I am trying to achieve the same rotational effect like Maya in my project.
I have some knowledge on quaternions and the trackball example.
Unfortunately I am still unable to wrap my head around the concept of using the quaternions to get the desired effect.
Basically I am still getting the same issue I had before with the 3d trackball. After flipping the object upside down, and then trying to rotate to the right, the object will rotate to the left. Well actually its my camera rotating around the focus point in the opposite direction.
The problem is that I am using the screen coordinates & trackball to get the old / new vectors and getting the angle of rotation from those two vectors. I will always get the wrong axis of rotation this way.
How should I go about solving this issue?
I don't know Maya so I can only guess that its rotation is like this: if you rotate left-right, it feels natural. Then if you rotate the object up-down 180 degrees, then rotate left-right again, it still feels natural.
If you are familiar with the concept of using a matrix to do transformations (like rotate, scale and translate), well a quaternion is just the same concept but it only allows rotations, so you might want to use it to constrain your transforms to just rotations. In practice, you can use either a matrix or a quaternion to do the same thing.
What you need to do is remember the current quaternion state for the object, then when the next frame of rotation occurs, multiply the new rotation with the old quaternion (in that order) to give you the next frame's quaternion. That will ensure that no matter what orientation the object is in, the next frame's rotation will be applied from the viewer's viewpoint. This is as opposed to some naive rotation where you just say "user is scrolling up/down, therefore alter the object's X-axis rotation", which causes that flipping.
Remember, like matrices, quaternions need to be multiplied in reverse order that the actions are actually applied, which is why I said to multiply the new operation by the existing quaternion.
To finish with an example. Let's say the user is going to perform 2 actions:
On frame 1, the user rotates the object 180 degrees about the X axis (up/down rotation).
On frame 2, the user rotates the object 90 degrees about the Y axis (left/right rotation).
Lets say the object has a quaternion Q. Every frame, you will reset the object to its default coordinates and apply the quaternion Q to rotate it. Now you might initialise it with the identity quaternion, but let's just say the initial quaternion is called Q0.
On frame 1, create a new quaternion R1 which is a "rotate 180 degrees about the X axis" quaternion (you can find some maths to compute such a quaternion). Pre-multiply the new operation by the existing quaternion: Q1 = R1 * Q0.
On frame 2, create a new quaternion R2 which is a "rotate 90 degrees about the Y axis" quaternion. Pre-multiply the new operation by the existing quaternion: Q2 = R2 * Q1.
On frame 1 you will use Q1 to display the object, and on frame 2 you will use Q2. You can simply keep applying any subsequent user actions to the quaternion and it will always be rotated in the viewer's frame of reference.
I think you have problems with changing coordinate system.
Suppose, you want to rotate object in X Axis, then in Y Axis, and then move it and scale. So, you should multiply your transformation maxtrix (at the beginning it equals to itentity matrix) to the rotation matrix (firstly to X, then to Y), then to translation matrix and at the end to scaling matrix. So, when your current matrix multiplies to the resulting matrix, your coordinate systems changes.
To avoid this problem you can use 2 methods:
1) to accumulate your resultig matrix as product of all previous matrices.
2) to use stack, where in the top will be the matrix, which equals to product of all matrices in the bottom of this matrix (in the stack).
P.S. I'm not sure, that it helps you. I never used quaternions in my projects.
Related
Say I have 2 objects, a camera and a cube, both on XZ plane, the cube has some arbitrary rotation, and camera is facing the cube.
now if a transformation R is applied to the camera such that it has a new rotation and position.
I want to move the cube in front of the camera using transformation R1, such that in the view it looks exactly as before R was applied, meaning relative distance, rotation and scale b/w the 2 objects remain same after both R and R1.
Following image gives a gist of the problem.
Assume that there's no scenegraph that we can use.
I've posed the problem mainly in 2D but I'm trying to solve it in 3D, so rotations can have all yaw, pitch and roll, and translations can be anywhere in 3D space.
EDIT:
I forgot to add what I have done so far.
I figured out how to maintain relative distance b/w camera and cube, I can project cube's position to get world to screen point, then unproject the screen point in new camera position to get new world position.
However for rotation, I have tried this
I thought I can apply same rotation as R in R1, this didn't work, it appears to work if rotation happens only in one axis, if rotation happens in more than one axes, it does not work.
I thought I can take delta rotation b/w camera and cube, and simply apply camera's rotation to the cube and then multiply delta rotation, this also didn't work
Let M and V be the model and view matrices before you move the camera, M2 and V2 be the matrices after you move the camera. To be clear: model matrix transforms the coordinates from object local coordinates into world coordinates; a view matrix transforms from world coordinates into clip-space camera coordinates. Consequently V*M*p transforms the position p into clip-space.
For the position on the screen to stay constant, we need V*M*p = V2*M2*p to be true for all p (that's assuming that the FOV doesn't change). Therefore V*M = V2*M2, or
M2 = inverse(V2)*V*M
If you apply the camera transformation on the right (V2 = V*R) then the above expression for M2 can be simplified:
M2 = inverse(R)*M
(that is you apply inverse(R) on the left of the model matrix to compensate).
