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2D Transformation | Rotation of objects

Last Updated : 17 Mar, 2023
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We have to rotate an object by a given angle about a given pivot point and print the new co-ordinates.
Examples: 

Input : {(100, 100), (150, 200), (200, 200), 
         (200, 150)} is to be rotated about 
          (0, 0) by 90 degrees
Output : (-100, 100), (-200, 150), (-200, 200), (-150, 200)

Example1

Input : {(100, 100), (100, 200), (200, 200)} 
        is to be rotated about (50, -50) by 
         -45 degrees
Output : (191.421, 20.7107), (262.132, 91.4214), 
         (332.843, 20.7107)

Example2

In order to rotate an object we need to rotate each vertex of the figure individually. 
On rotating a point P(x, y) by an angle A about the origin we get a point P'(x', y'). The values of x' and y' can be calculated as follows:-

Rotation Diagram

We know that, 
x = rcosB, y = rsinB
x' = rcos(A+B) = r(cosAcosB - sinAsinB) = rcosBcosA - rsinBsinA = xcosA - ysinA 
y' = rsin(A+B) = r(sinAcosB + cosAsinB) = rcosBsinA + rsinBcosA = xsinA + ycosA
Rotational Matrix Equation:-

\begin{bmatrix} x' \\ y' \end{bmatrix} =\begin{bmatrix} cosA & -sinA\\ sinA & cosA \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}            
 

CPP 
// C++ program to rotate an object by
// a given angle about a given point
#include <iostream>
#include <math.h>
using namespace std;

// Using macros to convert degree to radian
// and call sin() and cos() as these functions
// take input in radians
#define SIN(x) sin(x * 3.141592653589 / 180)
#define COS(x) cos(x * 3.141592653589 / 180)

// To rotate an object given as order set of points in a[]
// (x_pivot, y_pivot)
void rotate(float a[][2], int n, int x_pivot, int y_pivot,
            int angle)
{
    int i = 0;
    while (i < n) {
        // Shifting the pivot point to the origin
        // and the given points accordingly
        int x_shifted = a[i][0] - x_pivot;
        int y_shifted = a[i][1] - y_pivot;

        // Calculating the rotated point co-ordinates
        // and shifting it back
        a[i][0] = x_pivot
                  + (x_shifted * COS(angle)
                     - y_shifted * SIN(angle));
        a[i][1] = y_pivot
                  + (x_shifted * SIN(angle)
                     + y_shifted * COS(angle));
        cout << "(" << a[i][0] << ", " << a[i][1] << ") ";
        i++;
    }
}

// Driver Code
int main()
{
    // 1st Example
    // The following figure is to be
    // rotated about (0, 0) by 90 degrees
    int size1 = 4; // No. of vertices
    // Vertex co-ordinates must be in order
    float points_list1[][2] = { { 100, 100 },
                                { 150, 200 },
                                { 200, 200 },
                                { 200, 150 } };
    rotate(points_list1, size1, 0, 0, 90);

    // 2nd Example
    // The following figure is to be
    // rotated about (50, -50) by -45 degrees
    /*int size2 = 3;//No. of vertices
    float points_list2[][2] = {{100, 100}, {100, 200},
                                {200, 200}};
    rotate(points_list2, size2, 50, -50, -45);*/
    return 0;
}
Java 
// Java program to rotate an object by
// a given angle about a given point
public class rotation {

  static void rotate(double a[][], int n, int x_pivot,
                     int y_pivot, int angle)
  {
    int i = 0;
    while (i < n)
    {
      
      // Shifting the pivot point to the origin
      // and the given points accordingly
      int x_shifted = (int)a[i][0] - x_pivot;
      int y_shifted = (int)a[i][1] - y_pivot;

      // Calculating the rotated point co-ordinates
      // and shifting it back
      double x = Math.toRadians(angle);
      a[i][0] = x_pivot
        + (x_shifted * Math.cos(x)
           - y_shifted * Math.sin(x));
      a[i][1] = y_pivot
        + (x_shifted * Math.sin(x)
           + y_shifted * Math.cos(x));
      System.out.printf("(%f, %f) ", a[i][0],
                        a[i][1]);
      i++;
    }
  }

  // Driver Code
  public static void main(String[] args)
  {
    // 1st Example
    // The following figure is to be
    // rotated about (0, 0) by 90 degrees
    int size1 = 4; // No. of vertices

    // Vertex co-ordinates must be in order
    double points_list1[][] = { { 100, 100 },
                               { 150, 200 },
                               { 200, 200 },
                               { 200, 150 } };
    rotate(points_list1, size1, 0, 0, 90);

    // 2nd Example
    // The following figure is to be
    // rotated about (50, -50) by -45 degrees
    /*int size2 = 3;//No. of vertices
        double points_list2[][2] = {{100, 100}, {100, 200},
                                    {200, 200}};
        rotate(points_list2, size2, 50, -50, -45);*/
  }
}

// This code is contributed by karandeep1234
Python3 
# Python3 program to rotate an object by
# a given angle about a given point
import math

SIN=lambda x: int(math.sin(x * 3.141592653589 / 180))
COS=lambda x: int(math.cos(x * 3.141592653589 / 180))

