CS488 - Introduction to Computer Graphics - Lecture 4
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.
- 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 morphed into vectors.
- By trying to define addition so that algebra works.
- 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.
We can add them together
- 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
- This is point/vector addition.
- =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
Generators
- A basis for the vectors, (v1, v2, ... ), plus one point, O.
- Package them together (v1, v2, ..., O).
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.
- d(P1, P2) = d(P1 +2 v, P2 +2 v).
The familiar dot product (defined on vectors, Why?) is your baby.
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