Alternatively, ask yourself if you really need to keep the object coordinates in the world reference frame. It may be easier to not to apply the view matrix when rendering that object at all; that would effectively keep it relative to the camera at all times without any additional tweaks. That would have better numerical stability too.
I am building a camera class to look arround a scene. At the moment I have 3 cubes just spread arround to have a good impression of what is going on. I have set my scroll button on a mouse to give me translation along z-axis and when I move my mouse left or right I detect this movement and rotate arround y-axis. This is just to see what happens and play arround a bit. So I succeeded in making the camera rotate by rotating the cubes arround the origin but after I rotate by some angle, lets say 90 degrees, and try to translate along z axis to my surprise I find out that my cubes are now going from left to right and not towards me or away from me. So what is going on here? It seems that z axis is rotated also. I guess the same goes for x axis. So it seems that nothing actually moved in regard to the origin, but the whole coordinate system with all the objects was just rotated. Can anyone help me here, what is going on? How coordinate system works in opengl?
You are most likely confusing local and global rotations. Usual cheap remedy is to change(reverse) order of some of your transformation. However doing this blindly is trial&error and can be frustrating. Its better to understand the math first...
Old API OpeGL uses MVP matrix which is:
MVP = Model * View * Projection
Where Model and View are already multiplied together. What you have is most likely the same. Now the problem is that Model is direct matrix, but View is Inverse.
So if you have some transform matrix representing your camera in oder to use it to transform back you need to use its inverse...
MVP = Model * Inverse(Camera) * Projection
Then you can use the same order of transformations for both Model and Camera and also use their geometric properties like basis vectors etc ... then stuff like camera local movements or camera follow are easy. Beware some tutorials use glTranspose instead of real matrix Inverse. That is correct only if the Matrix contains only unit (or equal sized) orthogonal basis vectors without any offset so no scale,skew,offset or projections just rotation and equal scale along all axises !!!
That means when you rotate Model and View in the same way the result is opposite. So in old code there is usual to have something like this:
// view part of matrix
glMatrixMode(GL_MODELVIEW);
glLoadIdentity();
glRotate3f(view_c,0,0,1); // ugly euler angles
glRotate3f(view_b,0,1,0); // ugly euler angles
glRotate3f(view_a,1,0,0); // ugly euler angles
glTranslatef(view_pos); // set camera position
// model part of matrix
for (i=0;i<objs;i++)
{
glMatrixMode(GL_MODELVIEW);
glPushMatrix();
glTranslatef(obj_pos[i]); // set camera position
glRotate3f(obj_a[i],1,0,0); // ugly euler angles
glRotate3f(obj_b[i],0,1,0); // ugly euler angles
glRotate3f(obj_c[i],0,0,1); // ugly euler angles
//here render obj[i]
glMatrixMode(GL_MODELVIEW);
glPopMatrix();
}
note the order of transforms is opposite (I just wrote it here in editor so its not tested and can be opposite to native GL notation ... I do not use Euler angles) ... The order must match your convention... To know more about these (including examples) not using useless Euler angles see:
Understanding 4x4 homogenous transform matrices
Here is 4D version of what your 3D camera class should look like (just shrink the matrices to 4x4 and have just 3 rotations instead of 6):
reper4D
pay attention to difference between local lrot_?? and global grot_?? functions. Also note rotations are defined by plane not axis vector as axis vector is just human abstraction that does not really work except 2D and 3D ... planes work from 2D to ND
PS. its a good idea to have the distortions (scale,skew) separated from model and keep transform matrices representing coordinate systems orthonormal. It will ease up a lot of things latter on once you got to do advanced math on them. Resulting in:
MVP = Model * Model_distortion * Inverse(Camera) * Projection
I have a 4x4 transformation matrix. However, after trying out the transformation I noticed that movement and rotation of the Y axis is going the opposite way. The rest is correct.
I got this matrix from some other API so probably it is the difference of coordinate system. So, how can I flip an axis of transformation matrix?
If only translation I can add minus sign on the Y translation, but I have no idea about opposite rotation of only one axis since all the rotation is being represented in the same 3x3 area. I thought there might be some way that even affect both translation and rotation at the same time. (truly flipping the axis)
Edit: I'm pretty sure the operation you're looking for is changing coordinate systems while maintaining Z-up or Y-up. In this case, try setting all the elements of the second column (or row) of your matrix to their inverse.
This question would be better for the Math StackExchange. First, a really helpful read on rotation matrices.
The first problem is the matter of rotation order. I will be assuming the XYZ rotation order. We know the rotation matrices for each axis is as follows:
Given a matrix derived from the same rotation order, the resulting matrix would be as follows, where alpha is the X angle, beta is the Y angle, and gamma is the Z angle:
You can derive the individual components of each axis angle from this matrix. For example, you can derive the Y angle from -sin(beta) using some inverse trig. Given beta, you can derive alpha from cos(beta)sin(alpha). You can also derive gamma from cos(beta)sin(gamma). Note that the same number in the matrix can represent multiple values (e.g. sin(0)=0 and sin(180)=0).
