From 3D Points to Circuit States
Applying GA Point Registration to Electrical Circuit Analysis
🎯 Core Idea
Can we treat electrical measurements as points in n-dimensional space and apply geometric algebra point registration techniques for circuit analysis?
📐 The Analogy
3D Point Registration
• Multiple 3D points
• Find optimal transformation
• Align point clouds
→
Circuit Analysis
• Multiple n-D measurements
• Find system relationships
• Align measurement clouds
🔬 Mathematical Framework
Measurement Vector: m⃗(t) = [V₁(t), V₂(t), ..., Vₙ(t), I₁(t), I₂(t), ..., Iₙ(t)]
Point Cloud: M = {m⃗(t₁), m⃗(t₂), ..., m⃗(tₖ)} ∈ ℝ²ⁿ
Each time measurement creates a point in 2n-dimensional space (n voltages + n currents)
🔧 Potential Applications
System Identification
Find circuit parameters by registering measured vs. modeled clouds
Fault Detection
Compare normal vs. abnormal measurement patterns
State Estimation
Determine current system state from partial measurements
Control Design
Optimize control inputs based on desired measurement trajectories
💡 Key Insight
GA's natural handling of high-dimensional rotations and transformations could provide elegant solutions for:
- Nonlinear circuit behavior analysis
- Multi-domain coupling (electrical-thermal-mechanical)
- Time-varying system identification
🚀 Next Steps
Proof of Concept: Simple RC circuit with known parameters
Extension: Nonlinear elements, multi-port networks
Validation: Compare with traditional circuit analysis methods
🤔 Discussion
Does this geometric perspective reveal new insights about circuit behavior that traditional methods miss?