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OpenGL OpenGL Matrices, explanation?

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I tried to read this (http://www.sjbaker.org/steve/omniv/matrices_can_be_your_friends.html), but it's very confusingly written in my opinion. For example, he writes, "Well, if we neglect the translation part (the bottom row)", and the very next thing he writes is "After that, you just add the translation onto each point so that: " but he doesn't add the bottom row, he adds something else. And stuff like that seems to be abundant on that page... So I was wondering, could someone give me an example of how to use it?

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IMHO the author of the cited article will say the following:

A (affine) transformation matrix as used by OpenGL has a principle layout like

[ x_x y_x z_x t_x ]
[ x_y y_y z_y t_y ]
[ x_z y_z z_z t_z ]
[ 0 0 0 1 ]
. If only rotation and translation is used, then those x_x ... z_z denote the rotation, and those t_x ... t_z the translation. (BTW: Scaling and shearing will also be encoded into the x_x ... z_z.)

Then the author claims that such a matrix can be split into the rotational part and the translational part this way:

[ x_x y_x z_x t_x ] [ 1 0 0 t_x ] [ x_x y_x z_x 0 ]
[ x_y y_y z_y t_y ] == [ 0 1 0 t_y ] * [ x_y y_y z_y 0 ]
[ x_z y_z z_z t_z ] [ 0 0 1 t_z ] [ x_z y_z z_z 0 ]
[ 0 0 0 1 ] [ 0 0 0 1 ] [ 0 0 0 1 ]

And due to a mathematical rule that says
( M1 * M2 ) * M3 == M1 * ( M2 * M3 )
one can interpret that splitted matrix T * R, although being applied to a point p in a single step
( T * R ) * p
as a two-step transformation
== T * ( R * p )
that rotates the point (i.e. R * p ) and translates the result (i.e. translates the rotated point). The order of effects if important.

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I'm not exactly an expert when it comes to matrices, but this is my take on the page;

In most programming, you would expect an array of 9 elements (lets say int array[9]) to be laid out (if you were using it as a square), like so;

[0][1][2]
[3][4][5]
[6][7][8]

So basically, your data is stored in what is known as row-centric order. This just means that address 1 is next to address 2.

Most mathematicians, and OpenGL, treat the data like so;

[0][3][6]
[1][4][7]
[2][5][8]

which is called a column-centric matrix. Notice the way the data is laid out differently to the row-centric matrix.

Now, a 3x3 matrix, like above, only allows you to rotate, scale and shear your object.

To keep the math as simple as possible, the matrix needs to be kept as a square, but we need some way to store the position of the object in the matrix. To do this, we make the matrix 4x4, which gives us (keeping the OGL format);

[0] [4] [8] [12]
[1] [5] [9] [13]
[2] [6] [10] [14]
[3] [7] [11] [15]


In this way, the translation goes in elements [3],[7] and [11], as x,y and z respectively.

Have a look at the matrix the writer shows you. It looks something like this;

[1][0][0]
[0][1][0]
[0][0][1]
[0][0][0] <-- this is the part you need to notice


The author has made the matrix a 4x3 matrix, rather than keep to the 4x4 matrix he used earlier. When he says to add the translation onto each point, he means take each value of the new position, and add it to the relevant value in the last line of the matrix. I hope this helps with some of your confusion about this.

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I know that matrix math isn't that easy to understand, but ifyou understand it you have much less problems interpreting how 3D works. Trust me. So please forgive me, but the following is again some matrix math ;)

If you compose a transformation matrix with OpenGL's API on OpenGL's so-called MODELVIEW matrix stack, you do something like
glLoadIdentity();
glTranslatef(tx,ty,tz);
glRotatef(alpha,ax,ay,az);

If you wonder why I use glLoadIdentity: It is useful to initiate OpenGL's matrix stack with a defined value. All those glRotate and glTranslate routines are ever _multiplying_ self onto what is already onto the stack; see below.

Mathematically this is described by composing a transformation matrix
( I * T(tx,ty,tz) ) * R(alpha,ax,ay,az) =: M
Herein I denotes the identity matrix, R the rotation resulting from glRotatef, and T the translation resulting from glTranslatef.

Notice the order of matrices in the formula from left to right and the order of API calls from top to bottom is the same!