Now that you know alpha, beta, and gamma, you can reverse beta and remake the rotation matrix.
There's a good chance that there's a better way to do this using quaternions, but you should ask the Math StackExchange these kinds of language-agnostic questions.
Much shorter answer: if you are not careful with your frame orientation many things down your pipeline are likely to have a bad hair day. The reason is "parity", a.k.a. "frame orientation", a.k.a. "right-handedness" (or rarely left-handedness). Most 3D geometry tools and libraries that work together normally assume implicitly that all coordinate systems in play are right-handed (or at least consistently-handed). Inverting the orientation of just one axis in a coordinate system changes its orientation from right to left handed or viceversa.
So, suggestion for things to check & try in your problem:
Check that the frame you get from your API is right-handed. You do so
by computing the determinant of the 3x3 rotation part of your 4x4 transform matrix: it must be +1 or very close to it.
If it is -1, then flip one if its axis, i.e. change the sign of one of the columns of the 3x3 rotation.
Note carefully: I said "columns" because I assume that you apply a transform Q to a point x by multiplying as Q * x, x being a 4x1 column vector with the last component equal to one. If you use row vectors left-multiplied by Q you need flip a row.
If that determinant is +1, you have a bug someplace else.
I have a simple 3D LookAt vector, and I wish to rotate the player model (a simple cube) to show where the player/cube is looking at.
For sideways camera movement I've managed to figure it out and do the following:
glTranslatef(position.x, position.y, position.z);
glRotatef(atan2(lookAt.z, lookAt.x) * 180 / PI, 0, 1, 0);
Now I know that to get up-down camera movement to map to the rendered cube model, I need to rotate the cube around it's x and z axes as well, but I can't seem to figure out what formula to use for those two.
OpenGL will rotate the whole coordinate system (whole space, not only a cube) so after first rotation you just need to rotate only around z axis.
// first rotation
glRotatef(-atan2(lookAt.z, lookAt.x) * 180 / PI, 0, 1, 0);
// second rotation
float d = sqrt(pow(lookAt.x,2) + pow(lookAt.z,2));
float pitch = atan2(lookAt.y, d);
glRotatef(pitch * 180 / PI, 0, 0, 1);
First and second rotation:
I assume your model is looking along x axis (red arrow). I also assume lookAt is given relative to the position of the model.
If you're familiar with matrix math, matrices are an easier way to think about it. If you're not familiar with matrices, this series explains how to use them to solve common game development problems: https://www.youtube.com/playlist?list=PLW3Zl3wyJwWNQjMz941uyOIq3Nw6bcDYC Getting good with matrices is a good idea if you want to be a 3D game programmer.
For your problem, you want to make a translation/rotation matrix which will transform the box to the proper place for you. You can make a translation matrix and a rotation matrix individually, and then at the end take the product of the two. I'll try to break that down.
The translation matrix is simple, if your position is then your matrix will be
To construct a rotation matrix, you need to rotate the standard basis vectors the way you want. Then when you create a matrix from those rotated basis vectors, the matrix will rotate other vectors in the same way. As an example of that, take the standard basis vectors:
Now I'm going to rotate and around by 90 degrees clockwise:
Now put them into a matrix:
and you have R is a matrix that rotates things around by 90 degrees.
In your case you want to rotate stuff such that it faces a vector that you provide. That makes things easy, we can calculate our basis vectors from that vector. If your vector is then and we can solve for the other two basis vectors using cross products. You know that the character won't ever roll their view (right?) so we can use the global up vector as well. I'll call the global up vector . In your case you're using y as the "up" dimension so the global up vector will be
Then:
In the first line you do a cross product between the view vector and the up vector to get a vector orthogonal to both - this will serve as the third basis vector after it is normalized, which is the second line. In the third line another cross product generates the second basis vector. These three vectors represent what happens when the standard basis vectors are rotated the way you want them to be. Use them as the columns in a matrix like so:
Now the last step in the math is to make a final matrix that will do both translation and rotation, and this step is easy:
Then load that matrix into OpenGL with glLoadMatrix:
glLoadMatrixf(&M);
All of this gets explained in the video series I linked as well :)
I've been playing around for awhile now with directx and can't figure out how to rotate something without rotating it all the way around (0,0,0). The farther I get away from the center of my world the bigger circles in makes during its rotation.
Don't forget the matrix transformations apply to the coordinate system. Let's say you want to move your object upwards on an xy plane by 20 units, and rotate it by 90 degrees. If you rotate by 90 degrees first, you'll be rotating the entire plane by 90 degrees. This means 90 degrees is the new "upwards" when you translate up the y axis.
So, we translate first, so that our object's center is 0,0. Now when we rotate, we should be rotating around the center of the object. Of course, don't forgot to translate back, or clear the matrix somehow.
The order does matter when doing matrix transformations, as I'm sure you know. Usually, you should translate, scale, then rotate.
If you need a rotation α, around a point p located at (x₀,y₀,z₀), you create the matrix :
T(-x₀,-y₀,-z₀) * R(α) * T(x₀,y₀,z₀)
T means Translation and R means rotation. Also, depending on your convention such as row or column matrix, you may have to revert the order of operation.