Notes on Perspective Projection Matrix

Introduction to Matrices in Transformations

• Matrices are used to encode transformations:
  • Translating vectors
  • Rotating vectors
  • Scaling vectors
• Multiplying matrices by vectors changes their coordinates (x, y, z).

Projection Phase

• Beyond transformations, matrices can also represent projection phases.
• Purpose: Project 3D objects from world space to 2D screen space.
• Types of Projection:
  • Perspective Projection: Realistic depth perception.
  • Orthographic Projection: All objects appear the same size regardless of depth.

Goals of the Perspective Projection Matrix

• The perspective projection matrix serves three important functions:
  1. Aspect Ratio Adjustment:
     • Accounts for different screen resolutions and aspect ratios (e.g., 16:10, 16:9).
     • Adjust x and y values based on the height and width of the screen.
  2. Field of View (FOV):
     • Determines the angle of viewing (e.g., 60 degrees, 90 degrees).
     • Affects the scaling of x and y values based on the FOV angle.
  3. Normalization:
     • Maps all values into a range between -1 and 1 (Normalized Device Coordinates).
     • Important for achieving a consistent output across devices.

Aspect Ratio

• Defined as the ratio of height to width.
• For screen space conversion:
  • Multiply original x values by the aspect ratio scaling factor.

Field of View (FOV)

• Describes how much of the scene is visible at any given moment.
• Scaling factor based on the tangent of half the FOV angle:
  • Larger FOV results in smaller object sizes on the screen.
  • Inverse relationship: larger FOV → smaller objects, smaller FOV → larger objects.

Z Value Normalization

• Objects exist in world space with varying depth (z values).
• Need to normalize z values between two key planes:
  • zNear: Closest visible object.
  • zFar: Farthest visible object.
• Mapping z values into a range between 0 and 1:
  • Formula: (zFar / (zFar - zNear))

Constructing the Projection Matrix

• Combine all scaling factors:
  • Multiply x by aspect ratio and FOV scaling factor.
  • Multiply y by FOV scaling factor.
  • Multiply z by the normalization factor (lambda).

Matrix Representation

• The projection matrix can be represented in a tabular form.
• Each entry in the matrix corresponds to specific scaling and normalization factors.
  • e.g., Position (0, 0) = Aspect Ratio
  • e.g., Position (1, 1) = Inverse Tangent of FOV
  • e.g., Position (2, 2) = zFar/(zFar - zNear)

Perspective Divide

• The perspective divide occurs after the projection matrix multiplication.
• Important to save the original z value for later use (e.g., texture mapping).
• Store this value in the w component of the resulting vertex.

Implementation in Code

• Create a function to generate the projection matrix based on FOV, aspect ratio, zNear, and zFar.
• Implement a separate function for multiplying the projection matrix by vertex vectors and performing the perspective divide.

Key Takeaways

• Understanding matrix multiplication is essential for transforming vectors and performing projections.
• The projection matrix efficiently adjusts all necessary factors to simulate realistic 3D rendering on a 2D screen.