DSA - Geometric Algorithms

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Geometric algorithms are defined as a procedure used for solving problems related to geometric shapes and their properties. These algorithms can be applied to shapes like points, lines, polygons, and other geometric figures. We can find its applications in various fields such as computer graphics, computer-aided design, robotics, and geographical information systems.

Problems related to Geometric Algorithms

Some of the common problems that can be solved using geometric algorithms are −

  • It can be used to solve area of different polygons, such as triangle, rectangle and so on.

  • We can use geometric algorithm to find intersection of two lines.

  • Computing convex hull problem.

  • Finding a point inside different geometric shapes.

  • Geometric algorithms can solve triangulation of polygon problem.

Important Geometric Algorithms

Some of the important geometric algorithms are −

  • Graham Scan Algorithm

  • Sweep Line Algorithm

Graham Scan Algorithm

The Graham Scan Algorithm was developed by Ronald Graham in 1972. This algorithm is primarily used to solve Convex Hull problem and Maximum distance problem. Its application can be seen in other fields also, such as image processing, robotics, delivery systems and many more.

The following steps explain the working Graham Scan algorithm −

  • First, find the point with the lowest y-coordinate. If multiple points share the lowest y-coordinate, choose the one with the lowest x-coordinate.

  • Sort the remaining points based on their polar angle with respect to the lowest point.

  • After sorting the points, start with the lowest point and other two additional points. These three points form the initial part of the convex hull.

  • For each new point, determine whether travelling from the two preceding points constitutes a left turn or a right turn. If it's a right turn, the second-to-last point is not inside the convex hull.

  • Repeat this process until a left turn set is encountered.

Example

The following example practically demonstrates how Graham Scan algorithm works.

C C++ Java Python 
#include <stdio.h>
#include <stdlib.h>
// Structure for a point
struct Point2D {
    int x, y;
};
// Global variable for the initial point
struct Point2D initialPoint;
// Function to get the second top element
struct Point2D getSecondTop(struct Point2D Stack[], int* top) {
    return Stack[(*top)-1];
}
// Function to calculate square of distance between two points
int calcDistSq(struct Point2D p1, struct Point2D p2) {
    return (p1.x - p2.x)*(p1.x - p2.x) + (p1.y - p2.y)*(p1.y - p2.y);
}
// Function to find orientation of ordered triplet of points
int getOrientation(struct Point2D p, struct Point2D q, struct Point2D r) {
    int val = (q.y - p.y) * (r.x - q.x) - (q.x - p.x) * (r.y - q.y);
    if (val == 0) return 0;
    return (val > 0)? 1: 2;
}
// Function to compare two points
int comparePoints(const void *vp1, const void *vp2) {
    struct Point2D *p1 = (struct Point2D *)vp1;
    struct Point2D *p2 = (struct Point2D *)vp2;
    int o = getOrientation(initialPoint, *p1, *p2);
    if (o == 0)
        return (calcDistSq(initialPoint, *p2) >= calcDistSq(initialPoint, *p1))? -1 : 1;
    return (o == 2)? -1: 1;
}
// Function to compute the convex hull
void computeConvexHull(struct Point2D points[], int n) {
    int ymin = points[0].y, min = 0;
    for (int i = 1; i < n; i++) {
        int y = points[i].y;
        if ((y < ymin) || (ymin == y && points[i].x < points[min].x))
            ymin = points[i].y, min = i;
    }
    // Swapping
    struct Point2D temp = points[0];
    points[0] = points[min];
    points[min] = temp;
    // Initial point
    initialPoint = points[0];
    // Sort array of points
    qsort(&points[1], n-1, sizeof(struct Point2D), comparePoints);
    int m = 1;
    for (int i=1; i<n; i++) {
        while (i < n-1 && getOrientation(initialPoint, points[i], points[i+1]) == 0)
            i++;
        points[m] = points[i];
        m++;
    }
    if (m < 3) return;
    // Stack to store the points on the convex hull
    struct Point2D Stack[n];
    int top = -1;
    // Push the first three points into the stack
    Stack[++top] = points[0];
    Stack[++top] = points[1];
    Stack[++top] = points[2];
    // For the rest of the points
    for (int i = 3; i < m; i++) {
        while (getOrientation(getSecondTop(Stack, &top), Stack[top], points[i]) != 2)
            top--;
        Stack[++top] = points[i];
    }
    // Print the point
    while (top != -1) {
        struct Point2D p = Stack[top--];
        printf("(%d, %d)\n", p.x, p.y);
    }
}
int main() {
    struct Point2D points[] = {{0, 1}, {1, 2}, {2, 3}, {4, 5}, {0, 0}, {2, 1}, {3, 1}, {3, 3}};
    int n = sizeof(points)/ sizeof(points[0]);
    computeConvexHull(points, n);
    return 0;
}

