Rotating Calipers in C++

One geometric method for resolving a variety of computational geometry problems, especially those involving convex shapes, is the use of rotating calipers. This method is commonly used to calculate additional convex hull properties, such as the diameter of a convex polygon or the minimal enclosing rectangle.

A computational geometry method called rotating calipers that is used to solve convex form problems, especially when determining distances and maximizing characteristics. It usually entails utilizing methods like Andrew's monotone chain and Graham's scan to determine the convex hull of a set of points in C++. Rotating Calipers uses two pointers to effectively calculate the greatest distance between points on the hull, which may be used to indicate the diameter of the polygon after the convex hull has been constructed. This method improves algorithm performance in situations requiring convex polygons and lowers computing complexity. It is beneficial for a variety of geometric applications.

Features of Rotating Calipers

Several features of Rotating Calipers are as follows:

1. Convex Hull Computation: After determining the convex hull of a collection of points, the Rotating Calipers approach can be used. It enables effective investigation of the hull's vertices.
2. Extreme Point Queries: The algorithm's ability to locate a convex polygon's extreme points, such as its furthest points in particular directions, is helpful for a variety of computational geometry problems.
3. Minimum Enclosing Rectangle: You may determine the smallest rectangle that can contain the polygon by turning the calipers around the convex hull. A common name for this is the Minimum Area Rectangle problem.
4. Farthest Pair of Points: In spatial analysis and clustering, the procedure can be used to find the farthest pair of points in a set.
5. Point in Polygon Tests: Using their characteristics, rotating calipers can effectively determine if a point is inside a convex polygon.
6. Angle and Orientation Calculations: The method helps with graphical applications by enabling the computation of the angles and orientations of lines formed by a convex polygon's vertices.
7. Efficiency: For big datasets, the Rotating Calipers approach is efficient because it typically requires O(n) time for each query after taking O(nlogn) time to compute the convex hull.

Advantages of Rotating Calipers:

Several advantages of Rotating Calipers are as follows:

1. Reduced Complexity in 2D Geometric Computations: The algorithm reduces the complexity of tasks that would otherwise call for complex brute-force methods. For example, compared to O(n²) brute-force methods, it is possible to compute the maximum distance between two points in a convex hull more efficiently.
2. Works Well with Convex Hull Algorithms: Common convex hull methods, such as Jarvis March and Graham's scan are seamlessly integrated with rotating calipers. Convex hulls allow for simplified problem-solving because they may be applied directly without requiring major adjustment once they have been calculated.
3. Optimal Time Complexity: Rotating Calipers works in O(n) on the convex hull when paired with algorithms that operate in O(n log n) time, such as Andrew's monotone chain or Graham's scan. For problems requiring convex polygons or groups of points, this makes it extremely effective.

Disadvantages of Rotating Calipers:

Several disadvantages of Rotating Calipers are as follows:

1. Implementation Complexity: The Rotating Calipers method can be theoretically challenging to use, particularly when dealing with edge cases like degenerate polygons (such as, polygons that have collinear points). These situations require careful handling by developers, which may result in more complicated code.
2. Convex Hull Requirement: Only convex shapes can use the procedure. The convex hull of the input points must usually be calculated before applying rotating calipers. Convex hull computation (e.g., Jarvis March or Graham scan) adds coding effort and processing overhead.
3. Numerical Precision Issues: Algorithms in computational geometry that depend on floating-point computations, such as those that deal with slopes or angles, are vulnerable to accuracy problems. Calculating slopes, distances, and angles between points is a common task for rotating calipers. This can result in rounding mistakes, particularly when working with large coordinate values or near-collinear points.

Applications of Rotating Calipers:

Several applications of Rotating Calipers are as follows:

1. Finding the Diameter of a Convex Polygon (Farthest Points):

An algorithm is given a convex polygon and determines the diameter, or the pair of points that are the farthest apart. The distances between antipodal points can be calculated by spinning a pair of calipers around an n-sided polygon in O(n) time.

2. Smallest or Largest Area/Perimeter of a Rectangle Inside a Convex Polygon:

When determining the minimum or maximum area or perimeter of a rectangle that can be inscribed in a convex polygon, the procedure can be applied. The polygon can be "rotated", and the appropriate values can be effectively calculated by measuring the width and height of the rectangle with rotating calipers.

3. Computing the Minimum Bounding Box (Oriented Bounding Box):

The minimum area-bounding box around a group of points can be found using rotating calipers. The smallest rectangle that contains all of the points can be found by turning the calipers around the convex hull of the points.

4. Minimum Distance Between Two Convex Polygons:

Rotating Calipers can be used to quickly determine the smallest Euclidean distance between two convex polygons. To determine the minimum distance between any pair of points from the two polygons, the method rotates one polygon with respect to the other.

5. Width of a Convex Polygon (Smallest Distance Between Two Parallel Supporting Lines):

The smallest separation between two parallel lines touching a convex polygon on opposite sides is known as its breadth. By moving calipers along the edges of the polygon and taking measurements at perpendicular distances, spinning calipers may determine this width.

6. Finding All Antipodal Pair:

Vertex pairs that have a line segment linking them parallel to a convex hull edge are known as antipodal pairs. All of these pairs may be found in linear time by rotating calipers, which can be applied to a variety of geometric optimization problems.

Example:

Let us take an example to illustrate the Rotating Callipers in C++.

Output:

 
The maximum distance (diameter) between points in the convex polygon is: 3.16228   

Explanation:

The Rotating Calipers method for determining the greatest distance (diameter) between points in a convex polygon is implemented in the accompanying C++ code. To represent 2D coordinates, a Point structure is first defined, and then the < operator is overloaded for sorting. The cross function calculates the cross product to assist in figuring out how points are oriented. Andrew's monotone chain approach is used by the convexHull function to construct the convex hull from a set of points. Instead of performing pointless square root calculations, the squaredDistance function determines the squared distance between two places. To efficiently determine the greatest distance, the rotatingCalipers function uses two pointers to cycle around the convex hull. After that, the main function computes the convex hull of a set of points, initializes them, and shows the maximum distance. The use of rotating callipers in computational geometry is efficiently demonstrated by this implementation.