Brings great new insights to this topic. For example, I was introduced to rotation using the approach with the addition formula for sin, which is purly algebraic and geometrically not intuitive. But viewing it as a rotation of the basis vectors is genius.
"Rotations and reflections split the orthogonal transformations into two pieces" Exercise for the reader: what do those two pieces look like if we consider 2x2 matrices as points in R^4?
I learned the concept of homogenous coordinates in a mathematics class, so I really tend to think of this as an abstract notion, not as a concrete idea. I wish I had this class first so that I could have been able to grasp a fine understanding of this subject
for the shear transformation, i have a few questions - how can the formula for A be I + uvT , as I is a matrix and uvT is a vector? also, afterwards, how do we get the matrix that is shown for A, isn't uvT 0 in this example?
Very late reply, but uv^T is not a dot product but an n*n matrix, if u and v are column vectors. You can see this by looking at the dimensions: u is an n*1 matrix and v^T is a 1*n matrix so v^Tu is a 1*1 matrix and uv^T is an n*n matrix
1:10:14 - I think the order of the matrix multiplication is the other way around. For the matrix of the lower leg, it must be A_0 A_1 A_2 so the same for left upper leg, it must be A_0 A_1. The transformation matrices are each defined in their local coordinate system. In this case, the matrix multiplication happens from left to right.
5 videos into the playlist, I can already say that prof.Keenan Crane is a legend.
This presentation is really made with love. Thanks so much Keenan
greatest of all time course on CG
I love your replacing of formulaic matrix entries with drawings in "transpose as inverse". Thanks for the constant circling back to intuition.
fabulous !!
Funny how Keenan is better at teaching me Linear Algebra than my freshman professor way back when.
his 'okay' sounds cute!
Brings great new insights to this topic. For example, I was introduced to rotation using the approach with the addition formula for sin, which is purly algebraic and geometrically not intuitive. But viewing it as a rotation of the basis vectors is genius.
Wonderful lectures, thank you Crane!
dude you're awesome
Wow lots of knowledge in under 2 hours. Thank you !
"Rotations and reflections split the orthogonal transformations into two pieces" Exercise for the reader: what do those two pieces look like if we consider 2x2 matrices as points in R^4?
I learned the concept of homogenous coordinates in a mathematics class, so I really tend to think of this as an abstract notion, not as a concrete idea. I wish I had this class first so that I could have been able to grasp a fine understanding of this subject
very nice!!!
Small correction: at 14:35, should be `preserve x3` and not `preserve x1`.
2:11
Just wondering if the translation formula should be f(x)-f(y)=x-y (instead of f(x-y)=x-y, which for y=0 would imply f(x)=x)
In the polar decomposition isn't it assumed that the matrix A is a square matrix?
for the shear transformation, i have a few questions -
how can the formula for A be I + uvT , as I is a matrix and uvT is a vector?
also, afterwards, how do we get the matrix that is shown for A, isn't uvT 0 in this example?
Very late reply, but uv^T is not a dot product but an n*n matrix, if u and v are column vectors. You can see this by looking at the dimensions: u is an n*1 matrix and v^T is a 1*n matrix so v^Tu is a 1*1 matrix and uv^T is an n*n matrix
@@WannesMalfait i figured that out at some point, but thank you!
1:10:14 - I think the order of the matrix multiplication is the other way around. For the matrix of the lower leg, it must be A_0 A_1 A_2 so the same for left upper leg, it must be A_0 A_1. The transformation matrices are each defined in their local coordinate system. In this case, the matrix multiplication happens from left to right.