Welcome to the 3D geometry tutorial! This guide will walk you through the fundamentals of working with three-dimensional shapes, coordinates, and transformations. Whether you're a beginner or looking to deepen your understanding, you'll find practical examples and explanations here.

What is 3D Geometry?

3D geometry deals with the study of three-dimensional shapes and their properties. Unlike 2D geometry, which is flat, 3D geometry adds depth to the mix. It's essential in fields like computer graphics, robotics, and architecture.

Key Concepts:

  • Points: Defined by x, y, z coordinates (📍)
  • Lines: Formed by two points in space (📏)
  • Planes: Spanning infinitely in two directions (📐)
  • Solids: 3D shapes like cubes, spheres, and pyramids (📦)

3D Coordinate System

The 3D coordinate system extends the 2D Cartesian plane by adding a z-axis. This allows us to represent points in space accurately.

Example:

# A point in 3D space
point = (x, y, z)
3D Coordinate System

Transformations in 3D

Transformations are crucial for manipulating 3D objects. Common operations include:

  1. Translation (moving an object)
  2. Rotation (changing orientation)
  3. Scaling (resizing dimensions)
  4. Reflection (mirroring across a plane)

Each transformation can be represented using matrices. For instance, a rotation matrix around the z-axis looks like this:

Rz(θ) = [cosθ, -sinθ, 0;
         sinθ, cosθ, 0;
         0, 0, 1]

Applications of 3D Geometry

3D geometry has countless real-world applications. Here are a few examples:

  • Computer Graphics: Rendering realistic 3D environments 🎮
  • Engineering: Designing complex structures 🏗️
  • Game Development: Creating immersive game worlds 🕹️

For more advanced topics like 3D rendering pipelines or vector calculus, check out our Advanced 3D Geometry Guide.