Complete Guide to Quaternion Interpolation Methods

Brief Answer

To properly calculate an intermediate quaternion between two rotations, you should use Spherical Linear Interpolation (SLERP) rather than simple linear interpolation. SLERP interpolates along the shortest arc on the unit sphere in quaternion space, maintaining constant angular velocity and producing visually smooth rotations. The key is to normalize the interpolated result and handle the double-cover property of quaternions by choosing the shortest path between rotations.

Contents

• Understanding Quaternions and Rotations
• Why Linear Interpolation Fails
• Implementing Spherical Linear Interpolation (SLERP)
• Alternative Interpolation Methods
• Practical Implementation Considerations
• Code Examples and Best Practices
• Common Pitfalls and Solutions

Understanding Quaternions and Rotations

Quaternions are mathematical extensions of complex numbers that provide an efficient way to represent 3D rotations. A quaternion is typically represented as:

$$q = w + xi + yj + zk$$

where $w$ is the scalar component and $(x, y, z)$ is the vector component. For rotation quaternions, they must be unit quaternions, satisfying $w^2 + x^2 + y^2 + z^2 = 1$.

Key Insight: Quaternions represent rotations in 3D space through double covering - both $q$ and $-q$ represent the same rotation. This property is crucial when interpolating between rotations.

Unlike Euler angles, quaternions avoid gimbal lock and provide more efficient interpolation. However, their four-dimensional nature makes interpolation less intuitive than simple linear interpolation in 3D space.

Why Linear Interpolation Fails

When you tried linear interpolation of quaternion components, you likely encountered several issues:

1. Non-unit quaternions: Linear interpolation produces non-unit quaternions that don’t represent valid rotations.
2. Non-constant angular velocity: The rotation speed appears to vary during the interpolation.
3. Shortest path violation: Linear interpolation doesn’t guarantee taking the shortest path between rotations.

Consider two quaternions $q_1$ and $q_2$. Linear interpolation (LERP) would be:

$$q(t) = (1-t)q_1 + tq_2$$

This approach fails because it doesn’t account for the spherical geometry of quaternion space. The result is a quaternion that moves through the interior of the quaternion unit sphere rather than along its surface.

Visual representation:
  q₁
   \
    \
     \ (LERP path - not on sphere surface)
      \
       \
        q₂

Implementing Spherical Linear Interpolation (SLERP)

SLERP correctly interpolates along the surface of the unit sphere in quaternion space. The formula is:

$$\text{slerp}(q_1, q_2, t) = \frac{\sin((1-t)\Omega)}{\sin(\Omega)}q_1 + \frac{\sin(t\Omega)}{\sin(\Omega)}q_2$$

where $\Omega = \arccos(q_1 \cdot q_2)$ is the angle between the two quaternions.

Step-by-step SLERP implementation:

1. Calculate the dot product: $\cos \Omega = q_1 \cdot q_2 = w_1w_2 + x_1x_2 + y_1y_2 + z_1z_2$
2. Handle the double-cover property: If $\cos \Omega < 0$, negate one quaternion to ensure shortest path
3. Compute interpolation factors:
   • $k_1 = \frac{\sin((1-t)\Omega)}{\sin(\Omega)}$
   • $k_2 = \frac{\sin(t\Omega)}{\sin(\Omega)}$
4. Compute the result: $q(t) = k_1q_1 + k_2q_2$
5. Normalize the result: $q_{\text{normalized}} = \frac{q(t)}{|q(t)|}$

Special cases:

• When $\Omega \approx 0$ (quaternions are very close), use linear interpolation followed by normalization
• When $\Omega \approx \pi$ (quaternions are nearly opposite), you may need to handle this case specially

Here’s a Python implementation:

import math
import numpy as np

def slerp(q1, q2, t):
    """
    Spherical linear interpolation between quaternions q1 and q2
    """
    # Ensure inputs are numpy arrays
    q1 = np.array(q1)
    q2 = np.array(q2)
    
    # Calculate the dot product
    dot = np.dot(q1, q2)
    
    # If the dot product is negative, negate one quaternion to ensure shortest path
    if dot < 0:
        q2 = -q2
        dot = -dot
    
    # If quaternions are very close, use linear interpolation
    if dot > 0.9995:
        result = (1-t)*q1 + t*q2
        return result / np.linalg.norm(result)
    
    # Calculate the angle between quaternions
    theta_0 = math.acos(dot)
    sin_theta_0 = math.sin(theta_0)
    
    # Calculate interpolation factors
    theta = theta_0 * t
    sin_theta = math.sin(theta)
    
    # Compute the interpolated quaternion
    s0 = math.sin(theta_0 - theta) / sin_theta_0
    s1 = sin_theta / sin_theta_0
    
    result = s0 * q1 + s1 * q2
    return result / np.linalg.norm(result)  # Normalize

