1. 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 two points 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.

2. 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 of transformations, one followed by the other. Consider the 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

3. 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 c[i][k]=sum; false true false

4. Continue… for(i=0.0;i<m;++i) for(k=0.0;k<l;++k) a[i][k]=c[i][k]; false 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 true

5. for(i=0.0;i<m;++i) for(k=0.0;k<l;++k) a[i][k]=c[i][k]; false 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 true Continue…

6. for(i=0.0;i<m;++i) for(k=0.0;k<l;++k) a[i][k]=c[i][k]; false 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…

7. 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 on given position Print this message on given position Define translation matrix matrix a and b is multiplied and c holds result

8. connector Change the position of line and the value of c is assigned to a Draw a translated line on given 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 on given position Print this message on given position Display a translated, scaled and rotated line on screen. End of program Continue…

9. Program: /*composite transformation*/ /*program to scaling,translation,rotate a line about origin*/ //include header files #include #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 program void 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*/

10. //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]); }

11. /*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]); }

12. 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]; }