Intersection of Surfaces

Intersection of Surfaces

1)    Plot the surfaces defined by the cylinder z² + x²=8 and the paraboloid

z=0.2(x² +y² ) and show their intersection.

//program for plotting cylinder

deff('[z]=f1(x,y)','z=sqrt(8-x^2)') //define the function z=f1(x,y)

x= [-2.9:0.05:2.9]; y = x;                                  //Populate the vectors x and y.

z=feval(x,y,f1);                        //evaluate matrix z as per f1(x,y) for every value of x & y

surf(x,y,z','edgecolor',[0 0 1],'facecolor',[0 0 1]); //plot z as a surface to show the upper

//half of the cylinder

surf(x,y,-z','edgecolor',[0 0 1],'facecolor',[0 0 1]); //plot -z as a surface to show the lower

//half of the cylinder

//program for plotting paraboloid

deff('[z]=f2(x,y)','z=(x^2+y^2)/5')                   //define the function z=f2(x,y)

x= [-4:0.05:4]; y = [-4:0.05:4];                         //Populate the vectors x and y.

z=feval(x,y,f2);                        //evaluate matrix z as per f2(x,y) for every value of x & y

surf(x,y,z','edgecolor',[1 0 1],'facecolor',[1 0 1]); //plot z as a surface to show the           //paraboloid

OUTPUT:

2) Plot the surfaces defined by the cone (6-z)² = x² +y²and the plane

z=0 and show their intersection.

//Plotting of cone

deff('z=f1(x,y)','z=6-sqrt(x^2+y^2)')    //define the function z=f1(x,y) as a conical surface

x=-6:0.1:6 ; y=x ;   //populate the vectors x and y

fplot3d(x,y,f1)             //plot 3D graph of function z=f1(x,y) to show the conical surface

//Plotting of plane

deff('z=f2(x,y)','z=0')                           //define the function z=f2(x,y) as a plane surface

x=-6:0.1:6 ; y=x ;                                 //populate the vectors x and y

e=gce()                                                //get the handle of graphics

e.color_mode =23;                             //set the color of the plot

fplot3d(x,y,f2)             //plot 3D graph of function z=f2(x,y) to show the plane surface

OUTPUT:

3) Plot the surfaces defined by the cone (6-z)² = x² +y²and the plane

z=1 and show their intersection.

//Intersection of a conical surface with a plane to form a circle

//plot the conical surface

deff('z=f1(x,y)','z=6-sqrt(x^2+y^2)') //define the function z=f1(x,y) as a conical surface

x=-6:0.1:6 ;y=x ;                                  //populate the vectors x and y

fplot3d(x,y,f1) //plot 3D graph of function z=f1(x,y) to show the conical surface

//plot the plane

deff('z=f2(x,y)','z=1') //define the function z=f2(x,y) as a plane surface

x=-6:0.1:6 ;y=x ;                                  //populate the vectors x and y

e=gce()                                                //get the handle of graphics

e.color_mode =23;                             //set the color of the plot

fplot3d(x,y,f2) //plot 3D graph of function z=f2(x,y) to show the plane surface

OUTPUT:

4) Plot the surfaces defined by the cone (8-z)² = x² +y²and the plane

z=0.1x+0.3y+4 and show their intersection.

//Intersection of a conical surface and a plane to form an ellipse

//plot the conical surface

deff('z=f1(x,y)','z=8-sqrt(x^2+y^2)') //define the function z=f1(x,y) as a conical surface

x=[-6:0.1:6];y=x ;                                 //populate the vectors x and y

z=feval(x,y,f1);                        //evaluate matrix z as per f1(x,y) for every value of x & y

surf(x,y,z','edgecolor',[1 0 1],'facecolor',[1 0 1]);       //plot z as a surface to show the

//conical surface

//plot the plane

deff('z=f2(x,y)','z=.1*x+.3*y+4')          //define the function z=f2(x,y) as a plane surface

x=[-6:0.1:6];y=x ;                                 //populate the vectors x and y

e=gce()                                                //get the handle of graphics

e.color_mode =16;                             //set the color of the plot

fplot3d(x,y,f2); //plot 3D graph of function z=f2(x,y) to show the plane surface

OUTPUT:

5) Plot the surfaces defined by the cone (8-z)² = x² +y²and the plane

z=0.5(x+y) and show their intersection.

