May 2 at 12:04

To succinctly answer the "why" question, it's because a 4x4 matrix can describe rotation, translation, and scaling operations all at once. Being able to describe any of these in a consistent manner simplifies a lot of things.

Different kinds of transformations can be more simply represented with a different mathematical operations. As you note, translation can be done just by adding. Uniform scaling by multiplying by a scalar. But an appropriately crafted 4x4 matrix can do anything. So using 4x4's consistently makes code and interfaces much simpler. You pay some complexity in understanding these 4x4's, but then lots of things get easier and faster because of it.

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See this video to understand the concepts of model, view and projection.

4x4 matrices are not just used for translating a 3D object. But also for various other purposes.

See this to understand how the vertices in the world are represented as 4D Matrices and how they are transformed.

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This doesn't actually answer the OP question. –  concept3d Mar 20 at 7:30
    
Edited. Sounds good? –  cegprakash Mar 20 at 7:43

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