EXPERIMENTAL MATHEMATICAL WORKBENCHES
The Form Lab
Six tactile, real-time algorithmic simulators designed to explore the mathematics of form—subdivision stencils, implicit distance raymarching, parametric curve tension, microfacet PBR lobes, quad flow topology, and FABRIK inverse kinematics.
WORKBENCH 01 · SUBDIVISION SURFACES
Interactive Algorithmic Workbench
Catmull-Clark Subdivision Stepper
SUBDIVISION LEVEL: 1
Vertices: 26 · Quads: 24
χ = 26 - 48 + 24 = 2
Iteration Depth:Level 1
0 (Control)123 (Smooth)
ALGEBRAIC STENCIL:
v' = (F + 2E + (n - 3)v) / n
v' = (F + 2E + (n - 3)v) / n
WORKBENCH 02 · IMPLICIT DISTANCE FIELDS
Real-Time Implicit Raymarching
SDF Smooth Boolean & Distance Field Chamber
Inigo Quilez smin(a,b,k)
Click & drag gold probe to move secondary sphere
Smooth Blend Radius (k):0.40
WORKBENCH 03 · PARAMETRIC CURVE CALCULUS
Parametric Calculus
Bézier & Rational NURBS Curve Workbench
C² Continuous Polynomial
Drag control points P0, P1, P2, P3
Rational Weight (w₂):1.00
CONVEX HULL THEOREM:
Curve is mathematically guaranteed to stay strictly within the control cage polygon.
Curve is mathematically guaranteed to stay strictly within the control cage polygon.
WORKBENCH 04 · MICROFACET OPTICAL SHADING
Microfacet Optical Workbench
Cook-Torrance & GGX Reflectance Dial
Fresnel F₀ · Smith G₂ · GGX D
Roughness (α):0.35
Metallic vs Dielectric:Dielectric (Plastic)
Incident Light Angle:45°
WORKBENCH 05 · MESH TOPOLOGY & SINGULARITIES
Differential Mesh Syntax
Quad Flow & Pole Singularity Analyzer
Orbital Sphincter Loops
Concentric quad loops surround apertures (eyes, mouth) to enable radial expansion and contraction without diagonal shearing.
POINCARÉ-HOPF INDEX:
Singularities are mathematically unavoidable on closed 2-manifolds with genus g ≠ 1.
Singularities are mathematically unavoidable on closed 2-manifolds with genus g ≠ 1.
WORKBENCH 06 · SKELETAL INVERSE KINEMATICS
Kinematic Armatures
FABRIK Multi-Joint Inverse Kinematics Solver
Forward-Backward Reaching
Click & drag gold target crosshair to solve bone positions
Bone Segments:3 Bones
Bone Segment Length:65px
Solver Convergence Iterations:5 passes
ARISTIDOU & LASENBY (2011):
Replaces expensive trigonometric Jacobian matrix inversions with pure linear geometric point projections.
Replaces expensive trigonometric Jacobian matrix inversions with pure linear geometric point projections.
2026 AEO KNOWLEDGE GRAPH & INQUIRY TREE
Frequently Explored Structural Questions
6 Verified Semantic Answers