I have an issue of rotating my model. I have file that are composed of relative interger values for roll, pitch and yaw.
Thus every period of time I read those values and applie to my model :
rotate(current_yaw - previous_yaw, QVector3D(0,0,1)) ;
rotate(current_pitch - previous_pitch, QVector3D(0, 1, 0)) ;
rotate(current_roll - previous_roll, QVector3D(1,0,0)) ;
The rotate function multiplies the transformation/model matrix with a rotation matrix.
My issue is that when the values go back to 0, the model is not straight.
I tried doing one rotation only (only the yaw, then the roll, then the pitch) and it works fine. So the issue is with combining the rotations.
I know that order matters. I would like to do, for instance, roll - pitch - yaw and afterwards yaw - pitch -roll to get back to the initial position of the model.
But in case of thousands of non symetric values, where some angles may be positives and others negatives, I don't really know what to do.
Is there a way to apply all those rotation at the same time ?
Edit :
To try and apply all the rotations at the same time I tried using QQuaternion. Here is an example of what I though would work but still does not.
QQuaternion rotation = QQuaternion::fromEulerAngles(20, 20, 20) ;
m_world.setToIdentity();
m_world.rotate(rotation);
QQuaternion rotation_back = QQuaternion::fromEulerAngles(-20, -20, -20) ;
m_world.setToIdentity();
m_world.rotate(rotation_back);
The world matrix is not identity in the end.
Edit : of course the second quaternion should be (0,0,0)... I need to use angles not differences of angles. Works fine.
I am implementing a 3D engine for spatial visualisation, and am writing a camera with the following navigation features:
Rotate the camera (ie, analogous to rotating your head)
Rotate around an arbitrary 3D point (a point in space, which is probably not in the center of the screen; the camera needs to rotate around this keeping the same relative look direction, ie the look direction changes too. This does not look directly at the chosen rotation point)
Pan in the camera's plane (so move up/down or left/right in the plane orthogonal to the camera's look vector)
The camera is not supposed to roll - that is, 'up' remains up. Because of this I represent the camera with a location and two angles, rotations around the X and Y axes (Z would be roll.) The view matrix is then recalculated using the camera location and these two angles. This works great for pan and rotating the eye, but not for rotating around an arbitrary point. Instead I get the following behaviour:
The eye itself apparently moving further up or down than it should
The eye not moving up or down at all when m_dRotationX is 0 or pi. (Gimbal lock? How can I avoid this?)
The eye's rotation being inverted (changing the rotation makes it look further up when it should look further down, down when it should look further up) when m_dRotationX is between pi and 2pi.
(a) What is causing this 'drift' in rotation?
This may be gimbal lock. If so, the standard answer to this is 'use quaternions to represent rotation', said many times here on SO (1, 2, 3 for example), but unfortunately without concrete details (example. This is the best answer I've found so far; it's rare.) I've struggled to implemented a camera using quaternions combining the above two types of rotations. I am, in fact, building a quaternion using the two rotations, but a commenter below said there was no reason - it's fine to immediately build the matrix.
This occurs when changing the X and Y rotations (which represent the camera look direction) when rotating around a point, but does not occur simply when directly changing the rotations, i.e. rotating the camera around itself. To me, this doesn't make sense. It's the same values.
(b) Would a different approach (quaternions, for example) be better for this camera? If so, how do I implement all three camera navigation features above?
If a different approach would be better, then please consider providing a concrete implemented example of that approach. (I am using DirectX9 and C++, and the D3DX* library the SDK provides.) In this second case, I will add and award a bounty in a couple of days when I can add one to the question. This might sound like I'm jumping the gun, but I'm low on time and need to implement or solve this quickly (this is a commercial project with a tight deadline.) A detailed answer will also improve the SO archives, because most camera answers I've read so far are light on code.
Thanks for your help :)
Some clarifications
Thanks for the comments and answer so far! I'll try to clarify a few things about the problem:
The view matrix is recalculated from the camera position and the two angles whenever one of those things changes. The matrix itself is never accumulated (i.e. updated) - it is recalculated afresh. However, the camera position and the two angle variables are accumulated (whenever the mouse moves, for example, one or both of the angles will have a small amount added or subtracted, based on the number of pixels the mouse moved up-down and/or left-right onscreen.)