#include <iostream>
#include <vector>
#include <algorithm>
using namespace std;
// structure for a point 
struct Point2D {
    int x, y;
};
// initial point
Point2D initialPoint;
// Function to get the next top element 
Point2D getSecondTop(vector<Point2D> &Stack) {
    return Stack[Stack.size()-2]; 
}
// Function to calculate square of distance between two points
int calcDistSq(Point2D p1, Point2D p2) {
    return (p1.x - p2.x)*(p1.x - p2.x) + (p1.y - p2.y)*(p1.y - p2.y);
}
// Function to find orientation of ordered triplet of points
int getOrientation(Point2D p, Point2D q, Point2D r) {
    int val = (q.y - p.y) * (r.x - q.x) - (q.x - p.x) * (r.y - q.y);
    if (val == 0) return 0; 
    // checking clock or counterclock wise
    return (val > 0)? 1: 2; 
}
// Function to compare two points
int comparePoints(const void *vp1, const void *vp2) {
    Point2D *p1 = (Point2D *)vp1;
    Point2D *p2 = (Point2D *)vp2;
    int o = getOrientation(initialPoint, *p1, *p2);
    if (o == 0)
        return (calcDistSq(initialPoint, *p2) >= calcDistSq(initialPoint, *p1))? -1 : 1;
    return (o == 2)? -1: 1;
}
// Function to compute the convex hull 
void computeConvexHull(Point2D points[], int n) {
    int ymin = points[0].y, min = 0;
    for (int i = 1; i < n; i++) {
        int y = points[i].y;
        if ((y < ymin) || (ymin == y && points[i].x < points[min].x))
            ymin = points[i].y, min = i;
    }
    // swapping
    swap(points[0], points[min]); 
    // initial point
    initialPoint = points[0]; 
    // Sort array of points
    qsort(&points[1], n-1, sizeof(Point2D), comparePoints); 
    int m = 1; 
    for (int i=1; i<n; i++) {
        while (i < n-1 && getOrientation(initialPoint, points[i], points[i+1]) == 0)
            i++; 
        points[m] = points[i]; 
        m++; 
    }
    if (m < 3) return; 
    // stack to store the points on the convex hull
    vector<Point2D> Stack; 
    // Push the first three points into the stack
    Stack.push_back(points[0]); 
    Stack.push_back(points[1]);
    Stack.push_back(points[2]);
    // For the rest of the points
    for (int i = 3; i < m; i++) { 
        while (getOrientation(getSecondTop(Stack), Stack.back(), points[i]) != 2)
            Stack.pop_back(); 
        Stack.push_back(points[i]); 
    }
    // Print the point
    while (!Stack.empty()) { 
        Point2D p = Stack.back(); 
        cout << "(" << p.x << ", " << p.y <<")" << endl; 
        Stack.pop_back(); 
    }
}
int main() {
    Point2D points[] = {{0, 1}, {1, 2}, {2, 3}, {4, 5}, {0, 0}, {2, 1}, {3, 1}, {3, 3}};
    int n = sizeof(points)/ sizeof(points[0]); 
    computeConvexHull(points, n); 
    return 0;
}