Then you push a vertex into the API, e.g. (using the immediate mode)
glBegin(GL_TRIANGLES);
glVertex3f(px,py,pz);
glEnd();
with the above transformation being active that is hence applied to the vertex position, yielding in the transformed vertex position (I'm dropping the transformation arguments from here on since I'm too lazy to write them down, okay?)
p' := M * p == ( ( I * T ) * R ) * p

What does this mean? It means that OpenGL has a transformation on its stack, namely a composition of a translation and a rotation. This composed transformation is applied as a whole (notice the parantheses) to the vertex position.

Now let us inspect the effect of the transformation. Multiplying the identity matrix with another matrix has no effect, so that
== ( T * R ) * p

We already know that the parantheses play no role w.r.t. the result (see my previous post), so that we can _interpret_ the result (although it is not being applied in this way) as
== T * ( R * p )
what means nothing else than that the vertex position is rotated, and the rotated result is translated. Voila.

Now, isn't it the same as translating the vertex and rotating the result (i.e. the other order)? No, it isn't (in general)! You can simply construct an example: Say, you use
glTranslatef(0,1,0);
glRotatef(90,1,0,0);
glBegin(GL_TRIANGLES);
glVertex3f(0,1,0);
glEnd();
Using what is written above, OpenGL does
rotating [ 0,1,0 ] by 90 degrees around the x axis, yielding in [ 0,0,1 ]
translating [ 0,0,1 ] by [ 0,1,0 ] yielding in [ 0,1,1 ]

The other way
glRotatef(90,1,0,0);
glTranslatef(0,1,0); // <-- EDIT here was an error
glBegin(GL_TRIANGLES);
glVertex3f(0,1,0);
glEnd();
does
translating [ 0,1,0 ] by [ 0,1,0 ] yielding in [ 0,2,0 ]
rotating [ 0,2,0 ] by 90 degrees around the x axis, yielding in [ 0,0,2 ]
what obviously differs from the first result.

However, sometimes you apply a transformation that is already composed, using
glMultMatrixf(...);
instead of the particular transformations glRotatef, glTranslatef, ... Then, and that is the stuff of my previous post, one can decompose the transformation matrix and interpret it as
T * R
and that is, with the above explanations in mind, equivalent to
glTranslatef(...);
glRotatef(...);

_That_ is what the article said, as far as belonging to the cited section.

[Edited by - haegarr on August 10, 2008 1:17:01 PM]

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@webwraith: You're mainly correct, but not totally.

Considering that OpenGL uses column vectors, and furthur uses the 4th scalar of a (homogeneous) vector as the homogeneous co-ordinate, the translation must be in the right-most column but not in the bottom row. Hence, if using a 4x4 matrix, it looks like
[ 1  0  0  t_x ]
[ 0 1 0 t_y ]
[ 0 0 1 t_z ]
[ 0 0 0 1 ]
and for a 4x3 matrix, it looks like
[ 1  0  0  t_x ]
[ 0 1 0 t_y ]
[ 0 0 1 t_z ]

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Math is tricky. You must choose a methodological approach or you'll get lost in nowhere land in a blink of an eye. Chances are you give up learning math altogether if you choose the wrong approach in the beginning because in doing so, you'll eventually get to a point that you conclude "Well, Math is not my thing!". It is your thing my friend, but you need to choose a slow methodological approach or you'll get lost sooner than you expect it. You can't expect to learn 3D math (or any other field of science) by reading a tutorial or two, because Math is no picnic. You can't rush it. You need to invest a lot of time in learning it. It's tough, I know, but so is life and there is no easy way around either of them!

My apologies if that doesn't apply to you, in which case you can skip this post altogether, but if it does, please take a moment to consider it for your own good.

I don't care what those tutorials are advertising, but there is no fast way to learn Math or programming or any other scientific field. I hold a BSc. in Computer Science, have read a lot of math related books during the years and I do enjoy thinking about mathematical problems, so I'm neither the lazy guy nor do I lack proper trainings, but even I get stuck at math quite a lot, because as I said, math is no picnic.

