Mathematical Formulas for Video Game Animation: Complete Reference

Game animation relies on a core mathematical toolkit. Key formulas by category:

1. Interpolation & Blending

• LERP (Linear): P = A + t(B - A), t ∈ [0,1]
• SLERP (Quaternion spherical): q = q₀ (q₀⁻¹q₁)ᵗ – constant angular velocity
• NLERP (Normalized LERP): Faster approx for small angles
• Squad / Cubic spline: Smooth multi-key interpolation

2. Rotations

• Quaternions: q = (w, x, y, z), ||q||=1. Avoid gimbal lock. Composition: q₂q₁
• Rotation matrix (3×3): Orthonormal, det=1. From quaternion: standard conversion
• Euler angles: Intuitive but singularities; use only for UI/authoring

3. Curves & Motion Paths

• Bézier (cubic): B(t) = (1-t)³P₀ + 3(1-t)²t P₁ + 3(1-t)t² P₂ + t³P₃
• B-spline / NURBS: Local control, knot vectors
• Catmull-Rom: Interpolating spline, C¹ continuous

4. Inverse Kinematics (IK)

• CCD (Cyclic Coordinate Descent): Iterative joint rotation toward target
• FABRIK (Forward/Backward Reaching): Position-based, fast, stable
• Analytic (2-bone): Law of cosines → closed-form knee/elbow angle

5. Physics Integration

• Semi-implicit Euler: v₊ = v + aΔt; x₊ = x + v₊Δt (stable for springs)
• Verlet / Position-Based Dynamics: x₊ = 2x - x₋ + aΔt² (constraint-friendly)
• Runge-Kutta 4 (RK4): High accuracy for non-stiff systems

6. Procedural / Secondary Motion

• Spring-damper: F = -kx - cv (cloth, hair, tails)
• Noise (Perlin/Simplex): Organic jitter, wind
• Wave functions: sin(ωt + φ) for breathing, idle sway

7. Skinning

• Linear Blend Skinning (LBS): v' = Σ wᵢ Mᵢ v
• Dual Quaternion Skinning (DQS): Preserves volume, no candy-wrapper

Engines (Unreal, Unity, Godot, Frostbite) wrap these in animation graphs, state machines, and compute shaders.