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Tutorial Examples Overview

This page introduces nblade's tutorial examples to help you quickly master the basic concepts and operations of geometric algebra.

Example List

01_quickstart.py - Quick Start

Difficulty: ⭐ Beginner

Content: - Creating geometric algebra - Basis vectors and vector creation - Three basic products - Introduction to rotation operations

Run: 

python examples/tutorials/01_quickstart.py

Expected Output: 

============================================================
nblade Quick Start - Master Core Features in 5 Minutes
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【1. Creating Geometric Algebra】
Algebra Type: Geometric Algebra G(3, 0, 0) (3D)
Dimension: 3
Signature (p, q, r): (3, 0, 0)

【2. Basis Vectors and Vector Creation】
Basis vector e1 = 1.000000e1
...

Key Concepts: - nblade.Algebra.euclidean(n) - Create n-dimensional Euclidean algebra - alg.vector([...]) - Create vector - *, ^, | - Geometric product, outer product, inner product

02_basic_operations.py - Basic Operations

Difficulty: ⭐⭐ Basic

Content: - Detailed geometric product explanation - Detailed outer product explanation - Detailed inner product explanation - Relationship between the three products - Practical application examples

Run: 

python examples/tutorials/02_basic_operations.py

Key Concepts: - Core formula: ab = a·b + a∧b - Outer product represents the subspace spanned - Inner product represents projection relationship

03_vectors.py - Vector Operations

Difficulty: ⭐⭐ Basic

Content: - Various methods for creating vectors - Vector properties (norm, coefficients, etc.) - Algebraic operations (addition, subtraction, multiplication) - Geometric operations (projection, reflection) - Involution operations - High-dimensional vectors

Run: 

python examples/tutorials/03_vectors.py

Key Concepts: - alg.basis_vectors() - Get all basis vectors - v.norm() - Calculate norm - v.project_to(blade) - Projection - v.reflect_in(n) - Reflection

05_rotations.py - Rotation Tutorial

Difficulty: ⭐⭐⭐ Advanced

Content: - 2D rotation and complex numbers - 3D rotation and rotors - Combining multiple rotations - Rotation interpolation (SLERP) - Relationship between rotors and quaternions - Practical application examples

Run: 

python examples/tutorials/05_rotations.py

Key Concepts: - Rotor: R = exp(-B·θ/2) - Rotation formula: v' = R·v·R† - Combined rotation: R_combined = R2·R1

Learning Path

01_quickstart.py
       ↓
02_basic_operations.py
       ↓
03_vectors.py
       ↓
05_rotations.py
       ↓
   Application Examples

Frequently Asked Questions

Q: Why is the outer product result a bivector?

A: The outer product a ^ b represents the subspace (plane) spanned by two vectors. In geometric algebra, planes are represented by bivectors, not by normal vectors as in traditional vector analysis.

Q: What is the relationship between geometric product and inner/outer product?

A: Geometric product is the fundamental operation, decomposed into inner and outer products: ab = a·b + a∧b. Inner product is the symmetric part (scalar), outer product is the antisymmetric part (bivector).

Q: What is the relationship between rotors and quaternions?

A: Quaternions are the even subalgebra of geometric algebra. Quaternion units i, j, k correspond to bivectors e₂∧e₃, e₃∧e₁, e₁∧e₂.

Next Steps

After completing the tutorial examples, you can continue learning:

Or check the API Reference for more details.