I strongly suggest that you pick up a copy of "3D Math Primer for Graphics and Game Development" by Fletcher Dunn and Ian Parberry from your favorite library or bookstore and start reading it right away. It's the best elementary book on 3D math that I've come by. There IS a difference between reading 400 pages of 3D math and skimming over a couple of tutorials: you'd be much more knowledgeable when you invest more time and choose the right approach.

... and no, I'm not affiliated with the authors of this book in any way, shape or form. [smile]

Good luck!

[Edited by - Ashkan on August 10, 2008 7:28:14 PM]

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Quote:
Original post by haegarr
@webwraith: You're mainly correct, but not totally.
...

Thank you for pointing that out, but the example on the page is the same as the 4x3 matrix that you can see in my post. Perhaps the confusion then lies in whether the author of that page is using row- or column-centric matrices at that point?

[Edited by - webwraith on August 17, 2008 2:37:39 PM]

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Quote:
Original post by Rasmadrak
I disagree. :)
Col x Row.
Every reference I've ever seen on the topic uses the convention RowXCol. I'm pretty sure this convention is used more or less without exception, but if you can provide an example to the contrary I'd be interested to see it.

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Quote:
Original post by jyk
Quote:
Original post by Rasmadrak
I disagree. :)
Col x Row.
Every reference I've ever seen on the topic uses the convention RowXCol. I'm pretty sure this convention is used more or less without exception, but if you can provide an example to the contrary I'd be interested to see it.


This is also the form I have been taught, a matrix which has 4 rows (m) and 3 columns (n) is denoted an m*n or a 4*3 matrix.

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Quote:
Original post by webwraith
Do your references include OpenGL? :)
I assume this is in reference to my post? If so, then yeah, of course they include OpenGL :)

Again, I've never come across a reference (OpenGL or otherwise) that uses the convention Col-Row when referring to matrix dimensions or to individual matrix elements. Can you point me to a reference that contradicts this? (And I'm not being snide - if there is such a reference, I would really like to be aware of it!).

Just to eliminate potential confusion, note that we are talking here about a specific aspect of mathematical notation: whether the row or column is listed first when describing matrix dimensions or specifying a matrix element.

Note that vector notation (row or column) and matrix storage (row- or column-major) are entirely separate and unrelated issues. (I'm guessing this is where you're getting confused...)

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I didn't intend to derail this thread in such a way, but since it's at least partially on-topic... the choice of the Rows*Cols notation is not arbitrary.

Say you have two matrices A and B, where A is an m*n matrix and B is a k*l matrix. Then the matrix product A*B is defined iff n = k. When you write it down on paper, you'll see
 A * B
m*n k*l

It gets even more obvious when multiplying more than two matrices. The rule is sometimes formulated as "the inner dimensions of a matrix product must agree". So once you're used to it, everything flows naturally because the definitions are very consistent. [Also, think about how this works with vector-matrix or matrix-vector multiplication.]

Hope this helps in memorizing how things work.

