Composite Transformation: Composite objects are several objects and scripts bundled together exposed to the application as a single object. There are number of transformations performed on any object. The object undergoes sequence of transformation in series and the points of every transformation become input to the next transformation.

For example, a line can be scaled in the beginning then it is translated to a new position and subsequently rotated about origin. Then such a transformation can be represented by_

x1’, y1’, 1 x1, y1, 1 [Tr1] [Tr2] [Tr3]

x2’, y2’, 1 x2, y2, 1

Where, [Tr1] = [S ] = scaling matrix [Tr2] = [T] = translation matrix [Tr3] = [R] = rotation matrix

In this transformation a line is drawn between the twopoints by using ‘line_draw’ function. • This line is scaled by using [b] matrix the scaling

matrix and the operation of multiplication of [a]*[b] is performed by ‘matmult’ and ‘change’ functions resubstitite [c] matrix into [a] matrix and then line is redrawn by using the same ‘line_draw’ function.

• Then, further the line is translated by using ‘trans’ function and the operation is performed similarly by ‘matmult’ and ‘change’ function. The ‘line_draw’ function draws the line.

• The line is further rotated by an angle of ’25 degree’

using ‘rotate’ function and similar way operation is performed by ‘matmult’ and ‘change’ function.

Composite Transformations:A composite transformation is a sequence oftransformations, one followed by the other. Considerthe matrices and transformations in the following list:

Matrix A    Rotate by an angle Matrix B    Scale by a factor of sx and sy Matrix C    Translate in the x,y direction

FLOWCHART: TO DRAW COMPOSIT OBJECT. Start

float pi=22.0/7.0,l=3,m=2,n=3,a[2][3]={{115,80,1}{140,140,1}},b[20,20],c[20][20];float i,j,k,sum=0.0;double x,y,h=50,v=100,sx=1,sy=2;

float theta=25;double atheta=pi*theta/180.0;

line(a[0][0],a[0][1],a[1][0],a[1][1]);

outtextxy(400,300,”press any key to scale”);

b[0][0]=sx;b[0][1]=0.0;b[0][2]=0.0;b[1][0]=0.0;b[1][1]=sy;b[1][2]=0.0;b[2][0]=0.0;b[2][1]=0.0;b[2][2]=1.0;getch();

for(i=0.0;i<m;++i)

for(k=0;k<l;++k)

for(j=0;j<n;++j)

sum+=(a[i][j]*b[j][k]);

false

true

true

c[i][k]=sum;

false

true

false

Continue…

for(i=0.0;i<m;++i)

for(k=0.0;k<l;++k)

a[i][k]=c[i][k];

false

true

true

false

line(a[0][0],a[0][1],a[1][0],a[1][1]);

outtextxy(400,360,”press any key to translate”);

b[0][0]=1.0;b[0][1]=0.0;b[0][2]=0.0;b[1][0]=0.0;b[1][1]=1.0;b[1][2]=0.0;b[2][0]=h;b[2][1]=v;b[2][2]=1.0;getch();

for(i=0.0;i<m;++i)

for(k=0;k<l;++k)

c[i][k]=sum;

for(j=0;j<n;++j)

sum+=(a[i][j]*b[j][k]);

false

false

false

true

true

true

for(i=0.0;i<m;++i)

for(k=0.0;k<l;++k)

a[i][k]=c[i][k];

false

true

true

false

line(a[0][0],a[0][1],a[1][0],a[1][1]);

outtextxy(400,370,”press any key to rotate”);

x=sin(atheta); y=cos(atheta);

b[0][0]=y;b[0][1]=x;b[0][2]=0.0; b[1][0]=-x;b[1][1]=y;b[1][2]=0.0; b[2][0]=0.0;b[2][1]=0.0;b[2][2]=1.0; getch();

for(i=0.0;i<m;++i)

for(k=0;k<l;++k)

c[i][k]=sum;

for(j=0;j<n;++j)

sum+=(a[i][j]*b[j][k]);

false

false

false

true

true

true

Continue…

for(i=0.0;i<m;++i)

for(k=0.0;k<l;++k)

a[i][k]=c[i][k];

false

true

true

false

line(a[0][0],a[0][1],a[1][0],a[1][1]);

outtextxy(400,380,”press any key to close”);

