Lecture 3 Notes

Introduction

• Turret Example: Placing and aligning turret objects on uneven surfaces in a game.
• Main Focus: Understanding transformation matrices, vectors, and cross products.

Housekeeping

• Notification pings: Students needed to react to the ping.
• Lecture Naming: Episodes are named by stream index, not by days.

Transformation Between Coordinate Systems

• Context: Methods to transform objects between different spaces in games.
• Elementary Methods: Basic vector transformations are not sufficient for complete transformations.
• Transformation Matrices: Using matrices to store and manipulate transformation data.

Matrices

• Definition: Another way to store numerical data, like vectors but in a more structured form.
• Columns and Rows: Can be used to store vectors and more complex transformations.
• Types: Square matrices (2x2, 3x3, 4x4) and their properties.

Common Matrix Types

• Identity Matrix: For transformations with no change, maintaining components as is.
• Rotation Matrices: For storing rotation transformations.
• Column and Row Vectors: Ways to store one-dimensional data within matrices.
• Transformation Matrices in Games: Often 4x4 matrices, storing position, rotation, and scale.

Practical Use of Matrices in Unity

• Transformations: Unity functions like transform.localToWorldMatrix and transform.worldToLocalMatrix enable transformations.
• Functions: transform.TransformPoint, transform.InverseTransformPoint, and transform.TransformVector manipulate positions and directions according to object transforms.
• 4x4 Matrices in Unity: Handle detailed transformations, including scaling.

Cross Product

• Definition: Mathematical operation that results in a vector perpendicular to two other vectors.
• Anti-Commutative Property: Order affects results; a x b != b x a
• Utility: Generating basis vectors for coordinate systems.
• Visualization: Using right-hand or left-hand rule to determine direction.

Practical Examples

• Coordinate Basis Generation: Example using cross products to generate right and up vectors for a given forward vector.
• Game Development: Adjusting object orientations to maintain alignment with a ground surface using cross products and raycasting.

Application: Placing a Turret on Uneven Terrain

• Problem Statement: Align turret to surface normal while ensuring a forward direction.
• Steps:
  • Raycasting from camera to terrain surface.
  • Calculating normal vector of the hit surface.
  • Using cross product to determine right (x-axis) vector perpendicular to normal and view direction.
  • Calculating forward (z-axis) vector as cross product of right and up vectors.
  • Using quaternion look rotations to set turret transformation.
• Common Issues: Normalizing vectors, avoiding self-hitting in raycasts.

Coordinate Systems in Different Software

• Conventions: Variations in axis orientation and handedness.
• Unity Specifics: Left-handed coordinate system, y-axis as up.

Additional Useful Information

• Left vs Right Handed Rules: Impacts orientation visualization in 3D space.
• Practical Tips: Matrix and vector multiplication, transformation pipelines, and coordinate space mapping.

Assignments

• First Assignment: Rebuild and improve the turret placement system demonstrated in the lecture. Use cross products to correct orientation and handle special cases.
• Matrix-related Problem: Involves transforming positions using local offsets and understanding the full transformation pipeline in Unity.