Reflection:

It 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:

Reflection about the x-axis
Reflection about the y-axis
Reflection about an axis perpendicular to xy plane and passing through the origin
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.

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:

  #include <iostream.h>  #include <conio.h>  #include <graphics.h>  #include <math.h>  #include <stdlib.h>  #define pi 3.14  class arc  {  float x[10],y[10],theta,ref[10][10],ang;             float p[10][10],p1[10][10],x1[10],y1[10],xm,ym;  int i,k,j,n;  public:  void get();  void cal ();  void map ();  void graph ();  void plot ();  void plot1();  };  void arc::get ()  {  cout<<“n ENTER ANGLE OF LINE INCLINATION AND Y INTERCEPT”;  cin>> ang >> b;  cout <<“n ENTER NO OF VERTICES”;  cin >> n;  cout <<“n ENTER”;  for (i=0; i<n; i++)  {  cout<<“n x[“<<i<<“] and y[“<<i<<“]”;  }  theta =(ang * pi)/ 180;  ref [0] [0] = cos (2 * theta);  ref [0] [1] = sin (2 * theta);  ref [0] [2] = -b *sin (2 * theta);  ref [1] [0] = sin (2 * theta);  ref [1] [1] = -cos (2 * theta);  ref [1] [2] = b * (cos (2 * theta)+1);  ref [2] [0]=0;  ref [2] [1]=0;  ref [2] [2] = 1;  }  void arc :: cal ()  {  for (i=0; i < n; i++)  {  p[0] [i] = x [i];  p [1] [i] = y [i];  p [2] [i] = 1;  }  for (i=0; i<3;i++)  {  for (j=0; j<n; j++)  {  p1 [i] [j]=0;  for (k=0;k<3; k++)  }  p1 [i] [j] + = ref [i] [k] * p [k] [j];               }  for (i=0; i<n; i++)     {  x1 [i]=p1[0] [i];  y1 [i] = p1 [1] [i];      }  }  void arc :: map ()  {  int gd = DETECT,gm;  initgraph (&gd, &gm, ” “);              int errorcode = graphresult ();  /* an error occurred */  if (errorcode ! = grOK)      {   printf (“Graphics error: %s n”, grapherrormsg (errorcode));  printf (“Press any key to halt:”);  getch ();  exit (1); /* terminate with an error code */  }  }  void arc :: graph ()  {  xm=getmaxx ()/2;  ym=getmaxy ()/2;  line (xm, 0, xmm 2*ym);  }  void arc :: plot 1 ()  {  for (i=0; i <n-1; i++)  {  circle (x1[i]+xm, (-y1[i]+ym), 2);  line (x1[i]+xm, (-y1[i]+ym), x1[i+1]+xm, (-y1[i+1]+ym));  }  line (x1[n-1)+xm, (-y1[n-1]+ym), x1[0]+xm, (-y1[0]+ym));  getch();  }  void arc :: plot ()  {   for (i=0; i <n-1; i++)  {  circle (x1[i]+xm, (-y1[i]+ym, 2);  line (x1[i]+xm, (-y1[i]+ym), x[i+1]+xm, (-y1[i+1]+ym));  }  line (x[n-1]+xm, (-y1[n-1]+ym), x[0]+xm, (-y[0]+ym));  getch();  }  void main ()  {  class arc a;  clrscr();  a.map();  a.graph();  a.get();  a.cal();  a.plot();  a.plot1();  getch();  }  

Output:

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Computer Graphics Reflection

Reflection:

Types of Reflection:

Program to perform Mirror Reflection about a line:

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