If we scale x, y, w uniformly, nothing happens, as in homogeneous coordinates [x, y, w] and [ax, ay, aw] both map to [x/w, y/w] in 2D.
Norah_Knive
Just look at how scaling is done in 2DH co-ordinates in slide 33. The scaling matrix is chosen in such a way that it does not alter the third co-ordinate of the 2DH co-ordinate and it turns out to be 1. i.e. [x y 1] maps to [ax ay 1]
Senbei
This slide is helpful for understanding "translate in 2D is shear in 2D-H"
JavaSwing
Does using 2D-H (or 3D-H) also make a perspective projection a linear map?
Max
It does! That's another big benefit of using homogeneous coordinates.
However, it's still only linear in homogeneous coordinates - you have to divide xyz by w (a non-linear operation) after you're done transforming.
weiyuc
It is really a good visualization about the connection between translation and shear.
helloCrystal
I still feel like it is kind of hard to visualizing the 2D translate <-> shear in 2D-h. I can see that we have a large stack of bunnies cut out of paper with increasing size, but how should I think about the relationship?
Andy
The rabbit example is really helpful for understanding the concept!
borderwing
Homogeneous matrix is really useful when dealing with affine transformations.
JimL
Wonder if there is any case where we will use the ‘w’ of (x,y,w). For example, when w=2, the final result is scaled by 0.5(both x and y).
VegitableChicken
Love the way it visualizes the transformation of homogeneous matrix... Makes it easier to understand the concept.
If we scale x, y, w uniformly, nothing happens, as in homogeneous coordinates [x, y, w] and [ax, ay, aw] both map to [x/w, y/w] in 2D.
Just look at how scaling is done in 2DH co-ordinates in slide 33. The scaling matrix is chosen in such a way that it does not alter the third co-ordinate of the 2DH co-ordinate and it turns out to be 1. i.e. [x y 1] maps to [ax ay 1]
This slide is helpful for understanding "translate in 2D is shear in 2D-H"
Does using 2D-H (or 3D-H) also make a perspective projection a linear map?
It does! That's another big benefit of using homogeneous coordinates.
However, it's still only linear in homogeneous coordinates - you have to divide xyz by w (a non-linear operation) after you're done transforming.
It is really a good visualization about the connection between translation and shear.
I still feel like it is kind of hard to visualizing the 2D translate <-> shear in 2D-h. I can see that we have a large stack of bunnies cut out of paper with increasing size, but how should I think about the relationship?
The rabbit example is really helpful for understanding the concept!
Homogeneous matrix is really useful when dealing with affine transformations.
Wonder if there is any case where we will use the ‘w’ of (x,y,w). For example, when w=2, the final result is scaled by 0.5(both x and y).
Love the way it visualizes the transformation of homogeneous matrix... Makes it easier to understand the concept.