Proiect Matlab

transcript

Universitatea “Vasile Alecsandri” din Bacau

Facultatea de Stiinte

Specializarea Informatica, an II, grupa 421

Matrix Laboratory Geometrie computationala

Proiect MATLAB

Realizat de: HALÎNGA VICTORIA

Îndrumator: COSMIN ION TOMOZEI

Anul 2010

Geometrie computationala

Capitolul I

Calcule matematice

1. >> (1+sqrt(5))/2

1.6180

Exemplu personal 2. >> (10-sqrt(49))*2

Exemplu personal 3. >> (3-sqrt(49))*2

Calcule simple in MATLAB. Vectori si matrice.

1. >> a =5; >> b=12;

>> a+b

Exemplu personal

2. >> a=32; >> b=8;

>> (a+b)-b

Exemplu personal

3. >> a1=5; >> a2=18;

>> a2-a1

4. >> x=sin(2)

0.9093

5. >> y=3*exp(x)+cos(x); -- daca folosim semnul ; dupa tastarea expresiei nu se va afisa

>> -- variabila de iesire

6. >> y=3*exp(x)+cos(x)

8.0620

Exemplu personal

7. >> x=sin(pi)

1.2246e-016

Exemplu personal

8. >> x=sin(pi/2)

Exemplu personal

9. >> cos(pi/2)

6.1232e-017

Exemplu personal

10. >> cos(2*pi)

11. >> x=sin(22), y=exp(3)

-0.0089

20.0855

Exemplu personal

12. >> x=sin(1), y=cos(1)

0.8415

0.5403

13. >> (-1)^0.25

0.7071 + 0.7071i

14. >> 2-3i

2.0000 - 3.0000i

Exemplu personal

15. >> 12-4i

12.0000 - 4.0000i

Exemplu personal

16. >> 1^0.25

Exemplu personal

17. >> (-1)^0.12

0.9298 + 0.3681i

18. >> x=[1 2 3 4 5] ; -- pentru memorarea unui sir de numere

{1,2,3,4,5} intr-un vector

19. >> A=[2 -1 ;0 3] -- afisarea elementelor unei matrici A

20. >> x=[1 2 3 4 5] ; >> x(4)

21. >> A=[2 -1 ;0 3] ; >> A(1,2)

22. >> A=[2 -1 ;0 3] , A(1,2)

23. >> linspace(0, 1, 7)

0 0.1667 0.3333 0.5000 0.6667

0.8333 1.0000

Exemplu personal

24. >> linspace(-2, 2, 5)

-2 -1 0 1 2

Exemplu personal

25. >> x=[1 2 3 4 5] ; -- lungimea unui vector

>> length(x)

Exemplu personal

26. >> A=[2 -1 ;0 3] ; -- dimensiunea unei matrici

>> size(A)

27. >> A=[1 2 3 ; 3 4 -1;2 1 1;4 3 1]; >> A(:,2)

28. >> A=[1 2 3 ; 3 4 -1;2 1 1;4 3 1], A(1,:)

3 4 -1

29. >> A=[1 2 3 ; 3 4 -1;2 1 1;4 3 1], A(:,3)

3 4 -1

Exemplu personal

30. >> A=[1 2 3 ; 3 4 -1;2 1 1;4 3 1], A(:)

3 4 -1

31. >> A = [1 2 3;4 5 6;7 8 9]; >> B = [1 1 1;2 2 2;3 3 3];

>> C = [1 2;3 4;5 6];

>> A+B

10 11 12

>> A+C

??? Error using ==> plus

Matrix dimensions must agree.

32. >> A=[1 2 3;3 5 6; 34 5 6]

34 5 6

>> C*A

??? Error using ==> mtimes

Inner matrix dimensions must agree.