Alternative Interpolation Methods

While SLERP is the standard for interpolating between two quaternions, other methods may be appropriate in certain situations:

1. Normalized Linear Interpolation (NLERP)

A simpler alternative that normalizes the linearly interpolated quaternion:

$$q(t) = \frac{(1-t)q_1 + tq_2}{|(1-t)q_1 + tq_2|}$$

2. Squad (Spherical Quadrangle Interpolation)

For interpolating through multiple quaternions with smooth transitions:

$$\text{squad}(q_1, q_2, q_1', q_2', t) = \text{slerp}(\text{slerp}(q_1, q_2, t), \text{slerp}(q_1', q_2', t), 2t(1-t))$$

3. Comparison of Methods

Practical Implementation Considerations

Handling the Double-Cover Property

Since $q$ and $-q$ represent the same rotation, you should always choose the shortest path by checking the dot product:

if np.dot(q1, q2) < 0:
    q2 = -q2  # Use the conjugate instead

Numerical Stability

When the angle between quaternions is very small, $\sin(\Omega)$ approaches zero, potentially causing numerical instability. In such cases, fall back to normalized linear interpolation.

Performance Optimization

For performance-critical applications:

1. Precompute $\sin(\Omega)$ and store it
2. Use fast inverse square root approximations for normalization
3. Consider using SIMD instructions if available

Quaternion Normalization

Ensure all quaternions remain normalized throughout interpolation. Even with SLERP, floating-point errors can accumulate.

Code Examples and Best Practices

Here’s a more complete implementation with best practices:

import numpy as np

def quaternion_normalize(q):
    """Normalize a quaternion to unit length"""
    norm = np.linalg.norm(q)
    if norm == 0:
        return np.array([1, 0, 0, 0])  # Identity quaternion as fallback
    return q / norm

def quaternion_dot(q1, q2):
    """Calculate the dot product of two quaternions"""
    return np.dot(q1, q2)

def slerp_optimized(q1, q2, t):
    """
    Optimized SLERP implementation with proper handling of edge cases
    """
    # Ensure inputs are normalized
    q1 = quaternion_normalize(q1)
    q2 = quaternion_normalize(q2)
    
    # Calculate dot product
    dot = quaternion_dot(q1, q2)
    
    # If quaternions are nearly opposite or identical, handle specially
    if abs(dot) > 0.9995:
        # Use linear interpolation and normalize
        result = (1-t)*q1 + t*q2
        return quaternion_normalize(result)
    
    # Ensure we take the shortest path
    if dot < 0:
        q2 = -q2
        dot = -dot
    
    # Calculate interpolation parameters
    theta_0 = np.arccos(np.clip(dot, -1.0, 1.0))  # Clip for numerical stability
    sin_theta_0 = np.sin(theta_0)
    
    theta = theta_0 * t
    sin_theta = np.sin(theta)
    
    # Compute the interpolated quaternion
    s0 = np.sin(theta_0 - theta) / sin_theta_0
    s1 = sin_theta / sin_theta_0
    
    result = s0 * q1 + s1 * q2
    return quaternion_normalize(result)

Common Pitfalls and Solutions

Pitfall 1: Non-normalized Quaternions

Problem: Starting with non-unit quaternions or failing to normalize results.
Solution: Always normalize input quaternions and the result.

Pitfall 2: Ignoring the Double-Cover Property

Problem: Not checking if the dot product is negative, leading to the long path interpolation.
Solution: Always negate one quaternion if dot product < 0.

Pitfall 3: Division by Zero

Problem: When quaternions are identical or opposite, $\sin(\Omega) = 0$.
Solution: Fall back to normalized linear interpolation in these cases.

Pitfall 4: Incorrect Interpolation Parameter

Problem: Using t values outside [0,1] or misunderstanding what t represents.
Solution: Ensure t is between 0 (start rotation) and 1 (end rotation).

Conclusion

To properly calculate an intermediate quaternion between two rotations:

1. Use SLERP for mathematically correct interpolation with constant angular velocity
2. Handle the double-cover property by checking the dot product and choosing the shortest path
3. Normalize results to ensure valid rotation quaternions
4. Consider edge cases when quaternions are nearly identical or opposite
5. Choose the right method for your specific use case - SLERP for quality, NLERP for performance

The key difference between your previous attempts and the correct SLERP approach is that SLERP respects the spherical geometry of quaternion space, ensuring that interpolated rotations follow the shortest path at constant angular velocity. This produces the smooth, natural-looking motion you’re likely seeking for your application.