# To rotate an object
def rotate(a, n, x_pivot, y_pivot, angle):
    i = 0
    while (i < n) :
        # Shifting the pivot point to the origin
        # and the given points accordingly
        x_shifted = a[i][0] - x_pivot
        y_shifted = a[i][1] - y_pivot

        # Calculating the rotated point co-ordinates
        # and shifting it back
        a[i][0] = x_pivot + (x_shifted * COS(angle) - y_shifted * SIN(angle))
        a[i][1] = y_pivot + (x_shifted * SIN(angle) + y_shifted * COS(angle))
        print("({}, {}) ".format(a[i][0], a[i][1]),end=" ")
        i+=1
    


# Driver Code
if __name__=='__main__':
    # 1st Example
    # The following figure is to be
    # rotated about (0, 0) by 90 degrees
    size1 = 4 # No. of vertices

    # Vertex co-ordinates must be in order
    points_list1 = [[ 100, 100],
                    [ 150, 200],
                    [ 200, 200],
                    [ 200, 150],]  
    rotate(points_list1, size1, 0, 0, 90)

    # 2nd Example
    # The following figure is to be
    # rotated about (50, -50) by -45 degrees
    # size2 = 3#No. of vertices
    # points_list2 = [[100, 100],
    #                     [100, 200],
    #                     [200, 200]]
    # rotate(points_list2, size2, 50, -50, -45)
JavaScript 
// Javascript program to rotate an object by
// a given angle about a given point
const SIN = (x) => Math.sin(x * Math.PI / 180);
const COS = (x) => Math.cos(x * Math.PI / 180);

function rotate(a, n, x_pivot, y_pivot, angle) {
  let i = 0;
  while (i < n) {
    // Shifting the pivot point to the origin
    // and the given points accordingly
    const x_shifted = a[i][0] - x_pivot;
    const y_shifted = a[i][1] - y_pivot;

    // Calculating the rotated point co-ordinates
    // and shifting it back
    a[i][0] = x_pivot + (x_shifted * COS(angle) - y_shifted * SIN(angle));
    a[i][1] = y_pivot + (x_shifted * SIN(angle) + y_shifted * COS(angle));
    console.log(`(${a[i][0]}, ${a[i][1]}) `);
    i++;
  }
}

// Driver Code
// 1st Example
// The following figure is to be
// rotated about (0, 0) by 90 degrees
const size1 = 4; // No. of vertices

// Vertex co-ordinates must be in order
const points_list1 = [[ 100, 100],
                      [ 150, 200],
                      [ 200, 200],
                      [ 200, 150],];
rotate(points_list1, size1, 0, 0, 90);

// 2nd Example
// The following figure is to be
// rotated about (50, -50) by -45 degrees
// const size2 = 3; // No. of vertices
// const points_list2 = [[100, 100],
//                       [100, 200],
//                       [200, 200]];
// rotate(points_list2, size2, 50, -50, -45);
C# 
// C# Program to rotate an object by
// a given angle about a given point
using System; 
  
class rotation 
{ 
    // Function to rotate the given points 
    // about the pivot point by angle 
    static void rotate(double[,] a, int n, 
                       int x_pivot, int y_pivot, int angle) 
    { 
        int i = 0; 
        while (i < n) 
        { 
            // Shifting the pivot point to the origin 
            // and the given points accordingly 
            int x_shifted = (int)a[i, 0] - x_pivot; 
            int y_shifted = (int)a[i, 1] - y_pivot; 
  
            // Calculating the rotated point co-ordinates 
            // and shifting it back 
            double x = Math.PI * angle / 180.0; 
            a[i, 0] = x_pivot + (x_shifted * 
                                   Math.Cos(x) - y_shifted * 
                                   Math.Sin(x)); 
            a[i, 1] = y_pivot + (x_shifted * 
                                   Math.Sin(x) + y_shifted * 
                                   Math.Cos(x)); 
            Console.Write("({0}, {1}) ", 
                          a[i, 0], a[i, 1]); 
            i++; 
        } 
    } 
  
    // Driver Code 
    public static void Main(String[] args) 
    { 
        // 1st Example 
        // The following figure is to be 
        // rotated about (0, 0) by 90 degrees 
        int size1 = 4; // No. of vertices 
  
        // Vertex co-ordinates must be in order 
        double[,] points_list1 = { { 100, 100 }, 
                               { 150, 200 }, 
                               { 200, 200 }, 
                               { 200, 150 } }; 
        rotate(points_list1, size1, 0, 0, 90); 
  
        // 2nd Example 
        // The following figure is to be 
        // rotated about (50, -50) by -45 degrees 
        /*int size2 = 3;//No. of vertices 
        double[,] points_list2 = { { 100, 100 }, 
                                  { 100, 200 }, 
                                  { 200, 200 } }; 
        rotate(points_list2, size2, 50, -50, -45);*/ 
    } 
}

Output: 

(-100, 100), (-200, 150), (-200, 200), (-150, 200)

Time Complexity: O(N)
Auxiliary Space: O(1) 
References: Rotation matrix


 


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