- Geometrie Computationala

Geometrie computationala

Tema 1

1.a.t=0:pi/100:2*pi;

x=cos(t);

y=sin(t);

plot(x,y)

1.b.t=0:.01:1;

x=1-2.*t+t.^2;

y=2-2.*t.^2;

plot(x,y)

1.c.t=0:.01:1;

x=(1-t).^2+4.*(1-t).*t-t.^2;

y=-2.*(1-t).^2+t.^2;

plot(x,y)

2.a.t=0:.01:1;

x=6.*t-9.*t.^2+4.*t.^3;

y=4.*t.^3-3.*t.^2;

plot(x,y)

2.b.t=0:.01:1;

x=6-18.*t+12.*t.^2;

y=12.*t.^2-6.*t;

plot(x,y)

2.c.x(t)=6*t-9*t^2+4*t^3

y(t)=4*t^3-3*t^2

t=1/2

x(1/2)=6*1/2-9*1/2^2+4*1/2^3= =

y(1/2)=4*1/2^3-3*1/2^2= =

3.a.t=-1:.01:1;

x=(1-t.^2)./(1+t.^2);

y=(2.*t)./(1+t.^2);

plot(x,y)

3.b.

-curba Bezier in forma de parabola

Tema 2

1.a.[r,h]=meshgrid(0:.01:2*pi,0:3);

x=sin(r);

y=cos(r);

z=h;

surf(x,y,z)

1.b.[x,y]=meshgrid(-2:.02:2,-2:.01:2);

z=x. exp(−(x.^2+y.^2));

surf(x,y,z)

mesh(x,y,z)

1.c.[u,v]=meshgrid(0:.01:2*pi,0:.01:pi)

x=cos(u).*cos(v);

y=cos(u).*sin(v);

z=sin(v);

surf(x,y,z)

mesh(x,y,z)

1.b.[x,y]=meshgrid(-2:.02:2,-2:.01:2);

z=x.*exp(-(x.^2+y.^2));

syms x y

ezsurf(x.*exp(-(x.^2+y.^2)))

1.c.[u,v]=meshgrid(0:.01:2*pi,0:.01:pi);

x=cos(u).*cos(v);

y=cos(u).*sin(v);

z=sin(v);

syms x y z

ezsurf('cos(u).*cos(v)','cos(u).*sin(v)','sin(v)')

3.a.[u,v]=meshgrid(-1:.01:1,0:.01:1);

x=2-2.*u;

y=2-2.*u.^2;

z=v;

surf(x,y,z)

=> suprafata Bezier

3.b[u,v]=meshgrid(-1:.01:1,0:.01:1)

x=2-2.*u;

y=2-2.*u.^2;

z=1/2.*v;

surf(x,y,z)

3.c.[u,v]=meshgrid(-1:.01:1,0:.01:1);

x=2-2.*u;

y=2-2.*u.^2;

z=v;

surf(x,y,z)

contour3(x,y,z)

Tema 3

1.a.

c(t)=( x(t)=1-2*t+t^3; y(t)=t-2*t^2+3*t^3)

C(0)=b0

C(1)=b3

C'(0)=3*(b1-b0)

C'(1)=3*(b3-b2)

C(0)=(1 0)=b0

C(1)=(0 2)=b3

C'(t)=( x'(t)=-2+3*t^2; y'(t)=1-4*t+9*t^2)

C'(0)=(-2 1)

(-2 1)=3*(b1-b0)

(-2 1)=3*b1-3*(1 0)

3*b1=(3 0)+(-2 1)

b1=(1/3 1/3)

C'1=(1 6)

(1 6)=3*(b3-b2)

(1 6)=3*(0 2)-3*b2

3*b2=(0 6)-(1 6)

3*b=(-1 0)

b2=(-1/3 0)

b0=(1 0) ; b1=(1/3 1/3) ; b2=(-1/3 0) ; b3=(0 2) ;

1.b. Se deduce din pozitia punctelor ca dreapta nu are puncte de inflexiune.

1.c.t=0:.01:1

x=1-2.*t+t.^3;

y=t-2.*t.^2+3.*t.^3;

plot(x,y)

hold on

2.a.

2.b.

3.a.

3.b.

3.c.

3.d.

Tema 4

%input

bx = ones(4,1)*[1 2/3 2/3 0]; % coord X,Y,Z ale supr

by = ones(4,1)*[0 1/3 4 2];

bz = [0 1 2 3]'*ones(1,4);

%computation

u = linspace (0,1,20); %parametrii suprafetei

v = linspace (0,1,20);

[X,Y,Z] = bezs3eval(bx,by,bz,u,v); % coord. suprafetei

%output

surf(X,Y,Z);

shading interp

colormap(winter)

camlight

%alpha(S)

pause

%hold on

%axis of

%plot3(bx,by,bz,'b')

%plot3(bx',by',bz','b')