Lecture Notes: Scalars, Vectors, Dot Product, and Space Transformation

Scalars and Number Line

Scalars: One-dimensional vectors (numbers) on the number line (0, 1, 2, 3... -1, -2…) extending to positive/negative infinity.
Importance: Understanding the foundation for multidimensional concepts.

Number Line Extension: From one-dimensional number line to two dimensions with x and y axes.
Vectors: Fundamental in space transformation, velocities, positions, orientations, etc.

Definition: Operation between two vectors, resulting in a scalar.
Projection: Dot product involves the projection of one vector onto another, which holds if at least one vector is normalized.
Significance: Used to calculate distance, velocity, orientations; Dot(A, B) == Dot(B, A).
Geometric Interpretation: Gives scalar length along projection axis; important for vectors to be normalized.
Application: Useful in deriving many other mathematical formulas.
Signed Distance: Concept where projection could be negative based on vector orientation.

Given A and B (vectors), Dot Product = Projection of B onto A axis, requiring normalization for true distance.
Normalization Impact: Ensures the geometric meaning is accurate.

Assignment 1: Create a radial trigger to check if a point is within a certain radius in space.
Assignment 2: Develop a look trigger to activate when an object is within the player's field of view.

Setup: Use Unity's editor and gizmos to visualize and debug radius.
Visualization: Use functions like Handles.DrawWireDisc to draw the radius in the editor.
Checking Inside/Outside Radius: Calculate distance between object and radius to determine if it's inside or not. Visual cues (colors) indicate state.
Efficiency Tips: Use OnDrawGizmosSelected to limit gizmo rendering for performance.

Goal: Trigger an action when looking directly at an object within a specified threshold.
Setup: Define look direction using player’s transform (e.g., x-axis for 2D, z-axis for 3D).
Calculation: Use dot product to determine if the player's view direction is within the required preciseness of the object.
Threshold: Adjust based on desired preciseness; visualization shown using gizmo colors.
Applications: Useful for gameplay mechanics like interactive objects, AI behavior (e.g., enemies’ field of view).

Concept: Coordinate systems attached to objects. World space is the global reference frame; local space is relative and dependent on object transforms.
Transformations: Always consider the space a coordinate/vectors belong to avoid errors.
Basis Vectors: Vectors (right, up, forward) defining local space orientation.
Assignment 3: Write functions transforming coordinates between local and world space.

Local to World: Use basis vectors, multiply by local coordinates, add world position (formula and practical coding example provided).
World to Local: Subtract world position, project onto basis vectors using dot product.

Matrix Representation: 4x4 matrices encode rotation, position, and scale.
Components: Top-left 3x3 portion represents orientation, 4th column represents position, bottom row is usually fixed (0,0,0,1).
Transformations: Multiplying matrices with vectors transforms coordinates between spaces.
Matrix Inverses: Used for transforming in the opposite direction.
Practical Use: Unity functions like transform.TransformPoint and transform.InverseTransformPoint handle these transformations.

Definition: Operation returning a vector perpendicular to the input vectors.
Usage: Calculating normals, aligning objects, determining orientation (tangents and bitangents for surfaces).
Important Properties: Order of vectors matters, follows left-hand (Unity) or right-hand rule (general physics/math).
Applications: Placement on surfaces (e.g., placing objects with correct orientation on uneven terrain), defining surface normals in mesh data.

Gizmos and Handles: Useful tools for drawing and debugging in Unity Editor.
Adjustable Drawing: Colors, thickness, anti-aliasing options available to enhance legibility.
Practical Tips: Use Handles.DrawLine, Handles.DrawPolyLine, and color coding for clear visual feedback.

Note: Always verify space (local/global) for vectors when performing transformations to avoid logical errors.

Signed Distances: Useful in scenarios where direction impacts distance interpretation.
Handling Edge Cases: Handling parallel vectors or zero-length vectors in dot and cross products.
Practical Coding Implications: Efficiency in using Unity built-in functions versus custom implementations.

Upcoming: Trigonometry, space transformations in more detail, matrices in broader contexts.

Questions and feedback are highly encouraged to clarify concepts and address specific needs.