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2D Transformation | Rotation of objects

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  • Difficulty Level : Easy
  • Last Updated : 02 Dec, 2021

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}
 

C




// C program to rotate an object by
// a given angle about a given point
#include <math.h>
#include <stdio.h>
 
// 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
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));
        printf("(%f, %f) ", 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;
}

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;
}

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)

Output: 

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

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

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