Commenter JCooper states I'm suffering from gimbal lock, and I need to:
add another rotation onto your transform that rotates the eyePos to be
completely in the y-z plane before you apply the transformation, and
then another rotation that moves it back afterward. Rotate around the
y axis by the following angle immediately before and after applying
the yaw-pitch-roll matrix (one of the angles will need to be negated;
trying it out is the fastest way to decide which).
double fixAngle = atan2(oEyeTranslated.z,oEyeTranslated.x);
Unfortunately, when implementing this as described, my eye shoots off above the scene at a very fast rate due to one of the rotations. I'm sure my code is simply a bad implementation of this description, but I still need something more concrete. In general, I find unspecific text descriptions of algorithms are less useful than commented, explained implementations. I am adding a bounty for a concrete, working example that integrates with the code below (i.e. with the other navigation methods, too.) This is because I would like to understand the solution, as well as have something that works, and because I need to implement something that works quickly since I am on a tight deadline.
Please, if you answer with a text description of the algorithm, make sure it is detailed enough to implement ('Rotate around Y, then transform, then rotate back' may make sense to you but lacks the details to know what you mean. Good answers are clear, signposted, will allow others to understand even with a different basis, are 'solid weatherproof information boards.')
In turn, I have tried to be clear describing the problem, and if I can make it clearer please let me know.
My current code
To implement the above three navigation features, in a mouse move event moving based on the pixels the cursor has moved:
// Adjust this to change rotation speed when dragging (units are radians per pixel mouse moves)
// This is both rotating the eye, and rotating around a point
static const double dRotatePixelScale = 0.001;
// Adjust this to change pan speed (units are meters per pixel mouse moves)
static const double dPanPixelScale = 0.15;
switch (m_eCurrentNavigation) {
case ENavigation::eRotatePoint: {
// Rotating around m_oRotateAroundPos
const double dX = (double)(m_oLastMousePos.x - roMousePos.x) * dRotatePixelScale * D3DX_PI;
const double dY = (double)(m_oLastMousePos.y - roMousePos.y) * dRotatePixelScale * D3DX_PI;
// To rotate around the point, translate so the point is at (0,0,0) (this makes the point
// the origin so the eye rotates around the origin), rotate, translate back
// However, the camera is represented as an eye plus two (X and Y) rotation angles
// This needs to keep the same relative rotation.
// Rotate the eye around the point
const D3DXVECTOR3 oEyeTranslated = m_oEyePos - m_oRotateAroundPos;
D3DXMATRIX oRotationMatrix;
D3DXMatrixRotationYawPitchRoll(&oRotationMatrix, dX, dY, 0.0);
D3DXVECTOR4 oEyeRotated;
D3DXVec3Transform(&oEyeRotated, &oEyeTranslated, &oRotationMatrix);
m_oEyePos = D3DXVECTOR3(oEyeRotated.x, oEyeRotated.y, oEyeRotated.z) + m_oRotateAroundPos;
// Increment rotation to keep the same relative look angles
RotateXAxis(dX);
RotateYAxis(dY);
break;
}
case ENavigation::ePanPlane: {
const double dX = (double)(m_oLastMousePos.x - roMousePos.x) * dPanPixelScale;
const double dY = (double)(m_oLastMousePos.y - roMousePos.y) * dPanPixelScale;
m_oEyePos += GetXAxis() * dX; // GetX/YAxis reads from the view matrix, so increments correctly
m_oEyePos += GetYAxis() * -dY; // Inverted compared to screen coords
break;
}
case ENavigation::eRotateEye: {
// Rotate in radians around local (camera not scene space) X and Y axes
const double dX = (double)(m_oLastMousePos.x - roMousePos.x) * dRotatePixelScale * D3DX_PI;
const double dY = (double)(m_oLastMousePos.y - roMousePos.y) * dRotatePixelScale * D3DX_PI;
RotateXAxis(dX);
RotateYAxis(dY);
break;
}
The RotateXAxis and RotateYAxis methods are very simple:
void Camera::RotateXAxis(const double dRadians) {
m_dRotationX += dRadians;
m_dRotationX = fmod(m_dRotationX, 2 * D3DX_PI); // Keep in valid circular range
}
void Camera::RotateYAxis(const double dRadians) {
m_dRotationY += dRadians;
// Limit it so you don't rotate around when looking up and down
m_dRotationY = std::min(m_dRotationY, D3DX_PI * 0.49); // Almost fully up
m_dRotationY = std::max(m_dRotationY, D3DX_PI * -0.49); // Almost fully down
}
And to generate the view matrix from this:
void Camera::UpdateView() const {
const D3DXVECTOR3 oEyePos(GetEyePos());
const D3DXVECTOR3 oUpVector(0.0f, 1.0f, 0.0f); // Keep up "up", always.
