Final answer:
A limit rotation involves an infinitesimal angle of rotation, leading points to move through arbitrarily small distances. The distance of a point to the origin remains unchanged under rotation due to the Pythagorean theorem, and points farther from the center travel through larger arcs for the same rotation angle.
Step-by-step explanation:
To demonstrate how a limit rotation moves points through arbitrarily small distances, consider a point P at a distance r from the center of rotation. The point P will trace an arc as it rotates around the center. The length of this arc (s) can be calculated using the formula s = r Θ, where Θ represents the angle of rotation in radians. If the angle of rotation approaches zero, the arc length s approaches zero as well, showing that the point can be moved through arbitrarily small distances. In a limit rotation, the angle of rotation becomes infinitesimal.
When discussing the invariance of distance under rotations, the distance of a point P to the origin remains constant, regardless of the rotation of the coordinate system. This distance is given by the Pythagorean theorem as the square root of (x² + y²), which is invariant because rotation preserves distances in Euclidean geometry.
According to Figure 6.4, as points rotate through the same angle, a point that is farther from the center will move through a greater arc length. This demonstrates that while both points undergo a rotation by the same angle, their actual translational movement can be different, reinforcing the idea of rotational kinematics.