So just to clarify, all the rabbits in the upper left are the same exact shape when shown in regular 2D, but they just look larger as we move up the w axis since the x and y are scaled by w in 3 dimensions?
amilich
I think that's the case - if we scale x and y by a, and then move a units up the w axis, x and y would remain the same when projected back into 2D.
gracie
Seeing the rabbits scaled in a 3-D shear space made more sense here than trying to think about w as a shear on the other slide.
mlandis
I still can't intuitively grasp how a shear in 2d-h is a translate in 2d (though I took it by faith and got it working in the assignment). The image doesn't seem to help much either, it looks like it's being scaled as well.
arshbuch
Are all homogenous representations linear?
tulum
So would the rabbit change its size on the w=1 plane when shearing to the far right?
acb575
@mlandis - This link was helpful for me to understand a translation as a shear. It also has an animation that explains things more clearly for me
https://www.teamten.com/lawrence/graphics/homogeneous/
acb575
@arshbuch Homogenous representations allow us to represent non-linear transforms as linear transforms (in the form of matrix operations) in higher dimension space. So by definition, yes.
So just to clarify, all the rabbits in the upper left are the same exact shape when shown in regular 2D, but they just look larger as we move up the w axis since the x and y are scaled by w in 3 dimensions?
I think that's the case - if we scale x and y by a, and then move a units up the w axis, x and y would remain the same when projected back into 2D.
Seeing the rabbits scaled in a 3-D shear space made more sense here than trying to think about w as a shear on the other slide.
I still can't intuitively grasp how a shear in 2d-h is a translate in 2d (though I took it by faith and got it working in the assignment). The image doesn't seem to help much either, it looks like it's being scaled as well.
Are all homogenous representations linear?
So would the rabbit change its size on the w=1 plane when shearing to the far right?
@mlandis - This link was helpful for me to understand a translation as a shear. It also has an animation that explains things more clearly for me https://www.teamten.com/lawrence/graphics/homogeneous/
@arshbuch Homogenous representations allow us to represent non-linear transforms as linear transforms (in the form of matrix operations) in higher dimension space. So by definition, yes.