//Intersection of a conical surface with a plane to form a hyperbola

//Plotting of cone

deff('z=f1(x,y)','z=8-sqrt(x^2+y^2)')    // define the function z=f1(x,y) as a conical surface

x=[-6:0.1:6];y=x ;                                 //populate the vectors x and y

z=feval(x,y,f1); //evaluate matrix z as per f1(x,y) for every value of x & y

surf(x,y,z','edgecolor',[1 0 1],'facecolor',[1 0 1]);       //plot z as a surface to show the

//conical surface

//Plotting of plane

deff('z=f2(x,y)','z=.5*(x+y)')                // define the function z=f2(x,y) as a plane surface

x=[-6:0.1:6];y=x ;                                 //populate the vectors x and y

e=gce()                                                //get the handle of graphics

e.color_mode =23;                             //set the color of the plot

fplot3d(x,y,f2); //plot 3D graph of function z=f2(x,y) to show the plane surface

OUTPUT:

6) Plot the surfaces defined by the cylinder z² = x² - 8and the plane

z=0.2(x+y) and show their intersection.

//program for plotting cylinder

deff('[z]=f1(x,y)','z=sqrt(8-x^2)') // define the function z=f1(x,y) as a cylidrical surface

x= [-2.9:0.05:2.9]; y = x;          //populate the vectors x and y

z=feval(x,y,f1); //evaluate matrix z as per f1(x,y) for every value of x & y

surf(x,y,z','edgecolor',[0 0 1],'facecolor',[0 0 1]);       //plot z to show the upper half of the

//cylindrical surface

surf(x,y,-z','edgecolor',[0 0 1],'facecolor',[0 0 1]);//plot -z to show the lower half of the

//cylindrical surface

//program for plotting plane

deff('[z]=f2(x,y)','z=(x+y)/5')               // define the function z=f2(x,y) as a plane surface

x= [-4:0.05:4]; y = x;                            //populate the vectors x and y

z=feval(x,y,f2);                        //evaluate matrix z as per f2(x,y) for every value of x & y

surf(x,y,z','edgecolor',[1 0 1],'facecolor',[1 0 1]);       //plot z as a surface to show the

//plane surface

OUTPUT:

7) Plot the surfaces defined by the paraboloid z = 2(x²+y²) and the plane

z=6 and show their intersection.

//Plotting of paraboloid and sphere

deff('z=f1(x,y)','z=(x^2+y^2)/.5') // define the function z=f1(x,y) as a parabolic surface deff('z=f2(x,y)','z=6') // define the function z=f2(x,y) as a plane surface

x=-2:0.05:2 ;y=x ; //populate the vectors x and y

fplot3d(x,y,f1)//plot 3D graph of function z=f1(x,y) to show the parabolic surface

e=gce()                                    //get the handle of graphics

e.color_mode =23;                 //set the color of the plot

fplot3d(x,y,f2) //plot 3D graph of function z=f2(x,y) to show the plane surface

OUTPUT:

8) Plot the surfaces defined by z = 4sinx and z=5cosx and show their intersection.

deff('z=f1(x,y)','z=4*sin(x)') // define the function z=f1(x,y) as a sinusoidal surface

deff('z=f2(x,y)','z=5*cos(x)') // define the function z=f2(x,y) as a cosinusoidal surface

x=0:0.1:3.5 ;y=x ;                     //populate the vectors x and y

fplot3d(x,y,f1) //plot 3D graph of function z=f1(x,y) to show the sinusoidal surface

e=gce()                        //get the handle of graphics

e.color_mode =23; //set the color of the plot

fplot3d(x,y,f2) //plot 3D graph of function z=f2(x,y) to show the cosinusoidal surface

OUTPUT:

9) Plot the surfaces defined by the paraboloid z = 0.5(x²+y²) and the sphere x²+y²+z²=4 and show their intersection.

//Plotting of paraboloid and sphere

deff('z=f1(x,y)','z=(x^2+y^2)/2') // define the function z=f1(x,y) as a parabolic surface

deff('z=f2(x,y)','z=sqrt(-x^2-y^2+4)')// define the function z=f1(x,y) as a spherical surface

x=-2:0.1:2 ;y=x ;                                  //populate the vectors x and y

fplot3d(x,y,f1)//plot 3D graph of function z=f1(x,y) to show the parabolic surface

e=gce()                                                //get the handle of graphics

e.color_mode =23;                             //set the color of the plot

fplot3d(x,y,f2)//plot 3D graph of function z=f1(x,y) to show the spherical surface

OUTPUT: