15-06-2012, 12:51 PM
Mathematical Primitives , Introduction to Transformations
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Vector Spaces
A linear combination of vectors results in a new vector:
v = a1v1 + a2v2 + … + anvn
If the only set of scalars such that
a1v1 + a2v2 + … + anvn = 0
is a1 = a2 = … = a3 = 0
then we say the vectors are linearly independent
The dimension of a space is the greatest number of linearly independent vectors possible in a vector set
For a vector space of dimension n, any set of n linearly independent vectors form a basis .
Matrix Transformations
A sequence or composition of linear transformations corresponds to the product of the corresponding matrices
Note: the matrices to the right affect vector first
Note: order of matrices matters!
The identity matrix I has no effect in multiplication
Some (not all) matrices have an inverse.
A linear transformation:
Maps one vector to another
Preserves linear combinations
Thus behavior of linear transformation is completely determined by what it does to a basis
Turns out any linear transform can be represented by a matrix
Frame Buffers
A frame buffer may be thought of as computer memory organized as a two-dimensional array with each (x,y) addressable location corresponding to one pixel.
Bit Planes or Bit Depth is the number of bits corresponding to each pixel.
A typical frame buffer resolution might be
640 x 480 x 8
1280 x 1024 x 8
1280 x 1024 x 24