2D Transformation | Rotation of objects

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

C

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// C program to rotate an object by 
// a given angle about a given point
#include<stdio.h>
#include<math.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;
}

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CPP

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// 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;
}

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Output:

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

References: Rotation matrix

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