It is covered later that a linear transformation is a matrix. However, the examples provided are limited to rotation and translation; of which all the points within an original image move with the same transform. What does a shear transform do differently such that it "shears" (i.e. how does this effect occur while still being a linear transformation)?
aravmisra
@boomaa23 I found this series (I've linked to the topic on shearing, but there are articles on the affect of rotation/translation/etc on graphics) really helpful: https://www.mauriciopoppe.com/notes/computer-graphics/transformation-matrices/shearing/
I would definitely still recommend looking at the course resources as they are more robust than that link, but I think the brevity of those notes should hopefully address your question while juxtaposing rotation, translation, and shearing.
RishSharma7
From my understanding, it looks like the shear matrix actually allows us to keep the points of the image the same in regard to one axis (say the x-axis), and then diagonalizes the vertical elements of the picture to produce the effect shown on this slide, and that's done via relatively basic matrix multiplication and manipulation.
jerrymby
Is shear a different transform categorically? Because it is not a rigid body transform and I'm wondering if there is any matrix properties that describe a rigid transform.
llejj
In general, we can interpret linear transforms by how they distort space. If you imagine a two dimensional grid, any transformation that keeps grid lines parallel and the origin fixed is a linear transformation. So this includes shearing. Here's a great video on the topic: https://www.youtube.com/watch?v=kYB8IZa5AuE&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=3
mark-nguyen1
The new (x,y) can be represented in two ways:
Horizontal Shear (Shear along the x-axis):
[[1 0][Sx 1]] * [[x y]]
Vertical Shear (Shear along the y-axis):
[[1 Sy][0 1]] * [[x y]]
(assume each [a b] is a column in the matrix since I am not able to format correctly in this text)
It is covered later that a linear transformation is a matrix. However, the examples provided are limited to rotation and translation; of which all the points within an original image move with the same transform. What does a shear transform do differently such that it "shears" (i.e. how does this effect occur while still being a linear transformation)?
@boomaa23 I found this series (I've linked to the topic on shearing, but there are articles on the affect of rotation/translation/etc on graphics) really helpful: https://www.mauriciopoppe.com/notes/computer-graphics/transformation-matrices/shearing/
I would definitely still recommend looking at the course resources as they are more robust than that link, but I think the brevity of those notes should hopefully address your question while juxtaposing rotation, translation, and shearing.
From my understanding, it looks like the shear matrix actually allows us to keep the points of the image the same in regard to one axis (say the x-axis), and then diagonalizes the vertical elements of the picture to produce the effect shown on this slide, and that's done via relatively basic matrix multiplication and manipulation.
Is shear a different transform categorically? Because it is not a rigid body transform and I'm wondering if there is any matrix properties that describe a rigid transform.
In general, we can interpret linear transforms by how they distort space. If you imagine a two dimensional grid, any transformation that keeps grid lines parallel and the origin fixed is a linear transformation. So this includes shearing. Here's a great video on the topic: https://www.youtube.com/watch?v=kYB8IZa5AuE&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=3
The new (x
,y) can be represented in two ways:Horizontal Shear (Shear along the x-axis):
[[1 0][Sx 1]] * [[x y]]
Vertical Shear (Shear along the y-axis):
[[1 Sy][0 1]] * [[x y]]
(assume each [a b] is a column in the matrix since I am not able to format correctly in this text)