// Generate a rotation matrix via a quaternion
D3DXQUATERNION oRotationQuat;
D3DXQuaternionRotationYawPitchRoll(&oRotationQuat, m_dRotationX, m_dRotationY, 0.0);
D3DXMATRIX oRotationMatrix;
D3DXMatrixRotationQuaternion(&oRotationMatrix, &oRotationQuat);
// Generate view matrix by looking at a point 1 unit ahead of the eye (transformed by the above
// rotation)
D3DXVECTOR3 oForward(0.0, 0.0, 1.0);
D3DXVECTOR4 oForward4;
D3DXVec3Transform(&oForward4, &oForward, &oRotationMatrix);
D3DXVECTOR3 oTarget = oEyePos + D3DXVECTOR3(oForward4.x, oForward4.y, oForward4.z); // eye pos + look vector = look target position
D3DXMatrixLookAtLH(&m_oViewMatrix, &oEyePos, &oTarget, &oUpVector);
}
It seems to me that "Roll" shouldn't be possible given the way you form your view matrix. Regardless of all the other code (some of which does look a little funny), the call D3DXMatrixLookAtLH(&m_oViewMatrix, &oEyePos, &oTarget, &oUpVector); should create a matrix without roll when given [0,1,0] as an 'Up' vector unless oTarget-oEyePos happens to be parallel to the up vector. This doesn't seem to be the case since you're restricting m_dRotationY to be within (-.49pi,+.49pi).
Perhaps you can clarify how you know that 'roll' is happening. Do you have a ground plane and the horizon line of that ground plane is departing from horizontal?
As an aside, in UpdateView, the D3DXQuaternionRotationYawPitchRoll seems completely unnecessary since you immediately turn around and change it into a matrix. Just use D3DXMatrixRotationYawPitchRoll as you did in the mouse event. Quaternions are used in cameras because they're a convenient way to accumulate rotations happening in eye coordinates. Since you're only using two axes of rotation in a strict order, your way of accumulating angles should be fine. The vector transformation of (0,0,1) isn't really necessary either. The oRotationMatrix should already have those values in the (_31,_32,_33) entries.
Update
Given that it's not roll, here's the problem: you create a rotation matrix to move the eye in world coordinates, but you want the pitch to happen in camera coordinates. Since roll isn't allowed and yaw is performed last, yaw is always the same in both the world and camera frames of reference. Consider the images below:
Your code works fine for local pitch and yaw because those are accomplished in camera coordinates.
But when you rotate around a reference point, you are creating a rotation matrix that is in world coordinates and using that to rotate the camera center. This works okay if the camera's coordinate system happens to line up with the world's. However, if you don't check to see if you're up against the pitch limit before you rotate the camera position, you will get crazy behavior when you hit that limit. The camera will suddenly start to skate around the world--still 'rotating' around the reference point, but no longer changing orientation.
If the camera's axes don't line up with the world's, strange things will happen. In the extreme case, the camera won't move at all because you're trying to make it roll.
The above is what would normally happen, but since you handle the camera orientation separately, the camera doesn't actually roll.
Instead, it stays upright, but you get strange translation going on.
One way to handle this would be to (1)always put the camera into a canonical position and orientation relative to the reference point, (2)make your rotation, and then (3)put it back when you're done (e.g., similar to the way that you translate the reference point to the origin, apply the Yaw-Pitch rotation, and then translate back). Thinking more about it, however, this probably isn't the best way to go.
Update 2
I think that Generic Human's answer is probably the best. The question remains as to how much pitch should be applied if the rotation is off-axis, but for now, we'll ignore that. Maybe it'll give you acceptable results.
The essence of the answer is this: Before mouse movement, your camera is at c1 = m_oEyePos and being oriented by M1 = D3DXMatrixRotationYawPitchRoll(&M_1,m_dRotationX,m_dRotationY,0). Consider the reference point a = m_oRotateAroundPos. From the point of view of the camera, this point is a'=M1(a-c1).
