Bivector Forum

bivector-forum

Curated knowledge from the Geometric Algebra community

Version: 1.0.0
Trit: 0 (ERGODIC - knowledge transport)

Overview

bivector.net is the primary community hub for Geometric Algebra practitioners. This skill distills key insights, common pitfalls, and best practices.

Community Resources

Key Contributors

• Steven De Keninck (@enkimute) - ganja.js author
• Leo Dorst - "Geometric Algebra for Computer Science"
• Charles Gunn - PGA for graphics
• Eric Lengyel - Game engine applications

Common Patterns

1. Choosing the Right Algebra

2. Dual vs Homogeneous

PGA (Projective): Points are bivectors, planes are vectors

• Pro: Unified treatment of points at infinity
• Con: Counterintuitive grading

CGA (Conformal): Points are null vectors

• Pro: Circles/spheres as blades
• Con: Higher dimensional, slower

3. Sign Conventions

Different sources use different conventions:

4. Normalization

Always normalize motors/rotors after composition:

// WRONG: Drift over time
for (let i = 0; i < 1000; i++) {
  motor = motor.Mul(smallRotor);  // Accumulates error
}

// RIGHT: Normalize periodically
for (let i = 0; i < 1000; i++) {
  motor = motor.Mul(smallRotor);
  if (i % 100 === 0) motor = motor.Normalized;
}

5. Avoiding Common Pitfalls

Pitfall: Wrong Product

// WRONG: Using geometric product for join
var line = pointA * pointB;  // Gives rotor, not line!

// RIGHT: Use regressive product
var line = pointA & pointB;  // Vee = join

Pitfall: Forgetting Reverse

// WRONG: Missing reverse in sandwich
var transformed = motor * point;  // Only half the sandwich!

// RIGHT: Full sandwich
var transformed = motor * point * ~motor;
// Or use >>> operator
var transformed = motor >>> point;

Pitfall: Grade Extraction

// WRONG: Assuming grade by position
var scalar = result[0];  // May not be scalar!

// RIGHT: Use grade projection
var scalar = result.Grade(0);
var bivector = result.Grade(2);

Frequently Asked Questions

Q: PGA or CGA?

A: Start with PGA. It's simpler, faster, and covers most needs. Use CGA only when you need circles/spheres as first-class objects.

Q: Why are my transformations wrong?

A: Check:

1. Motor normalization
2. Sandwich product (both sides)
3. Transformation order (right-to-left)
4. Sign conventions

Q: How to debug multivector issues?

A:

1. Use console.log(mv.toString()) in ganja.js
2. Check grade with mv.Grade(k).Length
3. Verify normalization with mv.Mul(mv.Reverse).s

Q: Performance tips?

A:

1. Cache computed motors
2. Use specialized rotors/translators
3. Avoid repeated normalization
4. Pre-compute inverse when possible

GAlculator Tips

The online GAlculator is great for:

• Quick experiments
• Verifying formulas
• Learning GA intuitively

// Example session
> 1e1 * 1e2
= 1e12

> (1e1 + 1e2).Normalized
= 0.707e1 + 0.707e2

> (1 + 0.5e12).Exp()
= 0.878 + 0.479e12

Integration with Skills Ecosystem

From bivector.net → Skills

GF(3) Conservation

Community discussions often naturally balance:

• Questions (-1): Seeking understanding
• Answers (0): Sharing knowledge
• Examples (+1): Generating new applications

Discord Bot Commands

!algebra PGA3D     - Show algebra info
!product a b       - Compute geometric product
!explain wedge     - Explain wedge product
!example rotation  - Show rotation example

GF(3) Triads

bivector-forum (0) ⊗ ganja-wedge-game (+1) ⊗ conformal-ga (-1) = 0 ✓
bivector-forum (0) ⊗ ga-codegen (+1) ⊗ sheaf-cohomology (-1) = 0 ✓

Commands

# Open forum
open https://bivector.net/

# Open Discord
open https://discord.gg/vGY6pPk

# Open GAlculator
open https://enkimute.github.io/ganja.js/examples/galculator.html

References

• bivector.net
• GA primer (Dorst)
• PGA cheat sheet
• ganja.js documentation

Autopoietic Marginalia

The interaction IS the skill improving itself.

Every use of this skill is an opportunity for worlding:

• MEMORY (-1): Record what was learned
• REMEMBERING (0): Connect patterns to other skills
• WORLDING (+1): Evolve the skill based on use

Add Interaction Exemplars here as the skill is used.