Limit camera pitch

Question

How can I efficiently limit camera pitch when I have only camera quaternion? Do I have to convert to euler angles and then back to quaternion or is there any other way?

Answer 1

If the camera never has any roll (as is common in lots of games, such as first person shooters), then the solution is simple. If there is roll, then there's an additional step involved. I'll start with what to do if there is no roll, and generalize the solution to what to do if there is.

Let qc be the camera rotation. Let qy a rotation with the same yaw as qc, but with zero pitch. If there is no roll, the camera rotation is a yaw rotation followed by a pitch rotation:

qc = qp * qy

We can recover the pitch rotation qp as the rotation from qy to qc:

qp = qc * qy^-1

The trick, then, is to construct qy, so we can plug it into the above equation to solve for qp. Let vc be the unit vector pointing out of the lens of the camera, or the "forward vector". Let vy be the same vector, but projected to the horizontal plane and normalized. Finally, let v0 be the forward vector when the camera rotation qc is the identity rotation. The rotation that rotates v0 into vy is the yaw rotation. The angle can be given as:

yaw = asin(Norm(cross(v0, vy)))

The corresponding yaw rotation is:

qy = { cos(yaw/2), up * sin(yaw/2) }

Where "up" is the unit vector in the up direction, aka the axis for yaw rotations. Plug this into qp = qy^-1 * qc above to get pitch quaternion qp. Finally, get the pitch angle from qp as:

pitch = 2*asin(Dot(right, [qp[1], qp[2], qp[3]]))

Where "right" is the unit vector in the right direction, aka the axis for pitch rotations.

Like I said, things get more complicated if the camera also has roll, but the general strategy is the same. You formulate the camera rotation as a product of rotation components, then isolate the component you want (in this case, pitch). For example, if the euler sequence you use to define "pitch" is the common yaw-pitch-roll sequence, you define qc as:

qc = qr * qp * qy

We can define a variable qx to be the combined pitch and roll rotations:

qx = qr * qp

We can now write qc as:

qc = qx * qy

We already know how to solve for qx in this form, by retracing the steps we used above to solve for qp. Rearranging the definition for qx, we get:

qp = qr^-1 * qx

We just solved for qx, so to solve for the pitch rotation qp, we only need the roll qr. We can construct it using vectors as we did previously. Let vc be the forward vector again. The roll will be a rotation around this vector. Let vu be the camera's up vector (in world coordinates), and let vu0 be the camera's up vector with zero roll. We can construct vu0 by projecting the global up vector to the plane perpendicular to vc, then normalizing. The roll rotation qr is then the rotation from vu0 to vu. The axis of this rotation is the forward vector vc. The roll angle is

roll = asin(Dot(vc, cross(vu0, vu)))

The corresponding quaternion is:

qr = { cos(roll/2), forward * sin(roll/2) }

Where "forward" is the axis of roll rotations.

Answer 2

The pitch is just one component of the full rotation, so if you want to think about your rotation like that, you'd better store the pitch separately, possibly using Euler angles.

Just converting to Euler angles and back when you need to limit the movement might not work too well, since you'l need to remember the last frame (if you have any) to see if you passed the limit, and in what direction.

As I see it, the main point of quaternions is that you don't need to bother with limits like that. Everything just works without any singularities. Why exactly do you want to limit the pitch?