is a transformation which produces a mirror image of an object. The mirror image can be either about x-axis or y-axis. The object is rotated by180°.

Types of Reflection:

1. Reflection about the x-axis
2. Reflection about the y-axis
3. Reflection about an axis perpendicular to xy plane and passing through the origin
4. Reflection about line y=x

1. Reflection about x-axis: The object can be reflected about x-axis with the help of the following matrix

In this transformation value of x will remain same whereas the value of y will become negative. Following figures shows the reflection of the object axis. The object will lie another side of the x-axis.

Polymorphism in Java | Dynamic Method Dispatch

2. Reflection about y-axis: The object can be reflected about y-axis with the help of following transformation matrix

Here the values of x will be reversed, whereas the value of y will remain the same. The object will lie another side of the y-axis.

The following figure shows the reflection about the y-axis

3. Reflection about an axis perpendicular to xy plane and passing through origin:
In the matrix of this transformation is given below

In this value of x and y both will be reversed. This is also called as half revolution about the origin.

4. Reflection about line y=x: The object may be reflected about line y = x with the help of following transformation matrix

First of all, the object is rotated at 45°. The direction of rotation is clockwise. After it reflection is done concerning x-axis. The last step is the rotation of y=x back to its original position that is counterclockwise at 45°.

Example: A triangle ABC is given. The coordinates of A, B, C are given as

                    A (3 4)
                    B (6 4)
                    C (4 8)

Find reflected position of triangle i.e., to the x-axis.

Solution:

The a point coordinates after reflection

The b point coordinates after reflection

The coordinate of point c after reflection

a (3, 4) becomes a¹ (3, -4)
b (6, 4) becomes b¹ (6, -4)
c (4, 8) becomes c¹ (4, -8)

Program to perform Mirror Reflection about a line:

1. #include <iostream.h>  
2. #include <conio.h>  
3. #include <graphics.h>  
4. #include <math.h>  
5. #include <stdlib.h>  
6. #define pi 3.14  
7. class arc  
8. {  
9.     float x[10],y[10],theta,ref[10][10],ang;  
10.            float p[10][10],p1[10][10],x1[10],y1[10],xm,ym;  
11.     int i,k,j,n;  
12.     public:  
13.     void get();  
14.     void cal ();  
15.     void map ();  
16.     void graph ();  
17.     void plot ();  
18.     void plot1();  
19. };  
20. void arc::get ()  
21. {  
22.     cout<<"\n ENTER ANGLE OF LINE INCLINATION AND Y INTERCEPT";  
23.     cin>> ang >> b;  
24.     cout <<"\n ENTER NO OF VERTICES";  
25.     cin >> n;  
26.     cout <<"\n ENTER";  
27.     for (i=0; i<n; i++)  
28.     {  
29.         cout<<"\n x["<<i<<"] and y["<<i<<"]";  
30.     }  
31.     theta =(ang * pi)/ 180;  
32.     ref [0] [0] = cos (2 * theta);  
33.     ref [0] [1] = sin (2 * theta);  
34.     ref [0] [2] = -b *sin (2 * theta);  
35.     ref [1] [0] = sin (2 * theta);  
36.     ref [1] [1] = -cos (2 * theta);  
37.     ref [1] [2] = b * (cos (2 * theta)+1);  
38.     ref [2] [0]=0;  
39.     ref [2] [1]=0;  
40.     ref [2] [2] = 1;  
41. }  
42. void arc :: cal ()  
43. {  
44.     for (i=0; i < n; i++)  
45.     {  
46.         p[0] [i] = x [i];  
47.         p [1] [i] = y [i];  
48.         p [2] [i] = 1;  
49.     }  
50.     for (i=0; i<3;i++)  
51.     {  
52.         for (j=0; j<n; j++)  
53.         {  
54.             p1 [i] [j]=0;  
55.             for (k=0;k<3; k++)  
56.         }  
57.         p1 [i] [j] + = ref [i] [k] * p [k] [j];  
58.              }  
59. for (i=0; i<n; i++)  
60.    {  
61.     x1 [i]=p1[0] [i];  
62.     y1 [i] = p1 [1] [i];  
63.     }  
64. }  
65. void arc :: map ()  
66. {  
67.     int gd = DETECT,gm;  
68.     initgraph (&gd, &gm, " ");  
69.             int errorcode = graphresult ();  
70.     /* an error occurred */  
71.     if (errorcode ! = grOK)      
72.     {  
73.         printf ("Graphics error: %s \n", grapherrormsg (errorcode));  
74.         printf ("Press any key to halt:");  
75.         getch ();  
76.         exit (1); /* terminate with an error code */  
77.     }  
78. }  
79. void arc :: graph ()  
80. {  
81.     xm=getmaxx ()/2;  
82.     ym=getmaxy ()/2;  
83.     line (xm, 0, xmm 2*ym);  
84. }  
85. void arc :: plot 1 ()  
86. {  
87.     for (i=0; i <n-1; i++)  
88.     {  
89.         circle (x1[i]+xm, (-y1[i]+ym), 2);  
90.         line (x1[i]+xm, (-y1[i]+ym), x1[i+1]+xm, (-y1[i+1]+ym));  
91.     }  
92.         line (x1[n-1)+xm, (-y1[n-1]+ym), x1[0]+xm, (-y1[0]+ym));  
93.         getch();  
94. }  
95. void arc :: plot ()  
96. {   
97.     for (i=0; i <n-1; i++)  
98.     {  
99.         circle (x1[i]+xm, (-y1[i]+ym, 2);  
100.         line (x1[i]+xm, (-y1[i]+ym), x[i+1]+xm, (-y1[i+1]+ym));  
101.     }  
102.         line (x[n-1]+xm, (-y1[n-1]+ym), x[0]+xm, (-y[0]+ym));  
103.         getch();  
104. }  
105. void main ()  
106. {  
107.     class arc a;  
108.     clrscr();  
109.     a.map();  
110.     a.graph();  
111.     a.get();  
112.     a.cal();  
113.     a.plot();  
114.     a.plot1();  
115.     getch();  
116. }  

Output: