Transformation Chamber
How form changes across space and time—recursive subdivision stencils (Catmull-Clark, Loop), Dual Quaternion Skinning for volume preservation, Free-Form Deformation lattices, and FACS facial blendshapes.
Catmull-Clark Subdivision Stepper
v' = (F + 2E + (n - 3)v) / n
Indexed Transformation Operators (8)
Approximating vs. InterpolatingCatmull-Clark Subdivision Surface
The universal subdivision standard in 3D animation. Generalizes bi-cubic B-splines to arbitrary polyhedral control meshes, subdividing every polygon into clean quads while smoothing boundaries.
Face points: f = avg(face vertices). Edge points: e = (v1 + v2 + f1 + f2) / 4. Vertex points: v' = (F + 2E + (n-3)v) / n.Loop Subdivision Scheme
The triangular counterpart to Catmull-Clark. Based on quartic box splines, Loop subdivision splits each triangle into four sub-triangles while smoothing vertices toward a continuous limit surface.
Edge points: (3/8)(v1 + v2) + (1/8)(v3 + v4). Vertex points: (1 - n*β)v + β * sum(neighbors), where β = (1/n)(5/8 - (3/8 + (1/4)cos(2π/n))²).Doo-Sabin Biquadratic Subdivision
Generalizes biquadratic uniform B-splines. Corner-cutting stencil that replaces vertices with new polygons, creating dual polygonal meshes particularly useful for architectural paneling.
New points created for each vertex of every face by weighting all vertices of that face: P' = sum(alpha_i * P_i).Homogeneous 4x4 Affine Transformations
Unifies translation, rotation, scale, and shear into single 4x4 matrix multiplications in projective space ℝℙ³, enabling arbitrary hierarchical nesting of transformation chains.
\mathbf{p}' = \mathbf{M} \cdot \mathbf{p} = \begin{bmatrix} \mathbf{R}_{3\times 3} & \mathbf{t}_{3\times 1} \\ \mathbf{0}_{1\times 3} & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ 1 \end{bmatrix}Linear Blend Skinning (LBS / Matrix Palette)
The classic real-time character skinning method. Deforms mesh vertices as a weighted linear combination of transformed skeletal bone matrices. Susceptible to 'candy-wrapper' collapse under torsion.
\mathbf{v}' = \sum_{j=1}^B w_j \mathbf{M}_j \mathbf{B}_j^{-1} \mathbf{v}, \quad \sum_{j=1}^B w_j = 1Dual Quaternion Skinning (DQS)
Eliminates volume loss and joint collapse during character twisting. Represents rigid transformations as unit dual quaternions, ensuring constant volume preservation during extreme joint rotation.
\hat{q}_{\text{blend}} = \frac{\sum w_j \hat{q}_j}{\|\sum w_j \hat{q}_j\|}, \quad \hat{q} = q_0 + \epsilon q_dFree-Form Deformation (FFD & Harmonic Coordinates)
Embeds 3D meshes inside a deformable 3D grid cage or polyhedral cage. Moving cage control points smoothly warps the interior space using trivariate Bernstein polynomials or Laplace harmonic coordinates.
\mathbf{p}(s,t,u) = \sum_{i=0}^l \sum_{j=0}^m \sum_{k=0}^n B_i^l(s) B_j^m(t) B_k^n(u) \mathbf{P}_{i,j,k}Blendshapes, Morph Targets & Shape Keys
Additive vector offset morphing for facial performance and muscle bulging. Interpolates vertex deltas relative to a neutral base mesh, forming the foundation of modern speech and facial capture.
\mathbf{V}_{\text{final}} = \mathbf{V}_{\text{base}} + \sum_{k=1}^K w_k (\mathbf{V}_k - \mathbf{V}_{\text{base}}), \quad w_k \in [0, 1]