Output

stop

Continue…

Explanation: Start of program

Define data type, initialize variables

Draw a line according to coordinates

Print this message on given position

a and b are two matrix and c is resultant matrix,h and v are translating factors corresponding to x and y direction increments,sx and sy are scaling factors in x and y direction,theta and atheta are rotation angle

Define Scaling matrix

matrix a and b is multiplied and c holds result

Change the position of line and the value of c is assigned to a

Draw a scaled line ongiven position

Print this message on given position

Define translation matrix

matrix a and b is multiplied and c holds result

connector

Change the position of line and the value of c is assigned to a

Draw a translated line ongiven position

Print this message on given position

x and y are rotation angle

Define rotation matrix

matrix a and b is multiplied and c holds result

Change the position of line and the value of c is assigned to a

Draw a rotated line ongiven position

Print this message on given position

Display a translated, scaled and rotated line on screen.

End of program

Continue…

Program:

/*composite transformation*//*program to scaling,translation,rotate a line about origin*/

//include header files#include<graphics.h>#include<conio.h>#include<math.h>#define pi(22.0/7.0)

int driver,mode; /*data type defined that must be integer type variable*/

/*define variables that is used in program*/float pi=22.0/7.0; float l=3,m=2,n=3; float a[2][3]={{115,80,1},{140,140,1}};float b[20][20],c[20][20];double x,y,h,v,sx,sy;

/*define prototype of functions*/void line_draw(void);void scale(void);void matmult(void);void change(void);void trans(void);void rotate(void);

//main programvoid main(){ float theta; double atheta;

detectgraph(&driver,&mode); /*driver and modes are set*/ initgraph(&driver,&mode,"\\BGI"); /*graphic mode is initialized*/

line_draw(); //calling function call to line_draw function outtextxy(400,350,"press any key to scale"); sx=1,sy=2; /*scaling factor in x,y direction*/

//calling functing scale(); matmult(); change(); line_draw(); outtextxy(400,360,"press any key to translate"); h=50;v=100; /*h and v are x and y direction

increments*/

//calling functions trans(); matmult(); change(); line_draw(); outtextxy(400,370,"press any key to rotate");

/*rotation angle*/ theta=25; atheta=pi*theta/180.0;

x=sin(atheta); y=cos(atheta);

//calling functions rotate(); matmult(); change(); line_draw(); outtextxy(400,380,"press any key to close"); getch(); /*display output*/restorecrtmode(); /*restore the screen to the mode */} /*closing braces*/

/*line function, to draw aline*/void line_draw() { line(a[0][0],a[0][1],a[1][0],a[1][1]);}

/*scale function, to increase the size of line */void scale(){ b[0][0]=sx;b[0][1]=0.0;b[0][2]=0.0; b[1][0]=0.0;b[1][1]=sy;b[1][2]=0.0; b[2][0]=0.0;b[2][1]=0.0;b[2][2]=1.0; getch();}

/*translate function, to change the position of line in x and y direction*/void trans(){ b[0][0]=1.0;b[0][1]=0.0;b[0][2]=0.0; b[1][0]=0.0;b[1][1]=1.0;b[1][2]=0.0; b[2][0]=h;b[2][1]=v;b[2][2]=1.0; getch();}

/*rotate function, to rotate a line by an angle*/void rotate(){ b[0][0]=y;b[0][1]=x;b[0][2]=0.0; b[1][0]=-x;b[1][1]=y;b[1][2]=0.0; b[2][0]=0.0;b[2][1]=0.0;b[2][2]=1.0; getch();}

/*matrix multiplication function*/void matmult(){ float i,j,k,sum; for(i=0.0;i<m;++i) { for(k=0;k<l;++k) {

sum=0.0; for(j=0;j<n;++j) { sum+=(a[i][j]*b[j][k]); }

c[i][k]=sum; } }}

/*change function, to change the position of line*/void change(){ float i,k; for(i=0.0;i<m;++i) { for(k=0.0;k<l;++k) {

a[i][k]=c[i][k]; } }}