You want to change the orientation of the camera to M2 = D3DXMatrixRotationYawPitchRoll(&M_2,m_dRotationX+dX,m_dRotationY+dY,0). [Important: Since you won't allow m_dRotationY to fall outside of a specific range, you should make sure that dY doesn't violate that constraint.] As the camera changes orientation, you also want its position to rotate around a to a new point c2. This means that a won't change from the perspective of the camera. I.e., M1(a-c1)==M2(a-c2).
So we solve for c2 (remember that the transpose of a rotation matrix is the same as the inverse):
M2TM1(a-c1)==(a-c2) =>
-M2TM1(a-c1)+a==c2
Now if we look at this as a transformation being applied to c1, then we can see that it is first negated, then translated by a, then rotated by M1, then rotated by M2T, negated again, and then translated by a again. These are transformations that graphics libraries are good at and they can all be squished into a single transformation matrix.
#Generic Human deserves credit for the answer, but here's code for it. Of course, you need to implement the function to validate a change in pitch before it's applied, but that's simple. This code probably has a couple typos since I haven't tried to compile:
case ENavigation::eRotatePoint: {
const double dX = (double)(m_oLastMousePos.x - roMousePos.x) * dRotatePixelScale * D3DX_PI;
double dY = (double)(m_oLastMousePos.y - roMousePos.y) * dRotatePixelScale * D3DX_PI;
dY = validatePitch(dY); // dY needs to be kept within bounds so that m_dRotationY is within bounds
D3DXMATRIX oRotationMatrix1; // The camera orientation before mouse-change
D3DXMatrixRotationYawPitchRoll(&oRotationMatrix1, m_dRotationX, m_dRotationY, 0.0);
D3DXMATRIX oRotationMatrix2; // The camera orientation after mouse-change
D3DXMatrixRotationYawPitchRoll(&oRotationMatrix2, m_dRotationX + dX, m_dRotationY + dY, 0.0);
D3DXMATRIX oRotationMatrix2Inv; // The inverse of the orientation
D3DXMatrixTranspose(&oRotationMatrix2Inv,&oRotationMatrix2); // Transpose is the same in this case
D3DXMATRIX oScaleMatrix; // Negative scaling matrix for negating the translation
D3DXMatrixScaling(&oScaleMatrix,-1,-1,-1);
D3DXMATRIX oTranslationMatrix; // Translation by the reference point
D3DXMatrixTranslation(&oTranslationMatrix,
m_oRotateAroundPos.x,m_oRotateAroundPos.y,m_oRotateAroundPos.z);
D3DXMATRIX oTransformMatrix; // The full transform for the eyePos.
// We assume the matrix multiply protects against variable aliasing
D3DXMatrixMultiply(&oTransformMatrix,&oScaleMatrix,&oTranslationMatrix);
D3DXMatrixMultiply(&oTransformMatrix,&oTransformMatrix,&oRotationMatrix1);
D3DXMatrixMultiply(&oTransformMatrix,&oTransformMatrix,&oRotationMatrix2Inv);
D3DXMatrixMultiply(&oTransformMatrix,&oTransformMatrix,&oScaleMatrix);
D3DXMatrixMultiply(&oTransformMatrix,&oTransformMatrix,&oTranslationMatrix);
D3DXVECTOR4 oEyeFinal;
D3DXVec3Transform(&oEyeFinal, &m_oEyePos, &oTransformMatrix);
m_oEyePos = D3DXVECTOR3(oEyeFinal.x, oEyeFinal.y, oEyeFinal.z)
// Increment rotation to keep the same relative look angles
RotateXAxis(dX);
RotateYAxis(dY);
break;
}
I think there is a much simpler solution that lets you sidestep all rotation issues.
Notation: A is the point we want to rotate around, C is the original camera location, M is the original camera rotation matrix that maps global coordinates to the camera's local viewport.
Make a note of the local coordinates of A, which are equal to A' = M × (A - C).
Rotate the camera like you would in normal "eye rotation" mode. Update the view matrix M so that it is modified to M2 and C remains unchanged.
Now we would like to find C2 such that A' = M2 × (A - C2).
This is easily done by the equation C2 = A - M2-1 × A'.
Voilà, the camera has been rotated and because the local coordinates of A are unchanged, A remains at the same location and the same scale and distance.
As an added bonus, the rotation behavior is now consistent between "eye rotation" and "point rotation" mode.
You rotate around the point by repeatedly applying small rotation matrices, this probably cause the drift (small precision errors add up) and I bet you will not really do a perfect circle after some time. Since the angles for the view use simple 1-dimension double, they have much less drift.
