Maximum distance between two points in coordinate plane using Rotating Caliper's Method

Prerequisites:Graham Scan's Convex Hull, Orientation.
Given a set of N points in a coordinates plane, the task is to find the maximum distance between any two points in the given set of planes.

Examples:

Input: n = 4, Points: (0, 3), (3, 0), (0, 0), (1, 1) 
Output: Maximum Distance = 4.24264     
Explanation:
Points having maximum distance between them are (0, 3) and (3, 0) 

Input: n = 5, Points: (4, 0), (0, 2), (-1, -7), (1, 10), (2, -3) 
Output: Maximum Distance = 17.11724 
Explanation:
Points having maximum distance between them are (-1, 7) and (1, 10) 

Naive Approach: The naive idea is to try every possible pair of points from the given set and calculate the distances between each of them and print the maximum distance among all the pairs.

Below is the implementation of the above approach:

17.11724276862369

Time Complexity: O(N²), where N is the total number of points. 
Auxiliary Space: O(1)

Efficient Approach: The above naive approach can be optimized using Rotating Caliper’s Method.

Rotating Calipers is a method for solving a number of problems from the field of computational geometry. It resembles the idea of rotating an adjustable caliper around the outside of a polygon’s convex hull. Originally, this method was invented to compute the diameter of convex polygons. It can also be used to compute the minimum and maximum distance between two convex polygons, the intersection of convex polygons, the maximum distance between two points in a polygon, and many things more. 

To implement the above method we will use the concept of the Convex Hull. Before we begin a further discussion about the optimal approach, we need to know about the following:

• Unsigned Area Of Triangle: If we are given three points P1(x1, y1), P2(x2, y2) and P3(x3, y3) then

• is the signed area of triangle. If the area is positive then three points are in the clockwise order, Else they are in anti-clockwise order and if the area equals to zero then, the points are co-linear. If we take absolute value, then this will represent the unsigned area of the triangle. Here, unsigned basically means area without direction i.e., we just need the relative absolute value of the area. Therefore, we can remove (1/2) from the formula. Hence,

Relative Area of Triangle = abs((x2-x1)*(y3-y2)-(x3-x2)*(y2-y1))

• Antipodal Points: It is those points which are diametrically opposite to each other. But for us, antipodal points are those which are farthest from each other in the convex polygon. If we choose one point from the given set, then this point can only achieve it's maximum distance if and only if we can find it's antipodal point from the given set.

Below are the steps:

1. Two points having maximum distance must lie on the boundary of the convex polygon formed from the given set. Therefore, use Graham Scan's convex hull method to arrange points in counter-clockwise order.
2. We have N points, Initially start from point P1 and include those points from set of given points such that area of region always increases by including any points from the set.
3. Starting from point P1, Choose K = 2 and increment K while area(PN, P1, PK) is increasing and stop before it starts decreasing. Now the current point PK might be the antipodal point for P1. Similarly, find antipodal point for p2 by finding area(P1, P2, PK) and incrementing K form where we previously stopped and so on.
4. Keep updating the maximum distance for each antipodal points occurs in the above steps as the distance between initial point and point by including area was maximum.

Below is the implementation of the above approach:

17.11724276862369

Time Complexity: O(N*log N) 
Auxiliary Space: O(N)