Basically we were asked to create a matrix calls in C++ for my university course.
To start off, my class uses a column major constructor, but it places it in memory row major:
Matrix2::Matrix2(float R1C1, float R1C2, float R2C1, float R2C2)
{
//Row major order.
//Parameters written as column major (a normal matrix) but entered into memory as row major.
matrix[0] = R1C1;
matrix[1] = R2C1;
matrix[2] = R1C2;
matrix[3] = R2C2;
}
Which I believe is required for the unit test.
Matrices are defined as their identity matrix if no parameters are specified.
So, i rotate my matrix with this code:
void Matrix2::setRotateZ(float radians)
{
(*this) = (*this)*Matrix2( cosf(radians), -sinf(radians),
sinf(radians), cosf(radians));
}
I found this code on wikipedia and on my uni lecture slides. And it seems to go with column major matricies (which is how the data is meant to be input due to my constructor)
Also note that "setRotateZ" should actually be "rotateZ", as it accumulates rotation when called more than once.
My multiplication overload goes like this:
Matrix2 Matrix2::operator*(Matrix2& o)
{
Matrix2 newMatrix(0.0f, 0.0f, 0.0f, 0.0f);
newMatrix.matrix[0] = matrix[0] * o.matrix[0] + matrix[2] * o.matrix[1];
newMatrix.matrix[2] = matrix[0] * o.matrix[2] + matrix[2] * o.matrix[3];
newMatrix.matrix[1] = matrix[1] * o.matrix[0] + matrix[3] * o.matrix[1];
newMatrix.matrix[3] = matrix[1] * o.matrix[2] + matrix[3] * o.matrix[3];
return newMatrix;
}
Which just multiplies like a normal matrix (given that it's row major, 0 is R1C1 and 1 is R2C1 etc)
So pretty much, if I swap:
-sinf(radians)
and
sinf(radians)
in my setRotateZ function above, all works well, the matrix passes all tests including multiplication, rotation and addition.
However, as it is shown now, the matrix fails rotation.
As far as I know, swapping the sign functions should work for row major matrices only, but because mine swaps it in the constructor, entering it as a column major should work.
What am I doing wrong? This is bugging me, even though I've gotten it to work by swapping the values, I want to know what's going on.
I'm pretty much a matrix newb, so a little help will be appreciated!
NOTE: The unit test works by comparing the results of an operation with what they should evaluate to.
To start off, my class uses a column major constructor, but it places
it in memory row major
Well, the code you posted shows the exact opposite (wiki):
Matrix2(float R1C1, float R1C2, float R2C1, float R2C2)
// this ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ is ROW major!
{
matrix[0] = R1C1;
matrix[1] = R2C1; // <-- this is COLUMN major!
matrix[2] = R1C2; // <--
matrix[3] = R2C2;
}
Besides, in order to apply a rotation (the math is correct) you have to multiply the rotation matrix by the other matrix while you are doing the opposite.
Related
I have implemented a Gauss-Newton optimization process which involves calculating the increment by solving a linearized system Hx = b. The H matrx is calculated by H = J.transpose() * W * J and b is calculated from b = J.transpose() * (W * e) where e is the error vector. Jacobian here is a n-by-6 matrix where n is in thousands and stays unchanged across iterations and W is a n-by-n diagonal weight matrix which will change across iterations (some diagonal elements will be set to zero). However I encountered a speed issue.
When I do not add the weight matrix W, namely H = J.transpose()*J and b = J.transpose()*e, my Gauss-Newton process can run very fast in 0.02 sec for 30 iterations. However when I add the W matrix which is defined outside the iteration loop, it becomes so slow (0.3~0.7 sec for 30 iterations) and I don't understand if it is my coding problem or it normally takes this long.
Everything here are Eigen matrices and vectors.
I defined my W matrix using .asDiagonal() function in Eigen library from a vector of inverse variances. then just used it in the calculation for H ad b. Then it gets very slow. I wish to get some hints about the potential reasons for this huge slowdown.
EDIT:
There are only two matrices. Jacobian is definitely dense. Weight matrix is generated from a vector by the function vec.asDiagonal() which comes from the dense library so I assume it is also dense.
