This got me thinking about the other way around. For example, if you have a 2D object lying on the x-y plane in 3D space, and then rotate this object about the x-axis, the projection of result back onto the x-y plane is a non-uniform scaling of the original object. This can only really shrink the object, but the inverse (an enlargening) also makes some sense: start with a 2D object not on the x-y plane and then rotate along the x-axis anyways, comparing projections.

Staffjamesfobrien

Think about how homogenous coordinates allow you to express translation in N dimensions as a matrix in N+1 dimensions, but we know from SVD that any N+1 dimensional matrix can be expressed as rotation-scale-rotation.

The implication here is that translation in N-dimensional space can be expressed as rotation and scaling in N+1 dimensional space!

This got me thinking about the other way around. For example, if you have a 2D object lying on the x-y plane in 3D space, and then rotate this object about the x-axis, the projection of result back onto the x-y plane is a non-uniform scaling of the original object. This can only really shrink the object, but the inverse (an enlargening) also makes some sense: start with a 2D object not on the x-y plane and then rotate along the x-axis anyways, comparing projections.

Think about how homogenous coordinates allow you to express translation in N dimensions as a matrix in N+1 dimensions, but we know from SVD that any N+1 dimensional matrix can be expressed as rotation-scale-rotation.

The implication here is that translation in N-dimensional space can be expressed as rotation and scaling in N+1 dimensional space!