Advanced 3D Math Tutorial 🧮

Welcome to the advanced 3D math tutorial! This guide dives into the mathematical foundations critical for 3D graphics, game development, and computational geometry. Let's explore key concepts together.

1. Vector Mathematics 📈

Vectors are essential in 3D space. Here's what to master:

• Vector Addition/Subtraction: Use <center><img src="https://cloud-image.ullrai.com/q/vector_addition/" alt="vector_addition"/></center> to visualize operations.
• Dot Product: Calculates the angle between vectors. Formula: a · b = |a||b|cosθ
• Cross Product: Produces a perpendicular vector. Application: normal vector calculation

2. Coordinate Systems 🌍

Understanding different systems is crucial:

• Cartesian Coordinates: Standard 3D space with x, y, z axes
• Spherical Coordinates: Useful for directional calculations
• Cylindrical Coordinates: Simplifies radial symmetry problems

Use <center><img src="https://cloud-image.ullrai.com/q/coordinate_system/" alt="coordinate_system"/></center> to compare system transformations.

3. Matrix Transformations 🔄

Matrices enable complex transformations:

• Translation: Move objects in space
• Rotation: Change orientation using rotation matrices
• Scaling: Adjust size proportionally

Example:

# 3D rotation matrix around Y-axis
rotation_matrix = [
    [cosθ, 0, sinθ],
    [0, 1, 0],
    [-sinθ, 0, cosθ]
]

Visualize with <center><img src="https://cloud-image.ullrai.com/q/matrix_transformations/" alt="matrix_transformations"/></center>.

4. Applications in Real-World Scenarios 🌐

3D math powers modern technologies:

• Game Development: Physics simulations and character movements
• VR/AR: Spatial mapping and user interaction
• CAD Software: Precision modeling and engineering calculations

Explore more with our [/en/guides/3d-graphics] guide for practical examples!