A possible fix would be to store a dedicated yaw/pitch and relative position from the point when you enter that view mode, and using those to do the math. This requires a bit more bookkeeping, since you need to update those when moving the camera. Note that it will also make the camera move if the point move, which I think is an improvement.
If I understand correctly, you are satisfied with the rotation component in the final matrix (save for inverted rotation controls in the problem #3), but not with the translation part, is that so?
The problem seems to come from the fact that you treating them differently: you are recalculating the rotation part from scratch every time, but accumulate the translation part (m_oEyePos). Other comments mention precision problems, but it's actually more significant than just FP precision: accumulating rotations from small yaw/pitch values is simply not the same---mathematically---as making one big rotation from the accumulated yaw/pitch. Hence the rotation/translation discrepancy. To fix this, try recalculating eye position from scratch simultaneously with the rotation part, similarly to how you find "oTarget = oEyePos + ...":
oEyePos = m_oRotateAroundPos - dist * D3DXVECTOR3(oForward4.x, oForward4.y, oForward4.z)
dist can be fixed or calculated from the old eye position. That will keep the rotation point in the screen center; in the more general case (which you are interested in), -dist * oForward here should be replaced by the old/initial m_oEyePos - m_oRotateAroundPos multiplied by the old/initial camera rotation to bring it to the camera space (finding a constant offset vector in camera's coordinate system), then multiplied by the inverted new camera rotation to get the new direction in the world.
This will, of course, be subject to gimbal lock when the pitch is straight up or down. You'll need to define precisely what behavior you expect in these cases to solve this part. On the other hand, locking at m_dRotationX=0 or =pi is rather strange (this is yaw, not pitch, right?) and might be related to the above.
I've been working on a semi flight simulator. What I am trying to do is use a pitch roll and yaw to rotate an object. I have already looked online a lot, and although they explain what the problem is I have no idea how to implement the solution. So for example I do:
glRotatef(yaw,0,1,0);
glRotatef(pitch,1,0,0);
The yaw doesn't act properly, the pitch will work fine. And from what I have been reading it seems that the objects local axis has been changed so I need to find the object's new local axis and rotate around that. So I tried that with something like:
newpitch=pitch/57.29
VectorA(0,cos(newpitch)-sin(newpitch),sin(newpitch)+cos(newpitch));
glRotatef(yaw,vec.getXAxis(),vec.getYAxis(),vec.getZAxis());
glRotatef(pitch,1,0,0);
This seems to not work either.
I've also tried making a general rotation matrix and giving it both pitch and yaw and still the same problem. And I've tried using quaternions and the same problem still exists!
Here is my code for quaternions:
void Quat::eulerToQuat(float roll,float pitch,float yaw){
float radiansY = yaw/57.2;
float radiansZ = roll/57.2;
float radiansX = pitch/57.2;
float sY = sinf(radiansY * 0.5);
float cY = cosf(radiansY * 0.5);
float sZ = sinf(radiansZ * 0.5);
float cZ = cosf(radiansZ * 0.5);
float sX = sinf(radiansX * 0.5);
float cX = cosf(radiansX * 0.5);
w = cY * cZ * cX - sY * sZ * sX;
x = sY * sZ * cX + cY * cZ * sX;
y = sY * cZ * cX + cY * sZ * sX;
z = cY * sZ * cX - sY * cZ * sX;
}
Then I converted this into a matrix and use glMultMatrix(matrix) with the modelview matrix, and this has the same problem. So I'm confident it wouldn't be gimble lock =).
So in my code I do:
float matrix[4][4];
Quat this;
this.eularToQuat(roll,pitch,yaw);
this.toMatrix(matrix);
glMultMatrix(matrix);
I think you're referring to gimbal lock? You're right that each rotation modifies the axes around which subsequent local rotations will occur. In your case that affects the yaw because the OpenGL matrix stack works so that each thing you add to it occurs conceptually before whatever is already on the stack (ie, it's post multiplication in matrix terms).
Your solution, however, won't solve the problem even if implemented correctly. What you're trying to do is get the global y axis in local coordinate space so that you can rotate around the global y even after you've rotated around the global z, shifting the local axes. But that just buys you much the same problems as if you'd stuck with global axes throughout and applied the rotations in the other order. The second rotation will now interfere with the first rather than vice versa.