import java.util.*;
// class to represent points with x and y coordinates
class Cords {
    int xCoord, yCoord;
    // Constructor 
    Cords(int x, int y) {
        this.xCoord = x;
        this.yCoord = y;
    }
}
// class to find the convex hull 
public class ConvexHullFinder {
    // initial point
    Cords initialPoint;
    // Method to get the next to top element
    Cords getSecondTop(Stack<Cords> stack) {
        Cords temp = stack.pop();
        Cords result = stack.peek();
        stack.push(temp);
        return result;
    }
    // Method to swap two points
    void swapPoints(Cords p1, Cords p2) {
        Cords temp = p1;
        p1 = p2;
        p2 = temp;
    }
    // Method to calculate the square of the distance between two points
    int distanceSquare(Cords p1, Cords p2) {
        return (p1.xCoord - p2.xCoord)*(p1.xCoord - p2.xCoord) + (p1.yCoord - p2.yCoord)*(p1.yCoord - p2.yCoord);
    }
    // Method to find the orientation of an ordered triplet of points
    int findOrientation(Cords p, Cords q, Cords r) {
        int value = (q.yCoord - p.yCoord) * (r.xCoord - q.xCoord) - (q.xCoord - p.xCoord) * (r.yCoord - q.yCoord);

        if (value == 0) return 0; 
        // checking clock or counterclock wise
        return (value > 0)? 1: 2; 
    }
    // Method to check if two points have same slope 
    boolean sameSlope(Cords p1, Cords p2, Cords p3) {
        return (p2.yCoord - p1.yCoord) * (p3.xCoord - p2.xCoord) == (p3.yCoord - p2.yCoord) * (p2.xCoord - p1.xCoord);
    }
    // method to calculate convex hull 
    void calculateConvexHull(Cords points[], int n) {
        int yMin = points[0].yCoord, min = 0;
        for (int i = 1; i < n; i++) {
            int y = points[i].yCoord;
            if ((y < yMin) || (yMin == y && points[i].xCoord < points[min].xCoord)) {
                yMin = points[i].yCoord;
                min = i;
            }
        }
        // Swapping
        swapPoints(points[0], points[min]);
        // initial point
        initialPoint = points[0];
        // Sort array of points
        Arrays.sort(points, new Comparator<Cords>() {
            @Override
            public int compare(Cords p1, Cords p2) {
                if (sameSlope(initialPoint, p1, p2)) {
                    if (distanceSquare(initialPoint, p2) >= distanceSquare(initialPoint, p1)) {
                        return -1;
                    } else {
                        return 1;
                    }
                }
                if (findOrientation(initialPoint, p1, p2) == 2) {
                    return -1;
                } else {
                    return 1;
                }
            }
        });
        // Create a stack and push the first three points into it
        Stack<Cords> stack = new Stack<>();
        stack.push(points[0]);
        stack.push(points[1]);
        stack.push(points[2]);
        // keep removing top of stack while we get a clockwise turn
        for (int i = 3; i < n; i++) {
            while (stack.size() > 1 && findOrientation(getSecondTop(stack), stack.peek(), points[i]) != 2)
                stack.pop();
            stack.push(points[i]);
        }
        // print result
        while (!stack.empty()) {
            Cords p = stack.peek();
            System.out.println("(" + p.xCoord + ", " + p.yCoord + ")");
            stack.pop();
        }
    }
    public static void main(String[] args) {
        // points
        Cords points[] = {new Cords(0, 1), new Cords(1, 2), new Cords(2, 3), new Cords(4, 5),
                new Cords(0, 0), new Cords(2, 1), new Cords(3, 1), new Cords(3, 3)};
        int n = points.length;
        ConvexHullFinder finder = new ConvexHullFinder();
        finder.calculateConvexHull(points, n);
    }
}

from functools import cmp_to_key
import math

# Class to represent points with x and y coordinates
class Cords:
    def __init__(self, x, y):
        self.xCoord = x
        self.yCoord = y

