When it says rotate "around" some axis, does it mean that all points of the object that lie on that axis will remain in their place and the rest of the points will rotate in a circular motion around that axis ?
@BellaJ I think you are thinking it in the right way. But if we have a condition when the axis of rotation does not pass through the body then all the points of the body will move in a circular motion about that axis.
@BellaJ @Asterix Does the axis always pass through the origin?
The axis of rotation should always pass through the origin. Otherwise the rotation wouldn't be a linear transformation anymore (e.g. f(0) != 0). To rotate around an arbitrary axis, I believe you'd need to sandwich the rotation with translations on both ends.
It's interesting how in an n-dimensional space we seem to have transformations characterized by a single dimension (translation and scaling along a single axis), two dimensions (shearing, defined by the axis being sheared and the one linearly combined with it), an arbitrary number of dimensions (reflection through any lower-dimensional object), and n-2 dimensions (rotation around such an object). It's weird to think about 4D rotation about a 2D axis (what does that look like?), but it seems to hold based on how we think about rotation in 2D and 3D.
Are there any other common transformations categorized by, for example, 3 dimensions, or n-1 dimensions?
So to properly rotate something, we would first have to translate the points to the origin, perform the rotation, and then translate them back out?