Another way to convince yourself that what you're doing is wrong is to look at how much information you have. You're trying to describe the orientation of an object with two numbers. Two numbers isn't enough to describe any rotation whatsoever, so there's obviously some other rule in there to convert two numbers into a complete orientation. Whatever you do to modify that rule, you're going to end up limiting the orientations you can reach. But with an aeroplane you really want to be able to reach any orientation, so that's a fundamental contradiction.
The confusion comes because, if you have a suitable way of storing orientation, it's completely valid to work forward from that by saying 'what is the orientation if I modify that by rotating around local y by 5, then around local z by 10?', etc. The problem is trying to aggregate all those transformations into a single pair of rotations. It isn't possible.
The easiest solution if you're already generally up on OpenGL tends to be to store the orientation as a complete matrix. You accumulate pitch and yaw rotations by applying them as they occur to that matrix. You pass that matrix to OpenGL via glMultMatrix to perform your drawing.
It's not an optimal solution but a quick fix test solution would be to use glLoadMatrix and glGet to apply transformations by loading your matrix to and then retrieving it from the OpenGL matrix stack, separately from your drawing. It's not really what the stack is for so you'll probably get some performance problems and over time rounding errors will cause odd behaviour but you can fix those once you're persuaded by the approach. The OpenGL man pages give the formulas for all transformation matrices and you should look up matrix normalisation (you'll probably be using an orthonormal matrix whether you realise it or not, which should help with Google) to deal with cumulative rounding.
EDIT: with respect to the code you've posted while I was rambling, quaternions are another valid way of representing orientation and another thing that you can apply incremental updates to safely. They're also compact very easy to protect from rounding errors. However I think your problem may be that you aren't using quaternions as the storage for orientation, merely as an intermediate container. So adding them to the chain doesn't fix any of your problems.
EDIT2: a further bit of hand-waving explanation to push the idea that directly storing pitch and yaw isn't good enough: imagine that, from the point of view of the pilot, you apply a yaw of 90 degrees, then a pitch of 30 degrees, then a yaw of -90 degrees. Then you end up exactly as if you'd applied a roll of 30 degrees. But if you're just storing pitch and yaw then you've no way of storing roll. Furthermore, if you just add up the total yaw and total pitch you end up thinking you've applied a pitch of 30 degrees rather than a roll. So it doesn't matter what order you apply pitch and yaw, or whether you use global or local axes, you get the wrong result.
You should yaw, pitch and roll using one transformation. Cause when you don't, you'll pushing yourself towards gimbal lock. Excerpt:
Gimbal lock is the loss of one degree of freedom in a
three-dimensional space that occurs when the axes of two of the three
gimbals are driven into a parallel configuration, "locking" the system
into rotation in a degenerate two-dimensional space.
Consider this example of Gimbal locked airplane:
When the pitch (green) and yaw (magenta)
gimbals become aligned, changes to roll (blue) and yaw apply the same
rotation to the airplane
NOTICE: I have edited the question below which is more relevant to my real issue than the text right below, you can skip this if you but I'll leave it here for historic reasons.
To see if I get this right, a float in C is the same as a value in radians right? I mean, 360º = 6.28318531 radians and I just noticed on my OpenGL app that a full rotation goes from 0.0 to 6.28, which seems to add up correctly. I just want to make sure I got that right.
I'm using a float (let's call it anglePitch) from 0.0 to 360.0 (it's easier to read in degrees and avoids casting int to float all the time) and all the code I see on the web uses some kind of DEG2RAD() macro which is defined as DEG2RAD 3.141593f / 180. In the end it would be something like this:
anglePitch += direction * 1; // direction will be 1 or -1
refY = tan(anglePitch * DEG2RAD);
This really does a full rotation but that full rotation will be when anglePitch = 180 and anglePitch * DEG2RAD = 3.14, but a full rotation should be 360|6.28. If I change the macro to any of the following:
#define DEG2RAD 3.141593f / 360
#define DEG2RAD 3.141593f / 2 / 180
It works as expected, a full rotation will happen when anglePitch = 360.
What am I missing here and what should I use to properly convert angles to radians/floats?
IMPORTANT EDIT (REAL QUESTION):
I understand now the code I see everywhere on the web about DEG2RAD, I'm just too dumb at math (yeah, I know, it's important when working with this kind of stuff). So I'm going to rephrase my question:
I have now added this to my code:
#define PI 3.141592654f
#define DEG2RAD(d) (d * PI / 180)
Now, when working the pitch/yawn angles in degrees, which are floats, once again, to avoid casting all the time, I just use the DEG2RAD macro and the degree value will be correctly converted to radians. These values will be passed to sin/cos/tan functions and will return the proper values to be used in GLUT camera.
