13-05-2014, 02:34 PM
2-D Transformations
2-D Transformations.ppt (Size: 478 KB / Downloads: 10)
Homogeneous Coordinates
There are three types of co-ordinate systems
Cartesian Co-ordinate System
Left Handed Cartesian Co-ordinate System( Clockwise)
Right Handed Cartesian Co-ordinate System ( Anti Clockwise)
Polar Co-ordinate System
Homogeneous Co-ordinate System
We can always change from one co-ordinate system to another.
Advantages:
Mathematicians use homogeneous coordinates as they allow scaling factors to be removed from equations.
All transformations can be represented as 3*3 matrices making homogeneity in representation.
Homogeneous representation allows us to use matrix multiplication to calculate transformations extremely efficient!
Entire object transformation reduces to single matrix multiplication operation.
Combined transformation are easier to built and understand.
Matrices
Definition: A matrix is an n X m array of scalars, arranged conceptually in n rows and m columns, where n and m are positive integers. We use A, B, and C to denote matrices.
If n = m, we say the matrix is a square matrix.
We often refer to a matrix with the notation
A = [a(i,j)], where a(i,j) denotes the scalar in the ith row and the jth column
Note that the text uses the typical mathematical notation where the i and j are subscripts. We'll use this alternative form as it is easier to type and it is more familiar to computer scientists.
Transformations
A transformation is a function that maps a point (or vector) into another point (or vector).
An affine transformation is a transformation that maps lines to lines.
Why are affine transformations "nice"?
We can define a polygon using only points and the line segments joining the points.
To move the polygon, if we use affine transformations, we only must map the points defining the polygon as the edges will be mapped to edges!
We can model many objects with polygons---and should--- for the above reason in many cases.
2-D Transformations.ppt (Size: 478 KB / Downloads: 10)
Homogeneous Coordinates
There are three types of co-ordinate systems
Cartesian Co-ordinate System
Left Handed Cartesian Co-ordinate System( Clockwise)
Right Handed Cartesian Co-ordinate System ( Anti Clockwise)
Polar Co-ordinate System
Homogeneous Co-ordinate System
We can always change from one co-ordinate system to another.
Advantages:
Mathematicians use homogeneous coordinates as they allow scaling factors to be removed from equations.
All transformations can be represented as 3*3 matrices making homogeneity in representation.
Homogeneous representation allows us to use matrix multiplication to calculate transformations extremely efficient!
Entire object transformation reduces to single matrix multiplication operation.
Combined transformation are easier to built and understand.
Matrices
Definition: A matrix is an n X m array of scalars, arranged conceptually in n rows and m columns, where n and m are positive integers. We use A, B, and C to denote matrices.
If n = m, we say the matrix is a square matrix.
We often refer to a matrix with the notation
A = [a(i,j)], where a(i,j) denotes the scalar in the ith row and the jth column
Note that the text uses the typical mathematical notation where the i and j are subscripts. We'll use this alternative form as it is easier to type and it is more familiar to computer scientists.
Transformations
A transformation is a function that maps a point (or vector) into another point (or vector).
An affine transformation is a transformation that maps lines to lines.
Why are affine transformations "nice"?
We can define a polygon using only points and the line segments joining the points.
To move the polygon, if we use affine transformations, we only must map the points defining the polygon as the edges will be mapped to edges!
We can model many objects with polygons---and should--- for the above reason in many cases.