Orientation of an ordered triplet of points in the plane can be
- counterclockwise
- clockwise
- colinear
The following diagram shows different possible orientations of (a,b,c)
If orientation of (p1, p2, p3) is collinear, then orientation of (p3, p2, p1) is also collinear.
If orientation of (p1, p2, p3) is clockwise, then orientation of (p3, p2, p1) is counterclockwise and vice versa is also true.
Given three points p1, p2 and p3, find orientation of (p1, p2, p3).
Example:
Input: p1 = {0, 0}, p2 = {4, 4}, p3 = {1, 2}
Output: CounterClockWise
Input: p1 = {0, 0}, p2 = {4, 4}, p3 = {1, 1}
Output: Colinear
How to compute Orientation?
The idea is to use slope.
Slope of line segment (p1, p2): σ = (y2 - y1)/(x2 - x1)
Slope of line segment (p2, p3): τ = (y3 - y2)/(x3 - x2)
If σ > τ, the orientation is clockwise (right turn)
Using above values of σ and τ, we can conclude that,
the orientation depends on sign of below expression:
(y2 - y1)*(x3 - x2) - (y3 - y2)*(x2 - x1)
Above expression is negative when σ < τ, i.e., counterclockwise
Below is the implementation of above idea.
C++
// A C++ program to find orientation of three points #include <iostream> using namespace std; struct Point { int x, y; }; // To find orientation of ordered triplet (p1, p2, p3). // The function returns following values // 0 --> p, q and r are colinear // 1 --> Clockwise // 2 --> Counterclockwise int orientation(Point p1, Point p2, Point p3) { // See 10th slides from following link for derivation // of the formula int val = (p2.y - p1.y) * (p3.x - p2.x) - (p2.x - p1.x) * (p3.y - p2.y); if (val == 0) return 0; // colinear return (val > 0)? 1: 2; // clock or counterclock wise } // Driver program to test above functions int main() { Point p1 = {0, 0}, p2 = {4, 4}, p3 = {1, 2}; int o = orientation(p1, p2, p3); if (o==0) cout << "Linear"; else if (o == 1) cout << "Clockwise"; else cout << "CounterClockwise"; return 0; } |
Java
// JAVA Code to find Orientation of 3 // ordered points class Point { int x, y; Point(int x,int y){ this.x=x; this.y=y; } } class GFG { // To find orientation of ordered triplet // (p1, p2, p3). The function returns // following values // 0 --> p, q and r are colinear // 1 --> Clockwise // 2 --> Counterclockwise public static int orientation(Point p1, Point p2, Point p3) { // See 10th slides from following link // for derivation of the formula int val = (p2.y - p1.y) * (p3.x - p2.x) - (p2.x - p1.x) * (p3.y - p2.y); if (val == 0) return 0; // colinear // clock or counterclock wise return (val > 0)? 1: 2; } /* Driver program to test above function */ public static void main(String[] args) { Point p1 = new Point(0, 0); Point p2 = new Point(4, 4); Point p3 = new Point(1, 2); int o = orientation(p1, p2, p3); if (o==0) System.out.print("Linear"); else if (o == 1) System.out.print("Clockwise"); else System.out.print("CounterClockwise"); } } //This code is contributed by Arnav Kr. Mandal. |
Python3
# A Python3 program to find orientation of 3 points class Point: # to store the x and y coordinates of a point def __init__(self, x, y): self.x = x self.y = y def orientation(p1, p2, p3): # to find the orientation of # an ordered triplet (p1,p2,p3) # function returns the following values: # 0 : Colinear points # 1 : Clockwise points # 2 : Counterclockwise val = (float(p2.y - p1.y) * (p3.x - p2.x)) - \ (float(p2.x - p1.x) * (p3.y - p2.y)) if (val > 0): # Clockwise orientation return 1 elif (val < 0): # Counterclockwise orientation return 2 else: # Colinear orientation return 0 # Driver code p1 = Point(0, 0) p2 = Point(4, 4) p3 = Point(1, 2) o = orientation(p1, p2, p3) if (o == 0): print("Linear") elif (o == 1): print("Clockwise") else: print("CounterClockwise") # This code is contributed by Ansh Riyal |
C#
// C# Code to find Orientation of 3 // ordered points using System; public class Point { public int x, y; public Point(int x,int y) { this.x = x; this.y = y; } } class GFG { // To find orientation of ordered triplet // (p1, p2, p3). The function returns // following values // 0 --> p, q and r are colinear // 1 --> Clockwise // 2 --> Counterclockwise public static int orientation(Point p1, Point p2, Point p3) { // See 10th slides from following link // for derivation of the formula int val = (p2.y - p1.y) * (p3.x - p2.x) - (p2.x - p1.x) * (p3.y - p2.y); if (val == 0) return 0; // colinear // clock or counterclock wise return (val > 0)? 1: 2; } /* Driver program to test above function */<strong> public static void Main(String[] args) { Point p1 = new Point(0, 0); Point p2 = new Point(4, 4); Point p3 = new Point(1, 2); int o = orientation(p1, p2, p3); if (o == 0) Console.WriteLine("Linear"); else if (o == 1) Console.WriteLine("Clockwise"); else Console.WriteLine("CounterClockwise"); } } /* This code contributed by PrinciRaj1992 */ |
Output:
CounterClockwise
The concept of orientation is used in below articles:
Find Simple Closed Path for a given set of points
How to check if two given line segments intersect?
Convex Hull | Set 1 (Jarvisâs Algorithm or Wrapping)
Convex Hull | Set 2 (Graham Scan)
Source:
http://www.dcs.gla.ac.uk/~pat/52233/slides/Geometry1x1.pdf
This article is contributed by Rajeev Agrawal. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above
Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.
