Since we can express translation, a non-linear transformation, in a higher dimension space, I am curious if this method can be used for other non-linear transformation. Is translation the only one transformation that can be converted to linear?
keenan
@yongchi1 Great question. There are lots of transformations that can be represented in homogeneous coordinates; see my comments on the end of this slide.
ericchan
Based on your wording, I'm assuming there are some non-linear transformations that can not be represented in homogenous coordinates. I'm curious which transformtions those might be, and what causes them to be unable to be represented in homogenous coordinates. In addition, is there another possible way to represent them linearly then? Does representing 2d points with 4 values do anything useful?
keenan
@ericchan Sure, for instance the transformation $x \mapsto x^2 sin(x)$ is nonlinear, and cannot be represented in homogeneous coordinates.
I think, though, that you're more interested in understanding the space of all nonlinear transformations that can be represented linearly. For instance, by going up in dimension, applying a linear transformation, then going back down in dimension (where the projection is nonlinear, as with homogeneous coordinates). One terrific place to look is at Lie sphere geometry which represents a large family of transformations (including the Euclidean ones we studied, plus Moebius transformations and Laguerre transformations) as linear transformations in a higher-dimensional vector space. Here one can talk easily about points, spheres, and planes, and relationships between them. More broadly, the question you're asking is at the root of something called representation theory which seeks to represent different kinds of transformations (possibly "nonlinear") via linear transformations/matrices.
Since we can express translation, a non-linear transformation, in a higher dimension space, I am curious if this method can be used for other non-linear transformation. Is translation the only one transformation that can be converted to linear?
@yongchi1 Great question. There are lots of transformations that can be represented in homogeneous coordinates; see my comments on the end of this slide.
Based on your wording, I'm assuming there are some non-linear transformations that can not be represented in homogenous coordinates. I'm curious which transformtions those might be, and what causes them to be unable to be represented in homogenous coordinates. In addition, is there another possible way to represent them linearly then? Does representing 2d points with 4 values do anything useful?
@ericchan Sure, for instance the transformation $x \mapsto x^2 sin(x)$ is nonlinear, and cannot be represented in homogeneous coordinates.
I think, though, that you're more interested in understanding the space of all nonlinear transformations that can be represented linearly. For instance, by going up in dimension, applying a linear transformation, then going back down in dimension (where the projection is nonlinear, as with homogeneous coordinates). One terrific place to look is at Lie sphere geometry which represents a large family of transformations (including the Euclidean ones we studied, plus Moebius transformations and Laguerre transformations) as linear transformations in a higher-dimensional vector space. Here one can talk easily about points, spheres, and planes, and relationships between them. More broadly, the question you're asking is at the root of something called representation theory which seeks to represent different kinds of transformations (possibly "nonlinear") via linear transformations/matrices.