The code is really simple and the only difference that's causing the time change is the addition of the weight matrix. Here is a code snippet:
for (int iter=0; iter<max_iter; ++iter) {
// obtain error vector
error = ...
// calculate H and b - the fast one
Eigen::MatrixXf H = J.transpose() * J;
Eigen::VectorXf b = J.transpose() * error;
// calculate H and b - the slow one
Eigen::MatrixXf H = J.transpose() * weight_ * J;
Eigen::VectorXf b = J.transpose() * (weight_ * error);
// obtain delta and update state
del = H.ldlt().solve(b);
T <- T(del) // this is pseudo code, meaning update T with del
}
It is in a function in a class, and weight matrix now for debug purposes is defined as a class variable that can be accessed by the function and is defined before the function is called.
I guess that weight_ is declared as a dense MatrixXf? If so, then replace it by w.asDiagonal() everywhere you use weight_, or make the later an alias to the asDiagonal expression:
auto weight = w.asDiagonal();
This way Eigen will knows that weight is a diagonal matrix and computations will be optimized as expected.
Because the matrix multiplication is just the diagonal, you can change it to use coefficient wise multiplication like so:
MatrixXd m;
VectorXd w;
w.setLinSpaced(5, 2, 6);
m.setOnes(5,5);
std::cout << (m.array().rowwise() * w.array().transpose()).matrix() << "\n";
Likewise, the matrix vector product can be written as:
(w.array() * error.array()).matrix()
This avoids the zero elements in the matrix. Without an MCVE for me to base this on, YMMV...
I had followed this tutorial and got the output animation for a rigged model as expected. The tutorial uses assimp, glsl and c++ to load a rigged model from a file. However, there were things that I couldn't figure out.
First thing is assimp's transformation matrix are row major matrices and the tutorial uses a Matrix4f class which uses those transformation matrices just as they are i.e. row major order. The constructor of that Matrix4f class is as given:
Matrix4f(const aiMatrix4x4& AssimpMatrix)
{
m[0][0] = AssimpMatrix.a1; m[0][2] = AssimpMatrix.a2; m[0][2] = AssimpMatrix.a3; m[0][3] = AssimpMatrix.a4;
m[1][0] = AssimpMatrix.b1; m[1][3] = AssimpMatrix.b2; m[1][2] = AssimpMatrix.b3; m[1][3] = AssimpMatrix.b4;
m[2][0] = AssimpMatrix.c1; m[2][4] = AssimpMatrix.c2; m[2][2] = AssimpMatrix.c3; m[2][3] = AssimpMatrix.c4;
m[3][0] = AssimpMatrix.d1; m[3][5] = AssimpMatrix.d2; m[3][2] = AssimpMatrix.d3; m[3][3] = AssimpMatrix.d4;
}
However, in the tutorial for calculating the final node transformation, the calculations are done expecting the matrices to be in column major order, which is shown below:
Matrix4f NodeTransformation;
NodeTransformation = TranslationM * RotationM * ScalingM; //note here
Matrix4f GlobalTransformation = ParentTransform * NodeTransformation;
if(m_BoneMapping.find(NodeName) != m_BoneMapping.end())
{
unsigned int BoneIndex = m_BoneMapping[NodeName];
m_BoneInfo[BoneIndex].FinalTransformation = m_GlobalInverseTransform * GlobalTransformation * m_BoneInfo[BoneIndex].BoneOffset;
m_BoneInfo[BoneIndex].NodeTransformation = GlobalTransformation;
}
Finally, since the matrices calculated are in row major order, it is specified so while passing the matrices in the shader by setting GL_TRUE flag in the following function. Then, openGL knows it is in row major order as openGL itself uses column major order.
void SetBoneTransform(unsigned int Index, const Matrix4f& Transform)
{
glUniformMatrix4fv(m_boneLocation[Index], 1, GL_TRUE, (const GLfloat*)Transform);
}
So, how does the calculation done considering column major order
transformation = translation * rotation * scale * vertices
yield a correct output. I expected that for the calculation to hold true, each matrices should first be transposed to change to column order, followed by the above calculation and finally transposed again to obtain back row order matrix, which is also discussed in this link. However, doing so produced a horrible output. Is there something that I am missing here?
