CS488 - Introduction to Computer Graphics - Lecture 3

Comments and Questions


Geometry

The course notes give you the axioms and theorems. In class, I will concentrate on ideas and algorithms.

Euclid and the Greeks

Of course, it's completely unfair to give the Greeks all the credit.


Descartes

Points are pairs (triples, etc)

Points require vectors.


Vector Spaces, aka Linear Algebra.

We have some things that look like Descartes' points, though we don't know exactly what they are.

We can tell when they are equal

We can add them together

Where addition is, multiplication cannot be far behind

What are these things? We want the geometrical interpretation.

How?

We call them vectors.

Generators


Affine Spaces

Intuition is that when I add an origin, O, vectors define points.

Generators

Affine Combination

There is one form of point addition which makes obvious sense, called affine combination.

Affine Transformations

One point, Two frames

Two points, One frame


Euclidean Space

An affine space plus a metric, d(P1, P2), that has translational invariance

The familiar dot product (defined on vectors, Why?) is your baby.

The dot product also defines ahgles between vectors.

This is part of the `more' we got than compass.


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