Shortest Path Intuition Math Example 2
Follow the full solution, then compare it with the other examples linked below.
Example 2
hardWhat is the shortest path between two points in the Euclidean plane, and why? Then explain why the shortest path on the surface of a sphere is a great circle arc.
Solution
- 1 Step 1: In the Euclidean plane, the shortest path between two points is the straight line segment. By the triangle inequality, any other path has length the direct distance.
- 2 Step 2: On a sphere, straight lines in 3D pierce the surface and are not surface paths. Surface paths must stay on the sphere.
- 3 Step 3: Great circles are the intersections of the sphere with planes through the centre. They have the largest possible radius on the sphere and therefore the least curvature.
- 4 Step 4: By analogy with flat geometry, the path of least curvature (geodesic) on a sphere is a great circle arc. Airlines fly great circle routes for shortest international travel.
Answer
Plane: straight line. Sphere: great circle arc (geodesic).
The shortest path between two points is called a geodesic. In flat (Euclidean) geometry it is a straight line; on a sphere it is a great circle arc. These are generalisations of the same concept to different geometries.
About Shortest Path Intuition
The minimum-length route connecting two points, whose form depends on the geometry of the underlying space.
Learn more about Shortest Path Intuition →More Shortest Path Intuition Examples
Example 1 medium
A river runs along the [formula]-axis. Town [formula] is at [formula] and town [formula] is at [form
Example 3 easyWhat is the shortest distance from point [formula] to the origin? Justify that the straight line giv
Example 4 hardAn ant on the surface of a unit cube wants to travel from vertex [formula] to the opposite vertex [f