You are confusing two different things:
the layout the data has in memory (row vs. column major order)
the mathematical interpretation of the operations (things like multiplication order)
It is often claimed that when working with row major vs. column major, things have to be transposed and matrix multipication order hase to be reversed. But this is not true.
What is true is that mathematically, transpose(A*B) = transpose(B) * transpose(A). However, that is irrelevant here, because the matrix storage order is independent of, and orthogonal to, the mathematical interpretation of the matrices.
What I mean by this is: In math, it is exactly defined what a row and a column of a matrix is, and each element can be uniquely addressed by these two "coordinates". All the matrix operations are defined based on this convention. For example, in C=A*B, the element in the first row and the first column of C, is calculated as the dot product of the first row of A (transposed to a column vector) and the first column of B.
Now, the matrix storage order just defines how the matrix data is laid out in memory. As a generalization, we could define a function f(row,col) mapping each (row, col) pair to some memory address. We now could write or matrix functions using f, and we could change f to adapt row-major, column-major or something completely else (like a Z order curve, if we want some fun).
It doesn't matter what f we actually use (as long as the mapping is bijective), the operation C=A*B will always have the same result. What changes is just the data in memory, but we have also to use f to interpet that data. We could just write a simple print function, also using f, to print the matrix as the 2D array in columns x rows as a typical human would expect.
The confusion comes from this fact when you use a matrix in a different layout than the implementation of the matrix functions is designed on.
If you have a matrix library which is internally assuimg colum-major layout, and pass in data in row-major format, it is as if you transformed that matrix before - and only at this point, things get screwed up.
To confuse things even more, there is another issue related to this: the matrix * vector vs vector * matrix issue. Some people like to write x' = x * M (with v' and v being row vectors), while others like to write y' = N *y (with column vectors). It is clear that mathematically, M*x = transpose((transpose(x) * transpose(M)), so that people often also confuse this with row- vs column-major order effects - but it is also totally independent of that. It is just a matter of convention if you want to use the one or the other.
So, to finally answer your question:
The transformation matrices created there are written for the convention of multyplying matrix * vector, so that Mparent * Mchild is the correct matrix multiplication order.
Up to this point, the actual data layout in memory does not matter at all. It only begins to matter because now, we are interfacing a different API, with its own conventions. GL's default order is column-major. The matrix class in use is written for row-major memory layout. So you just transpose at this point, so that GL's interpretation of that matrix matches your other library's.
The alternative would be not convert them and account for that by incorporating the implicit operation created by this into the system - either by changing the multiplication order in the shader, or by adjusting the operations which created the matrix in the first place. However, I would not recommend going that path, because the resulting code will be totally unintuitive, because in the end, this would mean working with column-major matrices in a matrix class using a row-major interpretation.
Yes, the memory layout is similar for glm and assimp : data.html
But, according to the doc page : classai_matrix4x4t
The assimp matrix is always row-major whereas the glm matrix is always col-major meaning you need to create a transponse on conversion:
inline static Mat4 Assimp2Glm(const aiMatrix4x4& from)
{
return Mat4(
(double)from.a1, (double)from.b1, (double)from.c1, (double)from.d1,
(double)from.a2, (double)from.b2, (double)from.c2, (double)from.d2,
(double)from.a3, (double)from.b3, (double)from.c3, (double)from.d3,
(double)from.a4, (double)from.b4, (double)from.c4, (double)from.d4
);
}
inline static aiMatrix4x4 Glm2Assimp(const Mat4& from)
{
return aiMatrix4x4(from[0][0], from[1][0], from[2][0], from[3][0],
from[0][1], from[1][1], from[2][1], from[3][1],
from[0][2], from[1][2], from[2][2], from[3][2],
from[0][3], from[1][3], from[2][3], from[3][3]
);
}
PS: The abcd stands for row and 1234 stands for col in assimp.
I have a plane in my 3d space and I want to move it somewhere else, so I use glTranslate to do so.