# Class to find the convex hull
class ConvexHullFinder:
    # Initial point
    initialPoint = None

    # Method to get the next to top element
    def getSecondTop(self, stack):
        return stack[-2]

    # Method to swap two points
    def swapPoints(self, p1, p2):
        return p2, p1

    # Method to calculate the square of the distance between two points
    def distanceSquare(self, p1, p2):
        return (p1.xCoord - p2.xCoord)**2 + (p1.yCoord - p2.yCoord)**2

    # Method to find the orientation of an ordered triplet of points
    def findOrientation(self, p, q, r):
        value = (q.yCoord - p.yCoord) * (r.xCoord - q.xCoord) - (q.xCoord - p.xCoord) * (r.yCoord - q.yCoord)
        if value == 0: return 0
        return 1 if value > 0 else 2

    # Method to check if two points have same slope
    def sameSlope(self, p1, p2, p3):
        return (p2.yCoord - p1.yCoord) * (p3.xCoord - p2.xCoord) == (p3.yCoord - p2.yCoord) * (p2.xCoord - p1.xCoord)

    # Method to calculate convex hull
    def calculateConvexHull(self, points):
        n = len(points)
        ymin = points[0].yCoord
        min = 0
        for i in range(1, n):
            y = points[i].yCoord
            if (y < ymin) or (ymin == y and points[i].xCoord < points[min].xCoord):
                ymin = points[i].yCoord
                min = i
        # Swapping
        points[0], points[min] = self.swapPoints(points[0], points[min])
        # Initial point
        self.initialPoint = points[0]
        # Sort array of points
        def compare(p1, p2):
            if self.sameSlope(self.initialPoint, p1, p2):
                if self.distanceSquare(self.initialPoint, p2) >= self.distanceSquare(self.initialPoint, p1):
                    return -1
                else:
                    return 1
            if self.findOrientation(self.initialPoint, p1, p2) == 2:
                return -1
            else:
                return 1
        points = sorted(points, key=cmp_to_key(compare))
        # Create a stack and push the first three points into it
        stack = []
        stack.append(points[0])
        stack.append(points[1])
        stack.append(points[2])
        # Keep removing top of stack while we get a clockwise turn
        for i in range(3, n):
            while len(stack) > 1 and self.findOrientation(self.getSecondTop(stack), stack[-1], points[i]) != 2:
                stack.pop()
            stack.append(points[i])
        # Print result
        while stack:
            p = stack[-1]
            print("(", p.xCoord, ", ", p.yCoord, ")")
            stack.pop()

# Points
points = [Cords(0, 1), Cords(1, 2), Cords(2, 3), Cords(4, 5),
          Cords(0, 0), Cords(2, 1), Cords(3, 1), Cords(3, 3)]
finder = ConvexHullFinder()
finder.calculateConvexHull(points)

Output

(0, 1)
(4, 5)
(3, 1)
(0, 0)

Sweep Line Algorithm

The Sweep Line Algorithm is used to solve geometric and other problems involving dynamic sets of objects moving in a specific direction. Suppose a vertical line (say it sweep line) moving from left to right across the plane. As the sweep line encounters endpoints of line segments, we track which segments intersect at each point in time. By analyzing the interactions, we can efficiently solve various problems.

Following are the steps involved in Sweep Line Algorithm −

  • The sweep line starts at one end of the problem space and moves towards the other.

  • At regular intervals, the algorithm update a suitable data structure depending on the problem. Based on the current configuration of the sweep line position, it calculates distances, intersections, areas, or other desired outputs.

  • The final data structures or accumulated results provide the solution to the problem.

Example

Following is the example illustrating sweep line algorithm in various programming language.

