Quaternions can indeed be tricky to understand, but once you break down the concepts and visualize them, they start to make more sense. Let’s go step by step to explain what quaternions are and how we can visualize their parameters.

1. What are Quaternions?

A quaternion is a mathematical object used to represent rotations in 3D space. Unlike Euler angles (pitch, yaw, roll), quaternions avoid issues like gimbal lock and are more efficient for combining rotations.

A quaternion is made up of four components:

• x, y, z: These represent the axis of rotation in 3D space.
• w: This represents the amount of rotation around that axis (it’s the scalar component).

2. Quaternion Components:

A quaternion $q$ is usually written as:

\[q = w + xi + yj + zk\]

Where:

• x, y, z are the vector components, representing the axis of rotation in 3D space (they define a direction).
• w is the scalar component and represents the cosine of half the angle of rotation.

So, a quaternion is made up of:

• x, y, z: The vector part, which defines the axis of rotation.
• w: The scalar part, which defines how much rotation there is around the axis.

3. Visualizing the Parameters:

To visualize quaternions, it helps to break it down into two key parts:

1. Rotation Axis (x, y, z): This defines the line around which the rotation happens.

   • Think of a 3D vector. The direction of this vector is the axis of rotation. For example, if x = 1, y = 0, z = 0, the axis is the X-axis, and the object will rotate around the X-axis.

2. Amount of Rotation (w): The w parameter defines the angle of rotation.

   • The w component can be derived from the cosine of half the rotation angle. So, it represents how much rotation is around the axis.

   • The formula for w is:

     \[w = \cos(\frac{\theta}{2})\]

     where $\theta$ is the angle of rotation.

4. How Do x, y, z, w Work Together?

Let’s take an example to help visualize:

• Axis of rotation: Let’s say we rotate around the Y-axis.

  • The axis would be: x = 0, y = 1, z = 0.

• Angle of rotation: Suppose we want to rotate by 90 degrees.

  • First, convert 90 degrees into radians: $\theta = 90^\circ = \frac{\pi}{2} \, \text{radians}$.
  • Then, calculate $w = \cos(\frac{\pi}{4}) \approx 0.707$.

  • The quaternion becomes:

    \[q = (0.707, 0, 0.707, 0)\]

    So, you have an axis of rotation (x, y, z = 0, 1, 0) and the amount of rotation $w = 0.707$.

5. Visualizing the Rotation:

Think of it like this:

• The x, y, z components define the axis (the direction of the line around which rotation happens).
• The w component determines how much rotation is applied around that axis.

For example:

• x = 1, y = 0, z = 0, w = 0: This means the object is rotated 180° around the X-axis.
• x = 0, y = 1, z = 0, w = 0: This means the object is rotated 180° around the Y-axis.
• x = 0, y = 0, z = 1, w = 0: This means the object is rotated 180° around the Z-axis.

6. Visualizing the Quaternion as a Rotation:

To get an idea of how quaternions represent rotation:

• Imagine a spinning top. The direction in which the top spins is the axis (x, y, z).
• The amount of spin or how much it turns in a given time is the w (cosine of half the angle).
• So, quaternions are like describing a spinning top in 3D space: the axis of rotation and the speed or amount of spinning (w).

7. Why Use Quaternions Over Euler Angles?

Quaternions avoid some of the common problems that arise from Euler angles:

• Gimbal lock: When using Euler angles, you can lose one degree of freedom in rotation, leading to unexpected behavior. Quaternions don’t have this issue.
• Smooth interpolation: Quaternions allow smooth interpolation between rotations (called slerp, or spherical linear interpolation).

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Conclusion:

• x, y, z in a quaternion represent the axis of rotation.
• w in a quaternion represents the amount of rotation around the axis (cosine of half the angle).
• To visualize: Imagine the x, y, z as the direction you’re spinning around (like a top), and w as how much spin happens around that direction.

Hopefully, this gives you a clearer picture! It can take some time to fully grasp, but thinking of quaternions as describing a spinning object (with the axis and amount of rotation) can make the concept more intuitive. If you have more questions or need further clarification, feel free to ask!

Go!

commit 4702aa2352f1758c23475e3e7a9db318da313f42
Author: sea <zhangxinghai19890103@hotmail.com>
Date:   Mon May 26 00:40:00 2025 +0800

    go

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