In case you're wondering what the bottom row represents, it represents the w coordinate because we're in the homogenous coordinate system. So the u and v columns are vectors because their w coordinate is 0, and the o column is a point because it's w coordinate is a 1
jayc809
I can see that there is perhaps a relationship between the number of variables in the transformation matrix and the level of complexity of the transformation per se. For example, for scaling, there are only two variables. For a more complex operation such as rotation and translation, it requires four. And with a affine transformation, we have six unknowns. I am speculating that a projection transformation, where the relationship between all four points of the unit square (e.g. parallel lines, angles, etc.) can all be changed, would require 8 unknowns and potentially solving for a 4x4 transformation matrix.
mahumkhan1
As asked by a student in class, coordinate system transformations are essentially changes of basis. It is important to note that to bridge the difference there needs to be an additional column that includes the coordinates of the origin as the transformations happen relative to the origin and that is an important piece of information when doing transformations of matrices. So, this is not a necessarily new concept and is still a powerful tool when it comes to scaling transformations.
ninjab3381
I think it should have been clarified in the lecture that u and v are actually unit vectors so one can imagine that if you act on the point
represented by [1 0 1]^+, it will take it to to [ux uy 0] and then shift it by the origin [ox oy 1] to get the point at the "end of u" in the diagram.
NothernSJTU
I agree with the above statement that the transformation of the coordinate system is equivalent to the transformation of the space basis, which is also the difference between it and the direct transformation.
In case you're wondering what the bottom row represents, it represents the w coordinate because we're in the homogenous coordinate system. So the u and v columns are vectors because their w coordinate is 0, and the o column is a point because it's w coordinate is a 1
I can see that there is perhaps a relationship between the number of variables in the transformation matrix and the level of complexity of the transformation per se. For example, for scaling, there are only two variables. For a more complex operation such as rotation and translation, it requires four. And with a affine transformation, we have six unknowns. I am speculating that a projection transformation, where the relationship between all four points of the unit square (e.g. parallel lines, angles, etc.) can all be changed, would require 8 unknowns and potentially solving for a 4x4 transformation matrix.
As asked by a student in class, coordinate system transformations are essentially changes of basis. It is important to note that to bridge the difference there needs to be an additional column that includes the coordinates of the origin as the transformations happen relative to the origin and that is an important piece of information when doing transformations of matrices. So, this is not a necessarily new concept and is still a powerful tool when it comes to scaling transformations.
I think it should have been clarified in the lecture that u and v are actually unit vectors so one can imagine that if you act on the point represented by [1 0 1]^+, it will take it to to [ux uy 0] and then shift it by the origin [ox oy 1] to get the point at the "end of u" in the diagram.
I agree with the above statement that the transformation of the coordinate system is equivalent to the transformation of the space basis, which is also the difference between it and the direct transformation.