Transformation 2D | Rotation des objets
Nous devons faire pivoter un objet d'un angle donné autour d'un point de pivot donné et imprimer les nouvelles coordonnées.
Exemples :
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) 
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) 
Pour faire pivoter un objet, nous devons faire pivoter chaque sommet de la figure individuellement.
En faisant pivoter un point P(x y) d'un angle A autour de l'origine, nous obtenons un point P'(x' y'). Les valeurs de x' et y' peuvent être calculées comme suit : -
Nous savons que
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
Équation matricielle de rotation : -
début{bmatrix} x' \ y' end{bmatrix} = début{bmatrix} cosA & -sinA\ sinA & cosA end{bmatrix} début{bmatrix} x \ y end{bmatrix}
// C++ program to rotate an object by // a given angle about a given point #include #include 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);*/ } }
Sortir:
(-100 100) (-200 150) (-200 200) (-150 200)
Complexité temporelle : SUR)
Espace auxiliaire : O(1)
Références : Matrice de rotation