Now the real question, where I was really confused before but couldn't explain myself better:
angleYaw += direction * ROTATE_SPEED;
refX = sin(DEG2RAD(angleYaw));
refZ = -cos(DEG2RAD(angleYaw));
This code will be executed when I press the LEFT/RIGHT keys and the camera will rotate in the Y axis accordingly. A full rotation goes from 0º to 360º.
anglePitch += direction * ROTATE_SPEED;
refY = tan(DEG2RAD(anglePitch));
This is similar code and will be executed when I press the UP/DOWN keys and the camera will rotate in the X axis. But in this situation, a full rotation goes from 0º to 180º degrees and that's what's really confusing me. I'm sure it has something to do with the tangent function but I can't get my head around it.
Is there way I could use sin/cos (as I do in the yawn code) to achieve the same rotation? What is the right way, the most simple code I can add/fix and what makes more sense to create a full pitch rotation from 0º to 360º?
360° = 2 * Pi, Pi = 3.141593…
Radians are defined by the arc length of an angle along a circle of radius 1. The circumfence of a circle is 2*r*Pi, so one full turn on a unit circle has an arc length of 2*Pi = 6.28…
The measure of angles in degrees stem from the fact, that by aligning 6 equilateral triangles you span a full turn. So we have 6 triangles, each making up a 6th of the turn, so the old babylonians divided a circle into pieces of 1/(6*6) = 1/36, and to further refine it this was subdivded by 10. That's why we ended up with 360° in a full circle. This number is arbitrarily choosen, though.
So if there are 2*Pi/360° this makes Pi/180° = 3.141593…/180° which is the conversion factor from degrees to radians. The reciprocal, 180°/Pi = 180/3.141593…
Why on earth the old OpenGL function glRotate and GLU's gluPerspective used degrees instead of radians I cannot fathom. From a mathematical point of view only radians make sense. Which I think is most beautifully demonstrated by Euler's equation
e^(i*Pi) - 1 = 0
There you have it, all the important numbers of mathematics in one single equation. What's this got to do with angles? Well:
e^(i*alpha) = cos(alpha) + i * sin(alpha), alpha is in radians!
EDIT, with respect to modified question:
Your angles being floats is all fine. Why would you even think degress being integers I cannot understand. Normally you don't have to define PI yourself, it comes predefined in math.h, usually called M_PI, M_2PI, and M_PI2 for Pi, 2*Pi and Pi/2. You also should change your macro, the way it's written now can create strange effects.
#define DEG2RAD(d) ( (d) * M_PI/180. )
GLUT has no camera at all. GLUT is a rather dumb OpenGL framework I recommend not using. You probably refer to gluLookAt.
Those obstacles out of the way let's see what you're doing there. Remember that trigonometric functions operate on the unit circle. Let the angle 0 point towards the right and angles increment counterclockwise. Then sin(a) is defined as the amount of rightwards and cos(a) and the amount of forwards to reach the point at angle a on the unit circle. This is what the refX and refZ are getting assigned to.
refY however makes no sense written that way. tan = sin/cos so as we approach n*pi/2 (i.e. 90°) it diverges to +/- infinity. At least it explains your pi/180° cyclic range, because that's the period of tan.
I was first thinking that tan may have been used to normalize the direction vector, but didn't make sense either. The factor would have been 1./sqrt(sin²(Pitch) + 1)
I double checked: using tan there does the right thing.
EDIT2: I don't see where your problem is: The pitch angle is -90° to +90°, which makes perfect sense. Go get yourself a globe (of the earth): The east-west coordinates (longitude) go from -180° to +180°, the south-north coordinate (latitude) goes -90° to +90°. Think about it: Any larger coordinate range would create ambiguities.
The only good suggestion I offer you is: Grab some math text book and bend your mind around spherical coordinates! Sorry to tell you that way. Whatever you have works perfectly fine, you just need to understand sperical geometry.
You're using the terms Yaw and Pitch. Those are normally used in Euler angles. Now unfortunately Euler angles, which compelling at first, cause serious trouble later on (like gimbal lock). You should not use them at all. It may also be a good idea if you used some pencil/sticks/whatever to decompose the rotations you're intending with your hands to understand their mechanics.
