I guess whenever we want to expression some non-linear transformations, we need to add one more dimension?
A-star
Does homogeneous coordinates have anything to do with quaternions (https://en.wikipedia.org/wiki/Quaternion) ?
hesper
I guess this is probably because of the relationship between affinely independent and linearly independent from linear algebra?
Given points $x_1, ...,x_k$ in $\mathbb{R}^d$, define
$y_j =
\begin{bmatrix}x_{j}\
1\end{bmatrix}$ in $\mathbb{R}^{d+1}
$ (ahhh i am experiencing some technical sadness that i cannot put the 1 under $x_j$ ) in $\mathbb{R}^{d+1}$. Then $x_1, ...,x_k$ are affinely independent if and only if $y_1, ...,y_k$ are linearly independent.
motoole2
@xudongw This trick of adding one dimension to our vectors works for turning a translation operation into a linear one, but this trick does not make any arbitrary non-linear transformation into a linear one.
@A-star Both homogeneous coordinates and quaternions are expressed using four numbers, but they both represent completely different concepts. Moreover, the numerical operations one performs on homogeneous vectors and quaternions are different. More on this in today's lecture!
I guess whenever we want to expression some non-linear transformations, we need to add one more dimension?
Does homogeneous coordinates have anything to do with quaternions (https://en.wikipedia.org/wiki/Quaternion) ?
I guess this is probably because of the relationship between affinely independent and linearly independent from linear algebra? Given points $x_1, ...,x_k$ in $\mathbb{R}^d$, define $y_j = \begin{bmatrix}x_{j}\ 1\end{bmatrix}$ in $\mathbb{R}^{d+1} $ (ahhh i am experiencing some technical sadness that i cannot put the 1 under $x_j$ ) in $\mathbb{R}^{d+1}$. Then $x_1, ...,x_k$ are affinely independent if and only if $y_1, ...,y_k$ are linearly independent.
@xudongw This trick of adding one dimension to our vectors works for turning a translation operation into a linear one, but this trick does not make any arbitrary non-linear transformation into a linear one.
@A-star Both homogeneous coordinates and quaternions are expressed using four numbers, but they both represent completely different concepts. Moreover, the numerical operations one performs on homogeneous vectors and quaternions are different. More on this in today's lecture!