Geometric Transformations

Table of Contents

How Are Transformations Used in Computer Graphics?
Problem: How to move an object
Cartesian Coordinate Spaces
Moving an Object in a Coordinate System
Vectors & Vector Space
2D Object Definition
2D to 3D Object Definition
Moving Objects with Vectors
Adding Vectors Visually
Scalar Multiplication
Algebraic Properties of Vectors
The Dot Product
Rule for Dot Product
Finding the Length of a Vector
Finding the Angle Between Two Vectors
More Uses of the Dot Product
Basic Transformations of the Plane
Sets of Linear Equations and Matrices
Types of Transformations
Some Important Matrices
Two Other Matrices
Matrix-Vector Multiplication
Finding a Matrix
Combining Transformations
Homogenous Coordinates
What is ?
Homogenous Coordinate Transformations
Composition of 2D Transforms
Scale, Rotate, and Translate
Commutative and Non-Commutative Combinations of Transformations in 2D
3D Basic Transformations (right-handed coordinate system)
Homogeneous Coordinates
Finding a Matrix--Solutions
Addendum--Matrix Notation
But, There's a Problem...
Solution to Notational Problem
Homogeneous Coordinate Example
Matrix Notation and Composition


Geometric Transformations - 08 MAY 95

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