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: