CS488 - Introduction to Computer Graphics - Lecture 5
Geometry - Affine Spaces
Vector Spaces, aka Linear Algebra.
We have some things that look like Descartes' points, though we don't know
exactly what they are.
They are linear and we call them vectors.
- v =1 a1 v1 +1 a2 v2 is a vector
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.
- With respect to a basis (v1,
v2, ...) any vector has
coordinates (a1, a2, ... ), which we write v =1
a1 v1 +1 a2 v2 +1 ...
Transformations
Vectors can be transformed,
- Rigid transformations in 2D
- Unit
- Rotation
- Reflection
More complex in 3D
- Non-rigid transformations in 2D
- Scale
- Skew
These transformations are all linear.
- They can be represented as matrices. That is v' =1 Tv means the same as (a1', a2', ... ) = (a1, a2,
... ) M.
- The transformations can equally well operate on the basis v' =1 Tv
=1 a1 Tv1 +1 a2 Tv2 +1 ...
If you are Felix Klein you show that they are a group, and how they
separate into subgroups. But even though you are not Felix Klein your
response to this formulation is the same.
Affine Spaces
Intuition is that when I add an origin, O, vectors define points.
- Intuitively we can identify each point, P, with the vector that joins it to the
origin.
- Therefore we add one point O, and get as many points as there
are vectors
- 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).
- A general point has coordinates (a1, a2, a3, 1) which mean P =2
a1 v1 +1 a2 v2 +1 a3 v3 +2 O
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