3D Coordinate Mapping

In robotic systems and autonomous navigation, depth cameras (like RGB-D sensors) represent the surroundings as a 3D point cloud. A robot arm is situated at the origin $(0,0,0)$, and its software needs to calculate distances between detected objects to prevent collisions.

A depth sensor detects a cylindrical obstacle with a safety envelope modeled as a sphere centered at $C(3, -4, 2)\text{ meters}$ with a radius of $r = 1.5\text{ meters}$. A point obstacle is detected at $P(4, -2.5, 3)\text{ meters}$.

1. Find the distance between the robot arm at the origin $O(0,0,0)$ and the center of the spherical safety envelope $C$.
2. Determine the standard equation of the spherical safety envelope.
3. Calculate the distance from the point obstacle $P$ to the center of the safety envelope $C$. Determine if the point obstacle has breached the safety envelope (i.e., is it inside the sphere?).