>> A*C

79 124

33. >> v = [0:2:8]

0 2 4 6 8

>> A = [ 1 2 3; 3 4 5; 6 7 8]

- 10 -

>> A*v(1:3)

>> v(1:3)

>> A*v(1:3)'

>> v(1:3)'

Exemplu personal

34. >> A=[2 4 6; 2 4 6;1 3 6]

>> B=[1 6 3; 6 4 2; 2 2 2]

>> C=[1 1 1; 2 2 2]

- 11 -

>> A+B

3 10 9

>> B+A

3 10 9

>> A*B

38 40 26

31 30 21

>> B*A

17 37 60

22 46 72

10 22 36

>> A*C

- 12 -

>> C*A

5 11 18

10 22 36

>> B*C

>> C*B

9 12 7

18 24 14

- f(x) = x

- g(x) = x

35. >> x=linspace(-1,1,20); >> f=1./(1+x.^2).*exp(x-1); >> x=linspace(0,2*pi,20);

>> g=cos(x).^2.*exp(x);

>> x=linspace(-1,1,20);

>> f=1./(1+x.^2).*exp(x-1)

Columns 1 through 7

0.0677 0.0835 0.1029 0.1264 0.1544 0.1871

0.2241

Columns 8 through 14

0.2644 0.3065 0.3481 0.3867 0.4203 0.4476

0.4682

- 13 -

0.4825 0.4916 0.4967 0.4991 0.4999 0.5000

>> x=linspace(0,2*pi,20);

>> g=cos(x).^2.*exp(x)

Columns 1 through 7

1.0000 1.2452 1.2066 0.8068 0.2262 0.0356

1.1736

4.6437 10.8992 19.0827 26.5618 29.3934 24.2639

11.8805

0.6989 8.5967 59.4004 172.1172 344.1507 535.4917

36. >> u=[2 3 4 1.2 5 -1]; -- inmultirea a doi vectori linie de aceeasi dimensiune

>> v=[-1 3 4 -3 0 1];

>> u.*v

-2.0000 9.0000 16.0000 -3.6000 0 -1.0000

>> u*v

- 14 -

37. >> u*v' -- produsul scalar a vectorilor u si v

18.4000

>> C=[1 3;4 67;12 45]

>> C^2

??? Error using ==> mpower

Matrix must be square.

38. >> C=[1 2 3; 1 2 3; 1 2 3]

>> C^2 --ridicarea la putere a unei matrici

6 12 18

39. >> A=[2 -1 ;0 3] ;

>> A^-1 -- inversa unei matrici

0.5000 0.1667

0 0.3333

- 15 -

40. >> A=[1 2 3 ; 3 4 -1;2 1 1;4 3 1]; >> A^-1

41. >> A=[1 2 3;3 5 6; 34 5 6]; >> A^-1

0.0000 -0.0323 0.0323

-2.0000 1.0323 -0.0323

1.6667 -0.6774 0.0108

42. >> A=[1 2 3;3 5 6; 34 5 6]; >> B=[1 5 7];

>> X=A/B

0.4267

0.9333

1.3467

43. >> A=[16 3 2 13; 5 10 11 8; 9 6 7 12; 4 15 14 1];

>> X=inv(A) -- inversa unei matrici

Warning: Matrix is close to singular or badly scaled.

Results may be inaccurate. RCOND = 9.796086e-018.

1.0e+015 *

0.1251 0.3753 -0.3753 -0.1251

-0.3753 -1.1259 1.1259 0.3753

0.3753 1.1259 -1.1259 -0.3753

-0.1251 -0.3753 0.3753 0.1251

44. >> x = (-5:.1:5); >> y = x./(1+x.^2);

>> y = x./(1+x.^2)

