I am just learning about twists to represent 3D velocities (e.g. of a robot's end-effector), and I have two questions:
1) Wikipedia defines a twist as "an angular velocity around an axis and a linear velocity along this axis". To represent a twist mathematically, this requires defining a point in 3D space (3 coordinates), the direction of the axis passing through this point (2 coordinates), and the ratio of the linear and angular velocities. This is 6 numbers in total. However, it seems that you could also represent this same 3D velocity just by defining linear velocity about the axes (3 coordinates), and a rotational velocity about the three axes (3 coordinates), so that the linear and rotational velocities are defined independently. This is also 6 numbers in total. So why do we define 3D velocities as twists, rather than using my version? Is it just that there are some nice mathematical properties of twists that my version does not have? Or is it that my version is actually fundamentally wrong in some way?
2) Let's say I have an object with positive x- and y- linear velocity, but zero z- linear velocity. The object is also spinning around its z-axis, but there is no rotation about the x- and y- axes. You can think of this as a ball sliding across the floor, whilst spinning about its vertical axis. I am struggling to understand how it is possible to represent this 3D velocity using "an angular velocity around an axis and a linear velocity along this axis". In this example, it seems clear to me that the linear velocity is acting orthogonally to the axis of rotation of the angular velocity. The ball is rotating about its z-axis, and so it doesn't seem possible to also define the linear velocity in terms of this z-axis, because the linear velocity only has components in the x- and y- axes. What am I missing?