شما می توانید با تغییر معادله صفحه زیر  در سط 4،نمودار  سه بعدی  رشد یافته روی مسیر را روی آن صفحه مشاهده کنین.

f(p,q)=2*p  +2*q;

مثلا  می توانین  معادله صفحه در سط 4 را به فرم  2+ f(p,q)=2*p  +0.5*q;  تغییر دهین و نتیجه را مشاهده کنین.

حتی می تونین، به جای صفحه  معادله یک رویه را وارد کنین، مثلا سطر 4 را با معادله رویه f(p,q)=2*p  +.5*q+2*p*q; جایگزین کنین. و نتیجه شکل زیر را مشاهده کنین.

کد اجرایی:

clc;
clear;
close all;
syms p q X Y Z  yd(X) yl(X) f(p,q) ft(p,q) T xp(T) yp(T) zp(T)
f(p,q)=2*p  +2*q;
[p1,q1]=meshgrid( 0:.1:1 ,-1:.1:1 );
 surf(p1,q1,double(f(p1,q1)),FaceColor='g');alpha(.2);hold on;
 

t=linspace(0,1,100);
x=t;
y=sin(2*pi*x);yprim=2*pi*cos(2*pi*x);
plot3(x,y,f(x,y),'-r');hold on
n=length(t);
t1=linspace(0,1,200);
r=.08;
x1=r*sin(2*pi*t1);
y1=r*cos(2*pi*t1);
 
xM=[];
yM=[];
zM=[];

for s=1:length(t) -1

    %plot equLine for  slop and its 90 slop lines

    %% 
%     yprims =double((y(s+1)- y(s))/(x(s+1)- x(s)));
    yd(X)=yprim(s)*(X-x(s))+y(s);
    yl(X)=(-1./yprim(s))*(X-x(s))+y(s);
    tr=[x(s),x(s)+.00005 ];
    d1=diff( double(tr));
    d2=diff( double(yd(tr)));
    d3=diff( double(f(tr,yd(tr))));

    rd=[d1,d2,d3];
    xp(T)=x(s)+rd(1)*T;
    yp(T)=y(s)+rd(2)*T;
    zp(T)=double(f(x(s),y(s)))+rd(3)*T;
    tr1=double(solve((xp-x(s))^2+(yp-y(s))^2-r^2,T));
%     plot3(xp(tr1),yp(tr1) ,zp(tr1),'b-' );hold on
   


    %%

    d1=diff( double(tr));
    d2=diff( double(yl(tr)));
    d3=diff( double(f(tr,yl(tr))));

    r2=[d1,d2,d3];
%     r2=[1,double(yl(x(s)+1))-double(yl(x(s))),double(f(x(s)+1,double(yl(x(s)+1))))-double(f(x(s),double(yl(x(s)))))];
    xl(T)=x(s)+r2(1)*T;
    yl(T)=y(s)+r2(2)*T;
    zl(T)=double(f(x(s),y(s)))+r2(3)*T;
    tr2=(double(solve((xl-x(s))^2+(yl-y(s))^2-r^2,T)));
%     plot3(double(xl(tr2)),double(yl(tr2)) ,double(zl(tr2)),'c-' );hold on
    df=(double((yprim(s))))/abs(double((yprim(s))));
    
    if  df==-1
    tr2=tr2(end:df:1);
    end
    %%
%     plot3(x1+x(s),y1+y(s) ,double(f(x1+x(s),y1+y(s))),'b-' );hold on

    %%
    r3=double([subs(gradient(f ),[p,q],[x(s),y(s)])',-1]);
    xr=double(xl(tr2));
    yr=double(yl(tr2));
    zr=double(zl(tr2));

    xr0(T)=xr(1)+r3(1)*T;
    yr0(T)=yr(1)+r3(2)*T;
    zr0(T)=zr(1)+r3(3)*T; 
    
    xr1(T)=xr(2)+r3(1)*T;
    yr1(T)=yr(2)+r3(2)*T;
    zr1(T)=zr(2)+r3(3)*T; 
    tr3=[-.02,.02];
%     plot3(double([xr0(tr3),xr1([tr3(2),tr3(1)]),xr0(tr3(1))]),double([yr0(tr3),yr1([tr3(2),tr3(1)]),yr0(tr3(1))]),double([zr0(tr3),zr1([tr3(2),tr3(1)]),zr0(tr3(1))]));hold on
%     plot3(xr0(tr3),yr0(tr3),zr0(tr3),'*')
   
    xM=[xM;double([xr0(tr3),xr1([tr3(2),tr3(1)]),xr0(tr3(1))])];
    yM=[yM;double([yr0(tr3),yr1([tr3(2),tr3(1)]),yr0(tr3(1))])];
    zM=[zM;double([zr0(tr3),zr1([tr3(2),tr3(1)]),zr0(tr3(1))])];


  % peripendecular plane
  % peripendecular passpoints
    rpX=diff(double(xl(tr2))');
    rpY=diff(double(yl(tr2))') ;
    rpZ=diff(double(zl(tr2)'));
    B=[rpX,rpY,rpZ] ;
    
    C=double([subs(gradient(f ),[p,q],[x(s),y(s)])',-1]);
    C=cross(B,C);
    refPlane=C(1)*(X-x(s ))+C(2)*(Y-y(s))+C(3)*(Z-f(x(s ),y(s))) ;

    verPlaneCoff='XYZ';
    [~,ind]=find(C);
    ind=ind(end);
    in=find(~ismember([1,2,3],ind));

    ft(p, q)=subs(refPlane-C(ind)*sym(verPlaneCoff(ind)),[sym(verPlaneCoff(in(1))),sym(verPlaneCoff(in(2)))],[p,q]);
    ft=-(ft/C(ind));

      plot3(double([xr0(tr3),xr1([tr3(2),tr3(1)]),xr0(tr3(1))]),double([yr0(tr3),yr1([tr3(2),tr3(1)]),yr0(tr3(1))]),double([zr0(tr3),zr1([tr3(2),tr3(1)]),zr0(tr3(1))]),'r');hold on


% surf(double([xr0(tr3);xr1(tr3)]),double([yr0(tr3);yr1(tr3)]),double([zr0(tr3);zr1(tr3)]),FaceColor='g');hold on
% surf(double([xr0(tr3);xr1(tr3)]),double([yr0(tr3);yr1(tr3)]),double(ft(double([xr0(tr3);xr1(tr3)]),double([yr0(tr3);yr1(tr3)]))),FaceColor='b');hold on
plot3(double([xr0(tr3);xr1(tr3)])',double([yr0(tr3);yr1(tr3)])',double(ft(double([xr0(tr3);xr1(tr3)]),double([yr0(tr3);yr1(tr3)])))','b');hold on
plot3(double([xr0(tr3);xr1(tr3)]),double([yr0(tr3);yr1(tr3)]),double(ft(double([xr0(tr3);xr1(tr3)]),double([yr0(tr3);yr1(tr3)]))),'b');hold on



    view(3)
    xlabel('X');
    ylabel('Y');
    zlabel('Z');
%      axis equal
     pause(.0001)
end

axis equal
plot3(xM ,yM ,zM )
h=surf(xM ,yM ,zM ,'FaceColor','g');alpha(h,.1)