And by the way: There are also non-integer degrees. Just hop over to http://maps.google.com to see them in action (just select some place and let http://maps.google.com give you the link to it).
'float' is a type, like int or double. radians and degrees are units of measure, both of which can be represented with any precision you want. i.e., there's no reason you can't have 22.5 degrees, and keep that value in a float.
a full rotation in radians is 2*pi, about 6.283, whereas a full rotation in degrees is 360. You can convert between them by dividing out the starting unit's full circle, then multiplying by the desired unit's full circle.
for example, to get from 90 degrees to radians, first divide out the degrees. 90 over 360 is 0.25 (note this value is in 'revolutions'). Now multiply that 0.25 by 6.283 to arrive at 1.571 radians.
follow up
the reason you're seeing your pitch cycle twice as fast as it should is precisely because you're using tan(pitch) to compute the Y component. What you should have is that the Y component depends on sin(pitch). i.e., try changing
refY = tan(DEG2RAD(anglePitch));
to
refY = sin(DEG2RAD(anglePitch));
a technical detail: the numbers that go into the look matrix should all be in the range of -1 to +1, and if you were to inspect the values you're feeding to refY, and run your pitch outside of -45 to +45 degrees, you'd see the problem; tan() runs off to infinity at +/-90 degrees.
also, note that casting a value from int to float in no sense converts between degrees and radians. casting just gives you the nearest equivalent value in the new storage type. for example, if you cast the integer 22 to floating point, you get 22.0f, whereas if you cast 33.3333f to type int, you'd be left with 33. when working with angles, you really should just stick with floating point, unless you're constrained by working with an embedded processor or something. this is especially important with radians, where whole number increments represent leaps of (about) 57.3 degrees.
Assuming that your ref components are intended to be used as your look-at vector, I think what you need is
refY = sin(DEG2RAD(anglePitch));
XZfactor = cos(DEG2RAD(anglePitch));
refX = XZfactor*sin(DEG2RAD(angleYaw));
refZ = -XZfactor*cos(DEG2RAD(angleYaw));
This one has been eating up my entire night, and I'm finally throwing up my hands for some assistance. Basically, it's fairly straightforward to calculate the Pitch and Yaw from the View Matrix right after you do a camera update:
D3DXMatrixLookAtLH(&m_View, &sCam.pos, &vLookAt, &sCam.up);
pDev->SetTransform(D3DTS_VIEW, &m_View);
// Set the camera axes from the view matrix
sCam.right.x = m_View._11;
sCam.right.y = m_View._21;
sCam.right.z = m_View._31;
sCam.up.x = m_View._12;
sCam.up.y = m_View._22;
sCam.up.z = m_View._32;
sCam.look.x = m_View._13;
sCam.look.y = m_View._23;
sCam.look.z = m_View._33;
// Calculate yaw and pitch and roll
float lookLengthOnXZ = sqrtf( sCam.look.z^2 + sCam.look.x^2 );
fPitch = atan2f( sCam.look.y, lookLengthOnXZ );
fYaw = atan2f( sCam.look.x, sCam.look.z );
So my problem is: there must be some way to obtain the Roll in radians similar to how the Pitch and Yaw are being obtained at the end of the code there. I've tried several dozen algorithms that seemed to make sense to me, but none of them gave quite the desired result. I realize that many developers don't track these values, however I am writing some re-centering routines and setting clipping based on the values, and manual tracking breaks down as you apply mixed rotations to the View Matrix. So, does anyone have the formula for getting the Roll in radians (or degrees, I don't care) from the View Matrix?
Woohoo, got it! Thank you stacker for the link to the Euler Angles entry on Wikipedia, the gamma formula at the end was indeed the correct solution! So, to calculate Roll in radians from a "vertical plane" I'm using the following line of C++ code:
fRoll = atan2f( sCam.up.y, sCam.right.y ) - D3DX_PI/2;
As Euler pointed out, you're looking for the arc tangent of the Y (up) matrix's Z value against the X (right) matrix's Z value - though in this case I tried the Z values and they did not yield the desired result so I tried the Y values on a whim and I got a value which was off by +PI/2 radians (right angle to the desired value), which is easily compensated for. Finally I have a gryscopically accurate and stable 3D camera platform with self-correcting balance. =)
It seems the Wikipedia article about Euler angles contains the formula you're lookin for at the end.