• A0

# 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.

• Straightedge and compass.
• Congruence: two different triangles share a property called congruence.
• Felix Klein's Erlangen program. (circa 1870).

## Descartes

Points are pairs (triples, etc)

• Connects geometry to algebra.
• Computers do algebra, not geometry.
• Distance is defined: more strongly than the compass!

Points require vectors.

• Try to define addition so that algebra works.
1. Consider two points P1, P2. Both are ordered pairs, triples, etc
2. There is something we can add to P1 that gives P2 as a result. P1 + ?? = P2.

Intuitively we know its value!

3. To calculate the value; ?? = P2 - P1.

This is a calculation we know how to make.

4. In fact ?? is something more basic than a point.

We call it a vector.

• Analogies of vectors
1. Differences between addresses in computers. What is special about machine code that uses only differences between addresses?
2. Derivatives. What is integration?
3. Mouse motion.
• Makes geometry challenging as an interpretation of algebra!

## 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

• v1 =1 v2 if and only if x1 = x2 and y1 = y2.

• v1 +1 v2 means (x1, y1) +1 (x2, y2) =1 (x1 + x2, y1 + y2)
• They are closed under addition.
• There is a zero element 01 =1 (0, 0)

Where addition is, multiplication cannot be far behind

• a *1 v =1(a * x, a * y ), called scalar multiplication.
• They are closed under scalar multiplication.
• There is a unit element 1; there is a zero element 0.
• Scalars must be a field.
• Scalar multiplication distributes over addition: a *1 ( v1 +1 v2 ) =1 a *1 v1 +1 a *1 v2.

What are these things? We want the geometrical interpretation.

• Descartes called them points, BUT they are not like points.

How?

• You can move them around, like
• derivatives
• position independent code.

We call them vectors.

Generators

• How few vectors can I use to create all of them?
• The number is the dimension.
• A set having that number, with appropriate weasel words, is a basis.

## Affine Spaces

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

• Each point, P, can be - but is not - identified with the vector that joins it to the origin.
• P =2 O +2 v
• =2 and +2 are related to =1 as follows,
• First, P1 =2 O +2 v1 =2 P2 =2 O +2 v2 if and only if v1 =1 v2.
• Second, ( O +2 v1 ) +2 v2 =2 O +2 ( v1 +1 v2 )
• Define point subtraction
• v =1 P1 -2 P2 if and only if P1 =2 P2 +2 v
• What about point addition? P1 +? P2 =? (O +2 v1) +? (O +2 v2) =? (O +? O) +2 (v1 +1 v2)
• Pretty well impossible to make sense of, so we just disallow it.

Generators

• A basis for the vectors, (v1, v2, ... ), plus one point, O.
• Package them together (v1, v2, ..., O).

#### Affine Combination

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

• (1 - t) * P1 +3 t * P2 =2 (1 - t) * (O +2 v1) +3 t * (O +2 v2) =2 (1 - t ) * O +3 t * O +2 ((1 - t) * v1 +1 t * v2) =2 O +2 ((1 - t) * v1 +1 t * v2)
• This is often called point blending and defines the line passing through two points.

#### 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

• Metric is symmetric by definition
• also non-negative and obeying the triangle inequality.
• Translational invariance means: d(P1, P2) = d(P1 +2 v, P2 +2 v).

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

• d(P1, P2) = ( P1 -2 P2 ) /dot ( P1 -2 P2 ) = | P1 -2 P2 |
• This gives you what Euclid was able to do with a compass, and more.

The dot product also defines ahgles between vectors.

• cos(s) = v1 \dot v2 / (|v1| |v2|),
• s is the angle between v1 and v2.

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