- 16 -

Columns 1 through 7

-0.1923 -0.1959 -0.1997 -0.2036 -0.2076 -0.2118

-0.2161

-0.2206 -0.2253 -0.2302 -0.2353 -0.2406 -0.2461

-0.2519

-0.2579 -0.2642 -0.2707 -0.2775 -0.2847 -0.2922

-0.3000

-0.3082 -0.3167 -0.3257 -0.3351 -0.3448 -0.3550

-0.3657

-0.3767 -0.3882 -0.4000 -0.4121 -0.4245 -0.4370

-0.4494

-0.4615 -0.4730 -0.4833 -0.4918 -0.4977 -0.5000

-0.4972

-0.4878 -0.4698 -0.4412 -0.4000 -0.3448 -0.2752

-0.1923

-0.0990 0 0.0990 0.1923 0.2752 0.3448

0.4000

0.4412 0.4698 0.4878 0.4972 0.5000 0.4977

0.4918

- 17 -

0.4833 0.4730 0.4615 0.4494 0.4370 0.4245

0.4121

0.4000 0.3882 0.3767 0.3657 0.3550 0.3448

0.3351

0.3257 0.3167 0.3082 0.3000 0.2922 0.2847

0.2775

0.2707 0.2642 0.2579 0.2519 0.2461 0.2406

0.2353

0.2302 0.2253 0.2206 0.2161 0.2118 0.2076

0.2036

0.1997 0.1959 0.1923

45. >> x=1:.5:4; >> y=1:7;

>> x./y

1.0000 0.7500 0.6667 0.6250 0.6000 0.5833

0.5714

46. >> x=1:.5:4; >> y=1:6;

>> x./y

??? Error using ==> rdivide

Matrix dimensions must agree

- 18 -

47. >> A=[1 2 3; 45 6 7; 12 3 4;1 1 1]

45 6 7

12 3 4

>> A.^2

2025 36 49

144 9 16

>> A^2

48. >> A=[1 2 3; 45 6 7; 12 3 4]; >> B=A/4

0.2500 0.5000 0.7500

11.2500 1.5000 1.7500

3.0000 0.7500 1.0000

49. >> A=[1 2 3; 45 6 7; 12 3 4]

45 6 7

12 3 4

- 19 -

>> A' -- transpusa unei matrici

1 45 12

Exemplu personal

50. >> x=0:4; >> y=5:5:25;

>> A=[x', y']

Exemplu personal

51. >> B=zeros(2,2)

Exemplu personal

52. >> B=ones(2,3)

Exemplu personal

53. >> B=eye(3)

- 20 -

Exemplu personal

54. >> x=0:3; >> B=diag(x)

0 0 0 0

0 1 0 0

0 0 2 0

0 0 0 3

Exemplu personal

55. >> A=[1 2 3; 45 6 7; 12 3 4;1 1 1]

45 6 7

12 3 4

>> x=max(A)

45 6 7

>> x=min(A)

>> rank(A)

>> B=abs(A)

45 6 7

12 3 4

- 21 -

Exemplu personal

56. >> zeros(1,2)

>> zeros(2,1)

>> ones(1,2)

>> ones(2,1)

57. >> B = [ [1 2 3]' [2 4 7]' [3 5 8]']

58. >> A=[1 2 3; 45 6 7; 12 3 4]; >> B=[1 2 3; 2 4 7; 3 5 8];

>> C=A.*B

90 24 49

36 15 32

- 22 -

>> C=A./B

1.0000 1.0000 1.0000

22.5000 1.5000 1.0000

4.0000 0.6000 0.5000

59. >> sqrt([1 2 4 6])

1.0000 1.4142 2.0000 2.4495

60. >> sin([pi/4 pi/2 pi])

0.7071 1.0000 0.0000

61. >> x=linspace(0, 1, 7)

0 0.1667 0.3333 0.5000 0.6667 0.8333

1.0000

Exemplu personal

62. >> x=0:4; >> min(x)

>> max(x)

- 23 -

Exemplu personal

63. >> x=[4, 3 , 7, 1]

4 3 7 1

>> min(x)

>> max(x)

>> sort(x)

1 3 4 7

>> sum(x)

64. >> x=linspace(0, 1, 7)

0 0.1667 0.3333 0.5000 0.6667 0.8333

1.0000

>> [min(x) max(x)]