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      Another important thing that pipeline state object encompasses is the input layout description that defines how inputs to the vertex shader, which is the very first shader stage, should be read from the memory. Input layout may define several vertex streams that contain values of different formats and sizes:
      // Define input layout InputLayoutDesc &Layout = PSODesc.GraphicsPipeline.InputLayout; LayoutElement TextLayoutElems[] = {     LayoutElement( 0, 0, 3, VT_FLOAT32, False ),     LayoutElement( 1, 0, 4, VT_UINT8, True ),     LayoutElement( 2, 0, 2, VT_FLOAT32, False ), }; Layout.LayoutElements = TextLayoutElems; Layout.NumElements = _countof( TextLayoutElems ); Finally, pipeline state defines primitive topology type. When all required members are initialized, a pipeline state object can be created by IRenderDevice::CreatePipelineState() method:
      // Define shader and primitive topology PSODesc.GraphicsPipeline.PrimitiveTopologyType = PRIMITIVE_TOPOLOGY_TYPE_TRIANGLE; PSODesc.GraphicsPipeline.pVS = pVertexShader; PSODesc.GraphicsPipeline.pPS = pPixelShader; PSODesc.Name = "My pipeline state"; m_pDev->CreatePipelineState(PSODesc, &m_pPSO); When PSO object is bound to the pipeline, the engine invokes all API-specific commands to set all states specified by the object. In case of Direct3D12 this maps directly to setting the D3D12 PSO object. In case of Direct3D11, this involves setting individual state objects (such as rasterizer and blend states), shaders, input layout etc. In case of OpenGL, this requires a number of fine-grain state tweaking calls. Diligent Engine keeps track of currently bound states and only calls functions to update these states that have actually changed.
      Binding Shader Resources
      Direct3D11 and OpenGL utilize fine-grain resource binding models, where an application binds individual buffers and textures to certain shader or program resource binding slots. Direct3D12 uses a very different approach, where resource descriptors are grouped into tables, and an application can bind all resources in the table at once by setting the table in the command list. Resource binding model in Diligent Engine is designed to leverage this new method. It introduces a new object called shader resource binding that encapsulates all resource bindings required for all shaders in a certain pipeline state. It also introduces the classification of shader variables based on the frequency of expected change that helps the engine group them into tables under the hood:
      Static variables (SHADER_VARIABLE_TYPE_STATIC) are variables that are expected to be set only once. They may not be changed once a resource is bound to the variable. Such variables are intended to hold global constants such as camera attributes or global light attributes constant buffers. Mutable variables (SHADER_VARIABLE_TYPE_MUTABLE) define resources that are expected to change on a per-material frequency. Examples may include diffuse textures, normal maps etc. Dynamic variables (SHADER_VARIABLE_TYPE_DYNAMIC) are expected to change frequently and randomly. Shader variable type must be specified during shader creation by populating an array of ShaderVariableDesc structures and initializing ShaderCreationAttribs::Desc::VariableDesc and ShaderCreationAttribs::Desc::NumVariables members (see example of shader creation above).
      Static variables cannot be changed once a resource is bound to the variable. They are bound directly to the shader object. For instance, a shadow map texture is not expected to change after it is created, so it can be bound directly to the shader:
      PixelShader->GetShaderVariable( "g_tex2DShadowMap" )->Set( pShadowMapSRV ); Mutable and dynamic variables are bound via a new Shader Resource Binding object (SRB) that is created by the pipeline state (IPipelineState::CreateShaderResourceBinding()):
      m_pPSO->CreateShaderResourceBinding(&m_pSRB); Note that an SRB is only compatible with the pipeline state it was created from. SRB object inherits all static bindings from shaders in the pipeline, but is not allowed to change them.
      Mutable resources can only be set once for every instance of a shader resource binding. Such resources are intended to define specific material properties. For instance, a diffuse texture for a specific material is not expected to change once the material is defined and can be set right after the SRB object has been created:
      m_pSRB->GetVariable(SHADER_TYPE_PIXEL, "tex2DDiffuse")->Set(pDiffuseTexSRV); In some cases it is necessary to bind a new resource to a variable every time a draw command is invoked. Such variables should be labeled as dynamic, which will allow setting them multiple times through the same SRB object:
      m_pSRB->GetVariable(SHADER_TYPE_VERTEX, "cbRandomAttribs")->Set(pRandomAttrsCB); Under the hood, the engine pre-allocates descriptor tables for static and mutable resources when an SRB objcet is created. Space for dynamic resources is dynamically allocated at run time. Static and mutable resources are thus more efficient and should be used whenever possible.
      As you can see, Diligent Engine does not expose low-level details of how resources are bound to shader variables. One reason for this is that these details are very different for various APIs. The other reason is that using low-level binding methods is extremely error-prone: it is very easy to forget to bind some resource, or bind incorrect resource such as bind a buffer to the variable that is in fact a texture, especially during shader development when everything changes fast. Diligent Engine instead relies on shader reflection system to automatically query the list of all shader variables. Grouping variables based on three types mentioned above allows the engine to create optimized layout and take heavy lifting of matching resources to API-specific resource location, register or descriptor in the table.
