Three Dimensional Transformations in Graphics

Sep 26, 2024

Lecture Notes on Three Dimensional Transformations in Computer Graphics

Introduction

  • Discussion on three-dimensional transformations in computer graphics.
  • Previously covered: rotation, reflection, translation, scale, shear, orthographic and perspective projections.

Recap of Previous Discussions

  • Transformations Discussed:
    • Rotation
    • Reflection
    • Translation
    • Scale
    • Shear
  • Projective Transformations:
    • Orthographic Projections
    • Perspective Projections

Perspective Transformation

  • Focus on generalized formulation of perspective transformations.
  • Key Components:
    • Projection Plane (PP)
    • Center of Projection (COP)
    • Projection Reference Point (PRP)

Coordinate System

  • Right-handed system:
    • X-axis: Vertical
    • Z-axis: Moves to the right
    • Y-axis: Points towards the viewer
  • Projection plane is perpendicular to Z-axis and at a distance Zp from the origin.

Problem Statement

  • Given:
    • Coordinates of point P (x, y, z)
    • Zp, dx, dy, dz (direction cosines), q
  • Find:
    • Coordinates of the projected point P' (xp, yp).

Parametric Equation of the Line

  • Equation:
    • L = COP + T(P - COP)
    • T varies between 0 and 1.
    • 0: COP; 1: P.

Deriving the Projection Point

  • Intersection of line L with projection plane gives P'.
  • Equations for P' Coordinates:
    1. x' = Qdx + (x - Qdx)T
    2. y' = Qdy + (y - Qdy)T
    3. z' = Zp
  • Substitute z' = Zp to solve for T.

Generalized Transformation Matrix

  • M_general = Transformation matrix for perspective projection.
  • P' = M * P, P being the original 3D point coordinates.

Special Cases of Perspective Projection

  • Orthogonal Matrix:
    • Zp at origin, Q at infinity.
    • Direction vector: (0, 0, -1).
  • Perspective Projection Matrix:
    • Zp = Q = d, COP at origin.
    • Direction vector remains (0, 0, -1).

Parallel Projection

  • Distance from COP to projection plane = infinity.
  • Direction of projection is specified by direction cosines (dx, dy, dz).

Types of Parallel Projections

  1. Orthographic Projection:
    • Direction of projection normal to the projection plane.
  2. Oblique Projection:
    • Projector rays intersect the projection plane at an angle.

Orthographic Projections Explained

  • Includes:
    • Top view
    • Front elevation
    • Side elevation
  • Examples of visualizing structures using orthographic projections.

Isometric and Axonometric Projections

  • Axonometric Projection:
    • Planes of projection are not normal to the principal axes.
  • Isometric Projection:
    • Normal makes equal angles with all principal axes.

Exercises and Problems

  • Various exercises related to perspective and parallel projection transformations.
  • Emphasis on deriving and understanding transformation matrices.

Suggested Problems

  1. Illustrate vanishing points in three-point perspective projection.
  2. Obtain expressions for rotation about an arbitrary axis and reflection about an arbitrary plane.
  3. Prove rotation can be achieved by combining rotation and translation.
  4. Obtain the scale matrix along an arbitrary direction.

Conclusion

  • Transition to the next lecture focusing on 3D viewing and clipping in computer graphics.
  • Importance of solving exercises for better understanding of concepts.