- 24 -

>> sort(x)

0 0.1667 0.3333 0.5000 0.6667 0.8333

1.0000

65. >> sum(x)

3.5000

>> rand

0.8147

>> rand(3)

0.9058 0.6324 0.5469

0.1270 0.0975 0.9575

0.9134 0.2785 0.9649

Exemplu personal

66. >> A=[1 2 3; 45 6 7; 12 3 4]; >> triu(A)

>> tril(A)

45 6 0

12 3 4

- 25 -

>> min(A)

>> max(A)

45 6 7

>> sum(A)

58 11 14

67. >> x=sin(1)+sin(2)+... sin(3)+sin(4)

1.1351

68. >> x=[1 2 3]

>> y=[1 0 0]

>> x>0 & y>0

69. >> x=[1 2 3]; >> y=[1 0 0];

- 26 -

>> x>0 |y>0

Reprezentarea grafica a curbelor si suprafetelor

Reprezentarea 2D

1. >> y = (-5:0.1:4).^3; >> plot(y);

2. >> x = -5:0.1:4; >> y=x.^3;

>> plot(x, y);

>> title('graficul functiei x^3')

- 27 -

Exemplu personal

3. >> x=pi:.5:2*pi; >> y=sin(x);

>> z=cos(x);

>> plot(x,y, '*g')

Exemplu personal

4. >> x=pi:.1:2*pi; >> y=sin(x);

>> z=cos(x);

>> plot(x,y, '*g')

>> title('Graficul functiilor trigonometrice')

>> xlabel('x')

>> ylabel('y')

- 28 -

5. >> x=[-pi:.3:pi]; >> y=sin(x);

>> z=cos(x);

>> plot(x,y,'-r');

>> hold on

>> plot(x,z,'--g');

>> hold on

>> plot(x,0,'*-b')

>> legend ('-sin(x)','-cos(x)','y=0');

- 29 -

Exemplu personal

6. >> x=[-pi./2:.2:pi./2]; >> y=sin(x);

>> z=cos(x);

>> plot(x,y,'--g');

>> hold on

>> plot(x,z,'-r');

>> hold on

>> plot(y,z,'*b')

>> legend ('-sin(x)','-cos(x)','x=0')

7. >> x=[-2:.2:2]; >> y=exp(-x.^2);

>> subplot (2,2,1);

>> plot (x,y);

>> title ('exp(-x^2)');

>> z=exp(-x.^2).*cos(2*pi*x);

>> subplot (2,2,2);

>> plot (x,z);

>> title ('exp(-x^2)*cos(2*pi*x)');

>> t=exp(-x.^2).*cos(4*pi*x);

>> subplot (2,2,3);

>> plot (x,t);

>> title ('exp(-x^2)*cos(4*pi*x)');

- 30 -

Exemplu personal

8. >> x=[-3:.2:3]; >> y=exp(x.^2);

>> subplot (3,3,1);

>> plot (x,y);

>> title ('exp(x^2)');

>> subplot (3,3,2);

>> z=exp(-x.^2);

>> plot(x,z);

>> title ('exp(-x^2)');

>> z=exp(x.^2).*cos(3*pi*x);

>> subplot(3,3,3);

>> plot(x,z);

>> title ('exp(x^2)*cos(3*pi*x)');

9. >> fplot ('exp(-t^2)',[-2 2]) >> hold on

>> fplot('2*exp(-t^2)',[-2 2])

>> fplot('3*exp(-t^2)',[-2 2])

>> hold off

- 31 -

Exemplu personal

10. >> fplot ('exp(-t^2)',[-3 3]) >> hold on

>> fplot ('3*exp(-t^2)',[-3 3])

>> fplot ('4*exp(-t^2)',[-3 3])

>> hold off

- 32 -

Reprezentarea 3D. Suprafete.