C C++ Java Python 
#include <stdio.h>
#include <math.h>
// Structure to represent a coordinate
typedef struct {
   double a, b;
} Coordinate;
// Structure to represent a line segment
typedef struct {
   Coordinate start, end;
} Segment;
// Function to check if two line segments intersect
int checkIntersection(Segment s1, Segment s2) {
   // 
   return 0; 
}
// Function to find the intersection point of two line segments
Coordinate findIntersection(Segment s1, Segment s2) {
   // coordinates of the start and end points of the first line segment
   double a1 = s1.start.a, b1 = s1.start.b;
   double a2 = s1.end.a, b2 = s1.end.b;
   // coordinates of the start and end points of the second line segment
   double a3 = s2.start.a, b3 = s2.start.b;
   double a4 = s2.end.a, b4 = s2.end.b;
   // Calculate the denominator of the intersection formula
   double denominator = (b4 - b3) * (a2 - a1) - (a4 - a3) * (b2 - b1);
   double numerator1 = (a4 - a3) * (b1 - b3) - (b4 - b3) * (a1 - a3);
   double numerator2 = (a2 - a1) * (b1 - b3) - (b2 - b1) * (a1 - a3);
   // If the denominator is zero, the lines are parallel
   if (denominator == 0) {
      return (Coordinate){ INFINITY, INFINITY };
   }
   double u = numerator1 / denominator;
   double v = numerator2 / denominator;
   if (u >= 0 && u <= 1 && v >= 0 && v <= 1) {
      double a = a1 + u * (a2 - a1);
      double b = b1 + u * (b2 - b1);
      return (Coordinate){ a, b };
   }
   return (Coordinate){ INFINITY, INFINITY };
}
int main() {
   // line segments
   Segment s1 = { { 1, 2 }, { 3, 2 } };
   Segment s2 = { { 2, 1 }, { 2, 3 } };
   // If the line segments intersect
   if (checkIntersection(s1, s2)) {
      // Find the intersection point
      Coordinate c = findIntersection(s1, s2);
      printf("Intersection point: (%f, %f)\n", c.a, c.b);
   } else {
      printf("Segments do not intersect\n");
   }
   return 0;
}

#include <bits/stdc++.h>
#include <iostream>
using namespace std;
// structure to represent a coordinate
struct Coordinate {
   double a, b;
};
// structure to represent a line segment
struct Segment {
   Coordinate start, end;
};
// Function to check if two line segments intersect
bool checkIntersection(Segment s1, Segment s2)
{
    //
}
// Function to find the intersection point of two line segments
Coordinate findIntersection(Segment s1, Segment s2) {
   // coordinates of the start and end points of the first line segment
   double a1 = s1.start.a, b1 = s1.start.b;
   double a2 = s1.end.a, b2 = s1.end.b;
   // coordinates of the start and end points of the second line segment
   double a3 = s2.start.a, b3 = s2.start.b;
   double a4 = s2.end.a, b4 = s2.end.b;
   // Calculate the denominator of the intersection formula
   double denominator = (b4 - b3) * (a2 - a1) - (a4 - a3) * (b2 - b1);
   // Calculate the numerators of the intersection formula
   double numerator1 = (a4 - a3) * (b1 - b3) - (b4 - b3) * (a1 - a3);
   double numerator2 = (a2 - a1) * (b1 - b3) - (b2 - b1) * (a1 - a3);
   // If the denominator is zero, the lines are parallel
   if (denominator == 0) {
      return { INFINITY, INFINITY };
   }
   double u = numerator1 / denominator;
   double v = numerator2 / denominator;
   // If u and v are both between 0 and 1, the line segments intersect
   if (u >= 0 && u <= 1 && v >= 0 && v <= 1) {
      // Calculate the coordinates of the intersection point
      double a = a1 + u * (a2 - a1);
      double b = b1 + u * (b2 - b1);
      return { a, b };
   }
   // If u or v is not between 0 and 1, the line segments do not intersect
   return { INFINITY, INFINITY };
}
int main(){
   // line segments
   Segment s1 = { { 1, 2 }, { 3, 2 } };
   Segment s2 = { { 2, 1 }, { 2, 3 } };
   // If the line segments intersect
   if (checkIntersection(s1, s2)) {
      // Find the intersection point
      Coordinate c = findIntersection(s1, s2);
      // Print the intersection point
      cout << "Intersection point: (" << c.a << ", " << c.b << ")" << endl;
   } else {
      cout << "Segments do not intersect" << endl;
   }
   return 0;
}

