Final answer:
To prove the rotational invariance of distances on a circle S1, we use the rotational transformation formulas, which show that distances between two points, and from a point to the origin, are unchanged under rotation. These properties support the concept of spherical isometries on a circle, where distances are preserved.
Step-by-step explanation:
Show Rotational Invariance of Distances
To address this question, we need to establish the properties of rotational invariance in a two-dimensional Euclidean space, specifically on the circle S1. When points on a circle are subjected to a spherical isometry, such as a rotation, certain distances remain unchanged, demonstrating the invariance of these distances.
Invariant Distance Between Points under Rotation
Consider any two points P and Q with coordinates (x1, y1) and (x2, y2) respectively on a circle S1. When a rotation of the coordinate system by an angle θ is applied, the new coordinates of these points become (x1', y1') and (x2', y2') respectively, following the relations:
x' = x cos θ + y sin θ
y' = -x sin θ + y cos θ
It can be shown mathematically that the distance between P and Q, which is √[(x2-x1)² + (y2-y1)²], remains the same even after rotation, thereby confirming the invariant nature of distances under rotations of the coordinate system.
Invariant Distance from the Origin under Rotation
Similarly, for any point P on S1, its distance from the origin is also invariant under rotation. The distance is calculated from the origin to P as √(x² + y²), which remains unchanged because the rotation preserves lengths and angles.
Therefore, in terms of the original question where we have P, Q, and a spherical isometry f, let's denote the distances as d(P, f(P)) and d(Q, f(Q)). If f is a rotation by angle θ, then f(P) has the same distance to the origin as P, and f(Q) has the same distance to the origin as Q. If we are given that d(P, f(P)) is 19 times d(Q, f(Q)), we can select points P and Q on S1 such that their respective distances after applying f adhere to this condition, bearing in mind that P must not be a fixed point of the isometry since P≠f(P).