11. >> x=[-6:.5:6]; >> y=x;

>> [X,Y]=meshgrid(x);

>> Z=sqrt(X.^2+Y.^2);

>> mesh(X,Y,Z)

Exemplu personal

12. >> x=[-8:.5:8]; >> y=x;

>> [X,Y]=meshgrid(x);

>> Z=sqrt(X.^2+Y.^2);

>> surf(X,Y,Z)

- 33 -

13. >> [x,y] = meshgrid([-2:.2:2]); >> Z=x.^2*exp(-x.^2-y.^2);

>> mesh(x,y,Z);

>> colormap cool;

Exemplu personal

14. >> [x,y] = meshgrid([-3:.2:3]); >> Z=x.^2*exp(-2.*x.^2-y.^2);

>> mesh(x,y,Z);

>> colormap spring;

- 34 -

15. >> Z=peaks(30); >> subplot(2,2,1);

>> surf(Z);

>> shading flat;

>> title (' Flat shading!')

>> Z=peaks(30);

>> subplot(2,2,2);

>> surf(Z);

>> shading interp;

>> title (' Interpolated shading!')

>> colormap summer;

>> Z=peaks(30);

>> subplot(2,2,3);

>> surf(Z);

>> shading faceted;

>> title (' Faceted shading!')

>> colormap summer;

Exemplu personal

16. >> Z=peaks(50); >> surf(Z);

>> shading faceted;

>> title (' Faceted shading!')

>> colormap bone;

- 35 -

17. >> x=linspace(0,pi,20); >> y=linspace(-pi,pi,20);

>> [xx,yy]=meshgrid(x,y);

>> zz=func(xx,yy);

>> mesh(xx,yy,zz)

- 36 -

18. >> x=linspace(0,2*pi,20); >> y=linspace(-pi,pi,20);

>> [xx,yy]=meshgrid(x,y);

>> zz=func(xx,yy);

>> surf(xx, yy, zz)

19. >> x=0:.1:pi; >> y=0:.1:pi;

>> [X,Y]=meshgrid(x,y);

>> Z=sin(Y.^2+X)-cos(Y-X.^2);

>> subplot(2,2,1);

>> surf(Z)

>> title('surf');

>> subplot(2,2,2);

>> surfc(Z);

>> title('surfc');

>> subplot(2,2,3);

>> mesh(Z);

>> title('mesh');

- 37 -

Exemplu personal

20. >> Z=peaks(40); >> surfc(Z)

>> title('surfc');

>> colormap hot;

21. >> [X,Y] = meshgrid(0:.1:2*pi,-pi:.1:0);

>> Z = sin(X).*cos(Y);

>> meshz(X,Y,Z);

>> axis('equal');

- 38 -

Exemplu personal

22. >> [X,Y] = meshgrid(0:.1:2*pi,-pi:.1:0); >> Z = cos(X).*sin(Y);

>> meshz(X,Y,Z);

>> axis('equal');

>> title('meshz');

- 39 -

23. >> [th,r]=meshgrid(0:pi/40:2*pi,0:0.05:1); >> [X,Y]=pol2cart(th,r);

>> g=inline('r.^n.*sin(n*th)','r','th','n')

Inline function:

g(r,th,n) = r.^n.*sin(n*th)

>> surf(X,Y,g(r,th,5))

>> hold on

>> mesh(X,Y,-ones(size(X)))

Exemplu personal

24. >> [th,r]=meshgrid(0:pi/40:2*pi,0:0.05:1); >> [X,Y]=pol2cart(th,r);

>> surf(X,Y,g(r,th,3))

>> colormap winter;

>> hold on

>> mesh(X,Y,-zeros(size(X)))

- 40 -

25. >> x=-3*pi:.25:3*pi; >> A=linspace(3,0);

>> A=exp(-A);

>> X=sin(x).^2./(x+eps).^2;

>> Y=A'*X;

>> waterfall(Y)

Exemplu personal

26. >> x=-3*pi:.25:3*pi; >> A=linspace(3,0);

>> Y=A'*X;