      This post gives more details about the resource binding model in Diligent Engine.
      Setting the Pipeline State and Committing Shader Resources
      Before any draw or compute command can be invoked, the pipeline state needs to be bound to the context:
      m_pContext->SetPipelineState(m_pPSO); Under the hood, the engine sets the internal PSO object in the command list or calls all the required native API functions to properly configure all pipeline stages.
      The next step is to bind all required shader resources to the GPU pipeline, which is accomplished by IDeviceContext::CommitShaderResources() method:
      m_pContext->CommitShaderResources(m_pSRB, COMMIT_SHADER_RESOURCES_FLAG_TRANSITION_RESOURCES); The method takes a pointer to the shader resource binding object and makes all resources the object holds available for the shaders. In the case of D3D12, this only requires setting appropriate descriptor tables in the command list. For older APIs, this typically requires setting all resources individually.
      Next-generation APIs require the application to track the state of every resource and explicitly inform the system about all state transitions. For instance, if a texture was used as render target before, while the next draw command is going to use it as shader resource, a transition barrier needs to be executed. Diligent Engine does the heavy lifting of state tracking.  When CommitShaderResources() method is called with COMMIT_SHADER_RESOURCES_FLAG_TRANSITION_RESOURCES flag, the engine commits and transitions resources to correct states at the same time. Note that transitioning resources does introduce some overhead. The engine tracks state of every resource and it will not issue the barrier if the state is already correct. But checking resource state is an overhead that can sometimes be avoided. The engine provides IDeviceContext::TransitionShaderResources() method that only transitions resources:
      m_pContext->TransitionShaderResources(m_pPSO, m_pSRB); In some scenarios it is more efficient to transition resources once and then only commit them.
      Invoking Draw Command
      The final step is to set states that are not part of the PSO, such as render targets, vertex and index buffers. Diligent Engine uses Direct3D11-syle API that is translated to other native API calls under the hood:
      ITextureView *pRTVs[] = {m_pRTV}; m_pContext->SetRenderTargets(_countof( pRTVs ), pRTVs, m_pDSV); // Clear render target and depth buffer const float zero[4] = {0, 0, 0, 0}; m_pContext->ClearRenderTarget(nullptr, zero); m_pContext->ClearDepthStencil(nullptr, CLEAR_DEPTH_FLAG, 1.f); // Set vertex and index buffers IBuffer *buffer[] = {m_pVertexBuffer}; Uint32 offsets[] = {0}; Uint32 strides[] = {sizeof(MyVertex)}; m_pContext->SetVertexBuffers(0, 1, buffer, strides, offsets, SET_VERTEX_BUFFERS_FLAG_RESET); m_pContext->SetIndexBuffer(m_pIndexBuffer, 0); Different native APIs use various set of function to execute draw commands depending on command details (if the command is indexed, instanced or both, what offsets in the source buffers are used etc.). For instance, there are 5 draw commands in Direct3D11 and more than 9 commands in OpenGL with something like glDrawElementsInstancedBaseVertexBaseInstance not uncommon. Diligent Engine hides all details with single IDeviceContext::Draw() method that takes takes DrawAttribs structure as an argument. The structure members define all attributes required to perform the command (primitive topology, number of vertices or indices, if draw call is indexed or not, if draw call is instanced or not, if draw call is indirect or not, etc.). For example:
      DrawAttribs attrs; attrs.IsIndexed = true; attrs.IndexType = VT_UINT16; attrs.NumIndices = 36; attrs.Topology = PRIMITIVE_TOPOLOGY_TRIANGLE_LIST; pContext->Draw(attrs); For compute commands, there is IDeviceContext::DispatchCompute() method that takes DispatchComputeAttribs structure that defines compute grid dimension.
      Source Code
      Full engine source code is available on GitHub and is free to use. The repository contains two samples, asteroids performance benchmark and example Unity project that uses Diligent Engine in native plugin.
      AntTweakBar sample is Diligent Engine’s “Hello World” example.

       
      Atmospheric scattering sample is a more advanced example. It demonstrates how Diligent Engine can be used to implement various rendering tasks: loading textures from files, using complex shaders, rendering to multiple render targets, using compute shaders and unordered access views, etc.

      Asteroids performance benchmark is based on this demo developed by Intel. It renders 50,000 unique textured asteroids and allows comparing performance of Direct3D11 and Direct3D12 implementations. Every asteroid is a combination of one of 1000 unique meshes and one of 10 unique textures.

      Finally, there is an example project that shows how Diligent Engine can be integrated with Unity.

      Future Work
      The engine is under active development. It currently supports Windows desktop, Universal Windows and Android platforms. Direct3D11, Direct3D12, OpenGL/GLES backends are now feature complete. Vulkan backend is coming next, and support for more platforms is planned.
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