public class Main {
   // Class to represent a coordinate
   static class Coordinate {
      double a, b;
      Coordinate(double a, double b) {
         this.a = a;
         this.b = b;
      }
   }
   // Class to represent a line segment
   static class Segment {
      Coordinate start, end;
      Segment(Coordinate start, Coordinate end) {
         this.start = start;
         this.end = end;
      }
   }
   // Method to check if two line segments intersect
   static boolean checkIntersection(Segment s1, Segment s2) {
      return false; 
   }
   // Method to find the intersection point of two line segments
   static Coordinate findIntersection(Segment s1, Segment s2) {
      double a1 = s1.start.a, b1 = s1.start.b;
      double a2 = s1.end.a, b2 = s1.end.b;
      double a3 = s2.start.a, b3 = s2.start.b;
      double a4 = s2.end.a, b4 = s2.end.b;
      double denominator = (b4 - b3) * (a2 - a1) - (a4 - a3) * (b2 - b1);
      double numerator1 = (a4 - a3) * (b1 - b3) - (b4 - b3) * (a1 - a3);
      double numerator2 = (a2 - a1) * (b1 - b3) - (b2 - b1) * (a1 - a3);
      if (denominator == 0) {
         return new Coordinate(Double.POSITIVE_INFINITY, Double.POSITIVE_INFINITY);
      }
      double u = numerator1 / denominator;
      double v = numerator2 / denominator;
      if (u >= 0 && u <= 1 && v >= 0 && v <= 1) {
         double a = a1 + u * (a2 - a1);
         double b = b1 + u * (b2 - b1);
         return new Coordinate(a, b);
      }
      return new Coordinate(Double.POSITIVE_INFINITY, Double.POSITIVE_INFINITY);
   }
   public static void main(String[] args) {
      Segment s1 = new Segment(new Coordinate(1, 2), new Coordinate(3, 2));
      Segment s2 = new Segment(new Coordinate(2, 1), new Coordinate(2, 3));
      if (checkIntersection(s1, s2)) {
         Coordinate c = findIntersection(s1, s2);
         System.out.println("Intersection point: (" + c.a + ", " + c.b + ")");
      } else {
         System.out.println("Segments do not intersect");
      }
   }
}

import math
# Class to represent a coordinate
class Coordinate:
    def __init__(self, a, b):
        self.a = a
        self.b = b
# Class to represent a line segment
class Segment:
    def __init__(self, start, end):
        self.start = start
        self.end = end
# Function to check if two line segments intersect
def checkIntersection(s1, s2):
    # Implement intersection checking algorithm here
    return False  # Placeholder return statement
# Function to find the intersection point of two line segments
def findIntersection(s1, s2):
    a1, b1 = s1.start.a, s1.start.b
    a2, b2 = s1.end.a, s1.end.b
    a3, b3 = s2.start.a, s2.start.b
    a4, b4 = s2.end.a, s2.end.b
    denominator = (b4 - b3) * (a2 - a1) - (a4 - a3) * (b2 - b1)
    numerator1 = (a4 - a3) * (b1 - b3) - (b4 - b3) * (a1 - a3)
    numerator2 = (a2 - a1) * (b1 - b3) - (b2 - b1) * (a1 - a3)
    if denominator == 0:
        return Coordinate(math.inf, math.inf)
    u = numerator1 / denominator
    v = numerator2 / denominator
    if 0 <= u <= 1 and 0 <= v <= 1:
        a = a1 + u * (a2 - a1)
        b = b1 + u * (b2 - b1)
        return Coordinate(a, b)
    return Coordinate(math.inf, math.inf)
# Test the functions
s1 = Segment(Coordinate(1, 2), Coordinate(3, 2))
s2 = Segment(Coordinate(2, 1), Coordinate(2, 3))

if checkIntersection(s1, s2):
    c = findIntersection(s1, s2)
    print(f"Intersection point: ({c.a}, {c.b})")
else:
    print("Segments do not intersect")

Output

Segments do not intersect
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