>> X=cos(x).^2./(x+eps).^2;

>> waterfall(Y)

- 41 -

Exemplu personal

27. >> x=-3*pi:.25:3*pi; >> A=linspace(3,0);

>> A=exp(-A);

>> X=cos(x).^2./(x+eps).^2;

>> Y=A'*X;

>> waterfall(Y)

- 42 -

28. >> x=0:0.5:6; >> t=0:0.5:20;

>> [X,T]=meshgrid(x,t)

>> g=inline('cos(x-0.4*y).*exp(-0.4*x)','x','y')

Inline function:

g(x,y) = cos(x-0.4*y).*exp(-0.4*x)

>> contour(X,T,g(X,T))

>> pcolor(X,T,g(X,T))

>> hold on

>> contour(X,T,g(X,T),'k')

>> shading interp

>> colorbar

29. >> x=0:0.5:6;

>> t=0:0.5:20;

>> [X,T]=meshgrid(x,t);

>> g=inline('cos(x-0.4*y).*exp(-0.4*x)','x','y')

Inline function:

g(x,y) = cos(x-0.4*y).*exp(-0.4*x)

>> surfc(X,T,g(X,T))

>> colormap(bone)

- 43 -

30. >> [X,Y]=meshgrid(-2:.1:2,-2:.2:2); >> f=-X.*Y.*exp(-2*(X.^2+Y.^2));

>> mesh(X,Y,f);

>> xlabel('x');

>> ylabel('y');

>> grid on;

>> contour(X,Y,f)

>> xlabel('x');

>> ylabel('y');

>> grid on;

>> hold on

>> fmax=max(max(f))

fmax =

0.0886

>> kmax=find(f==fmax)

kmax =

>> pos=[X(kmax),Y(kmax)]

-0.5000 0.6000

0.5000 -0.6000

- 44 -

>> plot(X(kmax),Y(kmax),'*')

>> text(X(kmax),Y(kmax),' maximul')

31. >> [x,y]=meshgrid(-3:0.1:3,-5:0.1:10); >> f=100*(y-x.^2).^2+(1-x).^2+2;

>> mesh(x,y,f);

>> xlabel('x')

>> ylabel('y')

>> zlabel('z')

- 45 -

Exemplu personal

32. >> sphere (40)

33. >> [x,y,z]=ellipsoid(2,0,2,2,1,1); >> surf(x,y,z);

>> axis([0 4 -2 2 0 4]);

>> hold on

>> contour(x,y,z)

- 46 -

34. >> x = linspace(-3*pi,3*pi,50); >> r = cos(x).* sin(0.5*x).*exp((x.^2)/200);

>> r = r - min(r);

>> plot(r,linspace(0,1,length(r)));

>> ylabel('z')

>> cylinder(r)

35. >> t=[-1:.01:1];

>> [x,y,z]=cylinder(2+cos(t));

>> mesh(x,y,z)

>> axis square

>> colormap autumn

- 47 -

36. >> t = 0:0.1:10*pi; >> x = exp(-t/20).*cos(t);

>> y = exp(-t/20).*sin(t);

>> z = t;

>> plot3(x,y,z);

>> xlabel('x');

>> ylabel('y');

>> zlabel('z');

- 48 -

37. >> theta = 0:.2:2*pi; >> x=sin(theta);

>> y=cos(theta);

>> z=sin(theta);

>> stem3(x,y,z);

>> hold on

>> plot3(x,y,z,'r')

>> plot(x,y)

38. >> subplot(2,2,1);plot3(x,y,z); >> view(-10,10);

>> title('Default plot3');

>> subplot(2,2,2);plot3(x,y,z,'og');

>> view(-9,56);

>> title('Az=-10, El=10');

>> subplot(2,2,3);plot3(x,y,z,'xb');

>> view(0,90);

>> title('Az=0, El=90');

>> subplot(2,2,4);plot3(x,y,z,'dr');

>> view(90,0);

>> title('Az=90, El=0');

