But the question asks for how many points from the orbit are **reached** — meaning intersection points. - Richter Guitar

Understanding the Context

What Is an Orbital Intersection Point?

An orbital intersection point is a location in space where two or more orbiting bodies cross paths. These points arise when multiple trajectories converge at the same point at the same time — a rare geometric coincidence governed by the orbital dynamics of the system.

Counting Intersection Points: The Basics

Key Insights

For two planets or satellites with simple circular and coplanar orbits, no intersections occur because their paths never cross. However, once orbits are non-coplanar or have distinct semi-major axes and eccentricities, intersections (when they happen) depend on the specific orbital configuration.

• Minimum: In many stable systems (like non-intersecting planetary orbits), zero intersection points are reached.
  
• Zero (Typical Case): Two properly designed circular, coplanar orbits with no mutual perturbations usually do not intersect—their paths are parallel in space and never meet.
  
• One or More: If orbital parameters are carefully chosen—such as in spacecraft rendezvous, satellite constellations, or resonant orbital schemes—the number of intersection points increases, sometimes to one or several discrete points.

Factors That Determine Orbital Intersections

1. Orbital Inclination
   Orbits lying in different planes rarely intersect. Intersections require nearly co-planar paths.

Final Thoughts

2. Eccentricity and Apogee/Perigee
   Even slightly different eccentricities cause orbits to “fan out” in space, reducing—or eliminating—intersection likelihood.

3. Orbital Periods and Resonance
   Periodic orbits with commensurate periods (e.g., 2:1 resonance) can cause temporary closeness, but true intersection only occurs if paths converge.

4. Relative Position and Time
   Orbits must approximate exact spatial and temporal overlap at finite time intervals for intersection points to exist.

5. Perturbations and External Forces
   Gravitational nudges from other bodies, atmospheric drag, or solar radiation pressure alter orbits, making precise intersection rare without active control.

Real-World Examples of Orbit Intersections