Orientation of 3 ordered points
Given three points p1, p2 and p3, the task is to determine the orientation of these three points.
Orientation of an ordered triplet of points in the plane can be
- counterclockwise
- clockwise
- collinear
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.
Example:
Input: p1 = {0, 0}, p2 = {4, 4}, p3 = {1, 2}
Output: CounterClockWiseInput: p1 = {0, 0}, p2 = {4, 4}, p3 = {1, 1}
Output: Collinear
How to compute Orientation?
The idea is to use slope.
Slope of line segment (p1, p2): ? = (y2 - y1)/(x2 - x1)
Slope of line segment (p2, p3): ? = (y3 - y2)/(x3 - x2)
If ? > ?, the orientation is clockwise (right turn)
Using above values of ? and ?, 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 ? < ?, 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 collinear // 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; // collinear 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" ; cout << endl; p1 = { 0, 0 }, p2 = { 4, 4 }, p3 = { 1, 1 }; o = orientation(p1, p2, p3); if (o == 0) cout << "Linear" ; else if (o == 1) cout << "Clockwise" ; else cout << "CounterClockwise" ; cout << endl; p1 = { 1, 2 }, p2 = { 4, 4 }, p3 = { 0, 0 }; 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 collinear // 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 ; // collinear // 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 : Collinear 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 : # Collinear 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 collinear // 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; // collinear // 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 */ |
Javascript
<script> // javascript Code to find Orientation of 3 // ordered points class Point { constructor(x, y) { this .x = x; this .y = y; } } // To find orientation of ordered triplet // (p1, p2, p3). The function returns // following values // 0 --> p, q and r are collinear // 1 --> Clockwise // 2 --> Counterclockwise function orientation(p1, p2, p3) { // See 10th slides from following link // for derivation of the formula let val = (p2.y - p1.y) * (p3.x - p2.x) - (p2.x - p1.x) * (p3.y - p2.y); if (val == 0) return 0; // collinear // clock or counterclock wise return (val > 0) ? 1 : 2; } /* Driver program to test above function */ let p1 = new Point(0, 0); let p2 = new Point(4, 4); let p3 = new Point(1, 2); let o = orientation(p1, p2, p3); if (o == 0) document.write( "Linear" ); else if (o == 1) document.write( "Clockwise" ); else document.write( "CounterClockwise" ); // This code is contributed by Saurabh Jaiswal </script> |
CounterClockwise Linear Clockwise
Time Complexity: O(1)
Auxiliary Space: O(1)
The concept of orientation is used in below articles:
Approach#2: Using slope
This approach checks the orientation of 3 ordered points in the plane by calculating the slopes of the line segments formed by the points. If the slopes are equal, then the points are collinear. If the slope of the line segment formed by the first two points is less than the slope of the line segment formed by the last two points, then the orientation is counter-clockwise, otherwise it is clockwise.
Algorithm
1. Calculate slope of lines formed by (p1,p2) and (p2,p3)
2. If slopes are equal, then points are collinear
3. If slope of (p1,p2) < slope of (p2,p3), then points are in counter clockwise orientation
4. If slope of (p1,p2) > slope of (p2,p3), then points are in clockwise orientation
C++
#include <iostream> using namespace std; string orientation( int p1[], int p2[], int p3[]) { // Calculate slopes int slope1 = (p2[1] - p1[1]) * (p3[0] - p2[0]); int slope2 = (p3[1] - p2[1]) * (p2[0] - p1[0]); // Check orientation if (slope1 == slope2) { return "Collinear" ; } else if (slope1 < slope2) { return "CounterClockWise" ; } else { return "ClockWise" ; } } int main() { // Example usage int p1[] = { 0, 0 }; int p2[] = { 4, 4 }; int p3[] = { 1, 1 }; cout << orientation(p1, p2, p3) << endl; int p4[] = { 0, 0 }; int p5[] = { 4, 4 }; int p6[] = { 1, 2 }; cout << orientation(p4, p5, p6) << endl; return 0; } // This code is contributed by user_dtewbxkn77n |
Java
// Java code import java.io.*; class GFG { static String orientation( int p1[], int p2[], int p3[]) { // Calculate slopes int slope1 = (p2[ 1 ] - p1[ 1 ]) * (p3[ 0 ] - p2[ 0 ]); int slope2 = (p3[ 1 ] - p2[ 1 ]) * (p2[ 0 ] - p1[ 0 ]); // Check orientation if (slope1 == slope2) { return "Collinear" ; } else if (slope1 < slope2) { return "CounterClockWise" ; } else { return "ClockWise" ; } } public static void main (String[] args) { // Example usage int p1[] = { 0 , 0 }; int p2[] = { 4 , 4 }; int p3[] = { 1 , 1 }; System.out.println(orientation(p1, p2, p3)); int p4[] = { 0 , 0 }; int p5[] = { 4 , 4 }; int p6[] = { 1 , 2 }; System.out.println(orientation(p4, p5, p6)); } } // This code is contributed by Pushpesh Raj |
Python3
def orientation(p1, p2, p3): # Calculate slopes slope1 = (p2[ 1 ] - p1[ 1 ]) * (p3[ 0 ] - p2[ 0 ]) slope2 = (p3[ 1 ] - p2[ 1 ]) * (p2[ 0 ] - p1[ 0 ]) # Check orientation if slope1 = = slope2: return "Collinear" elif slope1 < slope2: return "CounterClockWise" else : return "ClockWise" # Example usage p1 = [ 0 , 0 ] p2 = [ 4 , 4 ] p3 = [ 1 , 1 ] print (orientation(p1, p2, p3)) p1 = [ 0 , 0 ] p2 = [ 4 , 4 ] p3 = [ 1 , 2 ] print (orientation(p1, p2, p3)) |
C#
using System; class Program { // Function to determine the orientation of three points static string Orientation( int [] p1, int [] p2, int [] p3) { // Calculate slopes int slope1 = (p2[1] - p1[1]) * (p3[0] - p2[0]); int slope2 = (p3[1] - p2[1]) * (p2[0] - p1[0]); // Check orientation if (slope1 == slope2) { return "Collinear" ; // Points are collinear } else if (slope1 < slope2) { return "CounterClockWise" ; // Points are in counter-clockwise order } else { return "ClockWise" ; // Points are in clockwise order } } static void Main() { // Example usage int [] p1 = { 0, 0 }; int [] p2 = { 4, 4 }; int [] p3 = { 1, 1 }; Console.WriteLine(Orientation(p1, p2, p3)); // Output: Collinear int [] p4 = { 0, 0 }; int [] p5 = { 4, 4 }; int [] p6 = { 1, 2 }; Console.WriteLine(Orientation(p4, p5, p6)); // Output: CounterClockWise } } |
Javascript
function orientation(p1, p2, p3) { // Calculate slopes let slope1 = (p2[1] - p1[1]) * (p3[0] - p2[0]); let slope2 = (p3[1] - p2[1]) * (p2[0] - p1[0]); // Check orientation if (slope1 == slope2) { return "Collinear" ; } else if (slope1 < slope2) { return "CounterClockWise" ; } else { return "ClockWise" ; } } // Example usage let p1 = [0, 0]; let p2 = [4, 4]; let p3 = [1, 1]; console.log(orientation(p1, p2, p3)); p1 = [0, 0]; p2 = [4, 4]; p3 = [1, 2]; console.log(orientation(p1, p2, p3)); |
Collinear CounterClockWise
Time Complexity: O(1)
Space Complexity: O(1)
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