The planes vertex data is: (0,0,0), (1,0,0), (1,1,0) and (0,1,0).
I translate the object to the position of (2,0,0) through the use of glTranslatef(2.0, 0.0, 0.0).
After the translation the point data is unchanged so if I was to want to collide with my plane the visual position is not its actual position.
Is there a way to get the point data from the MODELVIEW_MATRIX or at least a way to find out what the new values are after the glTranslate?
Don't respond with just add 2.0 to the actual values to move it because what if I want to the use glRotate etc. I still want the points locations.
If you really don't want to maintain your own transformation matrix, you can get the current modelview matrix with:
GLfloat mat[16];
glGetFloatv(GL_MODELVIEW_MATRIX, mat);
You can then apply this matrix to your vertices with a standard matrix multiplication. Keep in mind that the matrix is arranged in column-major order. With an input vector xIn, the transformed vector xOut is:
xOut[0] = mat[0] * xIn[0] + mat[4] * xIn[1] + mat[8] * xIn[2] + mat[12];
xOut[1] = mat[1] * xIn[0] + mat[5] * xIn[1] + mat[9] * xIn[2] + mat[13];
xOut[2] = mat[2] * xIn[0] + mat[6] * xIn[1] + mat[10] * xIn[2] + mat[14];
Keeping track of the current transformation matrix in your own code is really a better approach, IMHO. Aside from eliminating glGet() calls, which can be harmful to performance, it gets you on a path to using modern OpenGL (Core Profile), where the matrix stack and all related calls do not exist anymore.
You can create a matrix from your translation and rotation, so that you can use the matrix to transform the coordinates.
There're many libraries to help you create such matrix and transform coordinates.
The situation is as follows:
XMVECTOR posVec = XMLoadFloat3(&(pVertexInfos[j].pos));
// At this point posVec equals {6143.72119, -714.767151, -16615.9004, 0.000000000}
// and new newModelMatrix's rows are as follows:
// Row 0: {1.00000000, 0.000000000, 0.000000000, 0.000000000}
// Row 1: {0.000000000, 0.000000000, -1.00000000, 0.000000000}
// Row 2: {0.000000000, 1.00000000, 0.000000000, 0.000000000}
// Row 3: {0.000000000, 0.000000000, 0.000000000, 1.00000000}
posVec = XMVector3Transform(posVec, newModelMatrix);
// But then posVec equals **{6143.72119, -16615.9004, 714.767151, 1.00000000}**
According to my repeated pencil and paper calculations (Khan academy confirmed that I'm doing it right) and what the correct program execution is should equal {6143.72119, 16615.9004, -714.767151, 1.00000000}
Just in case I'm going crazy, here's a screenshot of the debugger before:
and after:
So what's going on here? According to my research XMVector3Transform should be doing exactly what I want, which is Matrix times Vector = Vector, but for some reason it looks like the negative signs get messed up. As you can imagine, this causes a pretty bad visual bug later on in the app (I confirmed that correctly hacking the operation resolves the problem).
Thank you in advance for any help,
Nico
There doesn't appear to be anything wrong with the result you're getting.
The matrix you've setup takes a position (X, Y, Z) and multiplies it by a matrix that swaps the Y and Z axes and negates the resulting Z.
As you multiply through the vector by the matrix, you go across one column at a time and work down, not along the row, the same way you would with matrix multiplication.
The resulting calculation should be:
X = (6143.72119 * 1.0f) + (-714.767151 * 0.0f) + (-16615.9004 * 0.0f) = 6143.72219
Y = (6143.72119 * 0.0f) + (-714.767151 * 0.0f) + (-16615.9004 * 1.0f) = -16615.9004
Z = (6143.72119 * 0.0f) + (-714.767151 * -1.0f) + (-16615.9004 * 0.0f) = 714.767151
This is exactly what the function gives you. Perhaps you're just misunderstanding how to do vector/matrix multiplication?
