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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

Translation

Scaling

Rotation

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

sl@cs.brown.edu