Definition

1. Noun:
   • The shortest path between two points on a curved surface: A geodesic line is the generalization of a straight line to a curved space. It represents the path of shortest distance between two points on a mathematically defined surface, such as a sphere or a saddle shape.

Usage Examples

• Noun:
  • On a sphere, a geodesic line is an arc of a great circle, like the path an airplane flies for long distances.
  • In the theory of relativity, the path of a freely moving object in curved spacetime is a geodesic line.

Advanced Usage

• "Geodesic" as an adjective: The term "geodesic" is often used adjectivally to describe structures or paths that follow this principle of shortest distance.
  • The geodesic dome is a structure based on a network of geodesic lines.
• In differential geometry: The geodesic line is defined by a specific differential equation that ensures it has zero tangential acceleration, meaning it is as "straight" as possible on the curved surface.

Variants and Related Words

• Geodesic (noun/adjective): Often used interchangeably with "geodesic line." As a noun, it refers to the shortest line. As an adjective, it describes anything related to such lines.
  • The satellite's orbit was calculated as a geodesic.
• Geodesy (noun): The scientific discipline concerned with measuring and understanding the Earth's geometric shape, which heavily utilizes the concept of geodesic lines.
• Great-circle arc: A specific example of a geodesic line on a sphere.

Synonyms

• Shortest path: The most direct route between two points on a surface.
• Minimum-distance curve: A curve that minimizes the distance between its endpoints on a given surface.

Related Phrases

• Geodesic distance: The length measured along a geodesic line between two points.
  • The geodesic distance between two cities determines the flight path.
• Geodesic equation: The mathematical equation whose solutions are geodesic lines.
  • Solving the geodesic equation for a given surface can be complex.

Related Concepts

• In a plane: On a flat plane, the geodesic line is simply a straight line segment.
• On a cylinder: A geodesic line can be a straight line, a circle, or a helix, depending on the endpoints.

Noun

1. (mathematics) the shortest line between two points on a mathematically defined surface (as a straight line on a plane or an arc of a great circle on a sphere)