内存编辑相机控制：如何将局部角速度转换为全局角速度？

Great question—this is a classic 3D coordinate transformation problem, and it's totally solvable once you wrap your head around how camera local space maps to global space. Let's break this down step by step:

First, we need to clarify two key coordinate systems (adjust these to match your program's specific axis setup):

• Global Space (World Coordinates): Let’s define this as having axes Xg (right), Yg (up), Zg (forward) — this is the fixed, universal space the program uses for all objects.
• Camera Local Space: The camera’s own coordinate system, with axes Xc (camera’s right), Yc (camera’s up), Zc (camera’s forward, what it’s looking at). When you want to "pitch up 1 rad/s", that’s a rotation around the camera’s Xc axis.

Angular velocity is a vector that encodes both the axis of rotation and the speed. To convert a local angular velocity vector to global space, you use the camera’s rotation matrix (or quaternion, if your program uses quaternions for orientation).

Using Rotation Matrices #

The camera’s rotation matrix R is a 3x3 orthogonal matrix where each column represents the local axis (Xc, Yc, Zc) expressed in global coordinates. To get the global angular velocity ω_global from the local ω_local:

ω_global = R × ω_local

(Note: This is matrix-vector multiplication, not a cross product.)

Using Quaternions #

If your program stores the camera’s orientation as a unit quaternion q, you can convert the local angular velocity using quaternion conjugation:

ω_global = q * ω_local * q⁻¹

Since q is a unit quaternion, its inverse q⁻¹ is just its conjugate (flip the sign of the imaginary components).

Let’s say you want to pitch the camera up at 1 rad/s — that’s a local angular velocity of ω_local = (1, 0, 0) (assuming pitch is around the Xc axis, positive = upward).

1. Case 1: Camera is aligned with global space
   The rotation matrix R is the identity matrix. Multiplying gives ω_global = (1, 0, 0) — so you’d set the global X-axis angular velocity to 1 rad/s, which makes intuitive sense.

2. Case 2: Camera has yawed 90° right (around Yg axis)
   The rotation matrix R would look like this:

   [ 0  0  1 ]
   [ 0  1  0 ]
   [-1  0  0 ]
   

   Multiplying R × (1,0,0) gives ω_global = (0, 0, -1) — so you’d set the global Z-axis angular velocity to -1 rad/s. This works because, from the world’s perspective, a camera facing right pitching up is rotating around the negative global Z axis.

1. Grab the camera’s current orientation:
   • If your program exposes a rotation matrix directly, use that.
   • If it uses Euler angles (yaw/pitch/roll), convert them to a rotation matrix first (avoid relying on Euler angles directly — they can cause gimbal lock issues).
   • If it uses quaternions, use the conjugation method above.
2. Define your local angular velocity vector:
   • Pitch (up/down): Rotation around Xc → (speed, 0, 0)
   • Yaw (left/right): Rotation around Yc → (0, speed, 0)
   • Roll (tilt): Rotation around Zc → (0, 0, speed)
     Adjust signs based on your program’s axis direction conventions (use the right-hand rule to verify: curl your fingers in the rotation direction, thumb points to the positive axis).
3. Perform the transformation: Multiply the rotation matrix by the local vector (or use quaternion conjugation) to get the global angular velocity components.
4. Apply the global angular velocity: Assign the resulting X, Y, Z components to the global angular velocity parameters you can edit via memory.

• Double-check axis conventions: If your program uses a different global space (e.g., Yg = forward, Zg = up), adjust your local vector and rotation matrix accordingly. A wrong axis mapping will result in inverted or incorrect camera movement.
• Unit consistency: Ensure all angular velocities use radians per second — if your program expects degrees, convert first (rad = deg × π/180).
• Gimbal lock: If you have to use Euler angles to build the rotation matrix, be aware of gimbal lock (when two axes align) and test edge cases (e.g., camera pointing straight up/down).

内容的提问来源于stack exchange，提问作者Kyle B