I'm working on a simple OpenGL world- and so far I've got a bunch of cubes randomly placed about and it's pretty fun to go zooming about. However I'm ready to move on. I would like to drop blocks in front of my camera, but I'm having trouble with the 3d angles. I'm used to 2d stuff where to find an end point we simply do something along the lines of:
endy = y + (sin(theta)*power);
endx = x + (cos(theta)*power);
However when I add the third dimension I'm not sure what to do! It seems to me that the power of the second dimensional plane would be determined by the z axis's cos(theta)*power, but I'm not positive. If that is correct, it seems to me I'd do something like this:
endz = z + (sin(xtheta)*power);
power2 = cos(xtheta) * power;
endx = x + (cos(ytheta) * power2);
endy = y + (sin(ytheta) * power2);
(where x theta is the up/down theta and y = left/right theta)
Am I even close to the right track here? How do I find an end point given a current point and an two angles?
Working with euler angles doesn't work so well in 3D environments, there are several issues and corner cases in which they simply don't work. And you actually don't even have to use them.
What you should do, is exploit the fact, that transformation matrixes are nothing else, then coordinate system bases written down in a comprehensible form. So you have your modelview matrix MV. This consists of a model space transformation, followed by a view transformation (column major matrices multiply right to left):
MV = V * M
So what we want to know is, in which way the "camera" lies within the world. That is given to you by the inverse view matrix V^-1. You can of course invert the view matrix using Gauss Jordan method, but most of the time your view matrix will consist of a 3×3 rotation matrix with a translation vector column P added.
R P
0 1
Recall that
(M * N)^-1 = N^-1 * M^-1
and also
(M * N)^T = M^T * N^T
so it seems there is some kind of relationship between transposition and inversion. Not all transposed matrices are their inverse, but there are some, where the transpose of a matrix is its inverse. Namely it are the so called orthonormal matrices. Rotations are orthonormal. So
R^-1 = R^T
neat! This allows us to find the inverse of the view matrix by the following (I suggest you try to proof it as an exersice):
V = / R P \
\ 0 1 /
V^-1 = / R^T -P \
\ 0 1 /
So how does this help us to place a new object in the scene at a distance from the camera? Well, V is the transformation from world space into camera space, so V^-1 transforms from camera to world space. So given a point in camera space you can transform it back to world space. Say you wanted to place something at the center of the view in distance d. In camera space that would be the point (0, 0, -d, 1). Multiply that with V^-1:
V^-1 * (0, 0, -d, 1) = (R^T)_z * d - P
Which is exactly what you want. In your OpenGL program you somewhere have your view matrix V, probably not properly named yet, but anyway it is there. Say you use old OpenGL-1 and GLU's gluLookAt:
void display(void)
{
/* setup viewport, clear, set projection, etc. */
glMatrixMode(GL_MODELVIEW);
glLoadIdentity();
gluLookAt(...);
/* the modelview matrix now holds the View transform */
At this point we can extract the modelview matrix
GLfloat view[16];
glGetFloatv(GL_MODELVIEW_MATRIX, view);
Now view is in column major order. If we were to use it directly we could directly address the columns. But remember that transpose is inverse of a rotation, so we actually want the 3rd row vector. So let's assume you keep view around, so that in your event handler (outside display) you can do the following:
GLfloat z_row[3];
z_row[0] = view[2];
z_row[1] = view[6];
z_row[2] = view[10];
And we want the position
GLfloat * const p_column = &view[12];
Now we can calculate the new objects position at distance d:
GLfloat new_object_pos[3] = {
z_row[0]*d - p_column[0],
z_row[1]*d - p_column[1],
z_row[2]*d - p_column[2],
};
There you are. As you can see, nowhere you had to work with angles or trigonometry, it's just straight linear algebra.
Well I was close, after some testing, I found the correct formula for my implementation, it looks like this:
endy = cam.get_pos().y - (sin(toRad(180-cam.get_rot().x))*power1);
power2 = cos(toRad(180-cam.get_rot().x))*power1;
endx = cam.get_pos().x - (sin(toRad(180-cam.get_rot().y))*power2);
endz = cam.get_pos().z - (cos(toRad(180-cam.get_rot().y))*power2);
This takes my camera's position and rotational angles and get's the corresponding points. Works like a charm =]