The Mechanics of Flesh: Dual Quaternion Skinning, Torsion, and Volume Preservation

1. The Anatomy of Candy-Wrapper Collapse

In Linear Blend Skinning (LBS), each mesh vertex $\mathbf{v}$ is deformed by a weighted sum of bone matrices $\mathbf{M}_j$:

Consider a human wrist twisting by $180^\circ$ along its forearm axis. The skin vertices at the wrist are weighted equally between the forearm bone ($0^\circ$ rotation) and the hand bone ($180^\circ$ rotation): $w_1 = 0.5, w_2 = 0.5$.

When two rotation matrices representing $0^\circ$ and $180^\circ$ rotations are averaged linearly: $$\mathbf{R}_{\text{blend}} = 0.5 \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} + 0.5 \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$

The resulting matrix is **completely singular**: its determinant is zero! The 3D cross-sectional volume of the character's wrist collapses to a single 1D line. This geometric disaster is known throughout the animation industry as the **candy-wrapper artifact**.

2. The Dual Quaternion Algebraic Solution

William Kingdon Clifford (1873) invented **dual numbers** $\mathbb{D} = \{ a + \epsilon b \mid a, b \in \mathbb{R}, \epsilon^2 = 0, \epsilon \neq 0 \}$.

A **dual quaternion** $\hat{q}$ extends this concept to 4D quaternions: $$\hat{q} = q_0 + \epsilon q_d$$ where $q_0$ is a unit quaternion representing pure 3D rotation, and $q_d = \frac{1}{2} \mathbf{t} q_0$ encodes the 3D translation vector $\mathbf{t}$.

A unit dual quaternion satisfies $\|\hat{q}\| = 1$ (which requires $q_0 \cdot q_0 = 1$ and $q_0 \cdot q_d = 0$). Unit dual quaternions form an exact double cover of the Special Euclidean Group $SE(3)$, the group of all rigid body transformations in 3D space.

3. Dual Quaternion Linear Blending (DLB)

In Dual Quaternion Skinning (Kavan et al. 2007), instead of blending non-orthogonal matrices, the GPU vertex shader linearly interpolates the dual quaternions of the influencing bones and normalizes the result:

Because normalization projects the interpolated result directly back onto the unit dual quaternion manifold, the blended transformation is **strictly guaranteed to be a pure rigid body rotation and translation**. The scale determinant is always exactly 1.0. The wrist twists gracefully, preserving total anatomical volume without a single artifact.