The inverse geodesic problem is a concept in differential geometry and related fields, particularly on curved surfaces or manifolds. It involves finding a geodesic—a shortest path between two points—given certain constraints, but in reverse. Instead of starting with two points and finding the geodesic between them (the direct geodesic problem), the inverse problem typically involves determining the initial conditions (e.g., a point and a direction or velocity) or the geometry of the space itself, given information about the geodesic’s endpoints or its properties.
For example, on a sphere, the direct geodesic problem would be: “Given points A and B, find the shortest path (a great circle arc) between them.” The inverse geodesic problem might be: “Given a point A and a distance and direction, find the point B that lies on the geodesic starting from A.” Alternatively, it could involve reconstructing the manifold’s properties (like curvature) based on observed geodesic paths.
In practical terms, this problem appears in fields like navigation, robotics, and general relativity, where understanding paths in curved spaces is crucial. It’s more complex than the direct problem because it often requires solving nonlinear differential equations (like the geodesic equations) backward, and solutions may not be unique depending on the geometry.