Computational Geometry Questions Long
In Computational Geometry, there are several types of geometric approximation problems that are commonly studied. These problems involve finding approximate solutions to geometric optimization or decision problems, where finding an exact solution may be computationally expensive or even impossible. Some of the different types of geometric approximation problems include:
1. Approximation of geometric shapes: This type of problem involves approximating a given geometric shape, such as a polygon or a curve, with a simpler shape that has fewer vertices or a simpler representation. For example, approximating a complex polygon with a simpler polygon that has fewer sides or approximating a smooth curve with a series of line segments.
2. Approximation of geometric distances: In this type of problem, the goal is to find an approximate solution to the distance between two or more geometric objects. For example, approximating the shortest distance between two points in a given set of points or approximating the minimum enclosing circle or rectangle for a set of points.
3. Approximation of geometric optimization problems: These problems involve finding approximate solutions to optimization problems in computational geometry. For example, approximating the maximum or minimum area of a geometric shape subject to certain constraints, such as finding the largest triangle that can be inscribed in a given polygon.
4. Approximation of geometric intersection problems: This type of problem involves finding approximate solutions to intersection problems between geometric objects. For example, approximating the intersection of two polygons or approximating the intersection of a line segment with a curve.
5. Approximation of geometric partitioning problems: These problems involve partitioning a given geometric space into simpler regions or subsets. For example, approximating the partitioning of a set of points into clusters or approximating the partitioning of a polygon into smaller polygons.
6. Approximation of geometric visibility problems: In this type of problem, the goal is to find an approximate solution to the visibility between points or objects in a given geometric space. For example, approximating the visibility polygon of a point in a polygonal environment or approximating the visibility between two points in a terrain.
These are just some of the different types of geometric approximation problems in Computational Geometry. Each problem type has its own set of algorithms and techniques that are used to find approximate solutions efficiently.