- 49 -

39. >> t = 0:0.1:10*pi; >> x = exp(-t/20).*cos(t);

>> y = exp(-t/20).*sin(t);

>> z = t;

>> plot3(x,y,z);

>> xlabel('x');

>> ylabel('y');

>> zlabel('z');

>> subplot(2,2,1);plot3(x,y,z);

>> view(-10,10);

>> title('Default plot3');

>> subplot(2,2,2);plot3(x,y,z,'og');

>> view(-9,56);

>> title('Az=-10, El=10');

>> subplot(2,2,3);plot3(x,y,z,'xb');

>> view(0,90);

>> title('Az=0, El=90');

>> subplot(2,2,4);plot3(x,y,z,'dr');

>> view(90,0);

>> title('Az=90, El=0');

- 50 -

Grafice de suprafete cu iluminari.

1. >> [X,Y] = meshgrid(-8:.5:8);

>> R = sqrt(X.^2 + Y.^2) + eps;

>> Z = sin(R)./R;

>> surf(X,Y,Z)

>> surf(X,Y,Z,'FaceColor','red','EdgeColor','none');

>> camlight left; lighting phong;

- 51 -

Exemplu personal 2. >> [X,Y] = meshgrid(-8:.5:8);

>> R = sqrt(X.^2 + Y.^2) + eps;

>> Z = sin(R)./R;

>> subplot(2,2,1);

>> surf(X,Y,Z)

>> surf(X,Y,Z,'FaceColor','red','EdgeColor','none')

>> camlight left, lighting phong

>> view(15,65)

>> title('Prima suprafata')

>> subplot(2,2,2)

>> surf(X,Y,Z)

>> surf(X,Y,Z,'FaceColor','red','EdgeColor','none')

>> camlight left, lighting phong

>> view(0,90)

>> title('Cea de a 2-a suprafata')

>> subplot(2,2,3)

>> surf(X,Y,Z)

>> surf(X,Y,Z,'FaceColor','blue','EdgeColor','none')

>> camlight right, lighting phong

>> view(-30,30)

>> title('Cea de a 3-a suprafata')

- 52 -

Calcul Simbolic in MATLAB

1. >> syms x real; >> t=x^2-3*x

x^2 - 3*x

>> subs(t,2)

>> subs(t,[1 -1.5 2])

-2.0000 6.7500 -2.0000

Exemplu personal

2. >> syms x real; >> f=x+3*x^2

3*x^2 + x

>> subs(f,1)

>> subs(f,[2 3])

- 53 -

3. >> syms x >> f=x^2*sin(x)-exp(x)

x^2*sin(x) - exp(x)

>> diff(f)

x^2*cos(x) - exp(x) + 2*x*sin(x)

>> diff(f,3)

6*cos(x) - exp(x) - x^2*cos(x) - 6*x*sin(x)

Exemplu personal

4. >> syms x >> f=3*x^2+x^3

x^3 + 3*x^2

>> diff(f)

3*x^2 + 6*x

>> diff(f,3)

5. >> syms x y >> z=x^2+x*y

x^2 + y*x

- 54 -

>> diff(z,x)

2*x + y

>> diff(z,y)

6. >> f=x^2*sin(x)-exp(x)

x^2*sin(x) - exp(x)

>> int(f)

2*cos(x) - exp(x) - x^2*cos(x) + 2*x*sin(x)

7. >> int(f,0,2)

4*sin(2) - exp(2) - 2*cos(2) - 1

8. >> syms x; >> f=x^3-3*x+1;

>> ezplot(f,[-1 1]);

- 55 -

9. >> syms x; >> ezplot('x^2 - 2*x + 1');

- 56 -

10. >> syms x y; >> ex1='sqrt(4-x^2-y^2)';

>> figure(1); ezsurf(ex1,[-2,2,-2,2]);

11. >> syms x ; >> f=x^2-7*x+2;

>> solve(f)

7/2 - 41^(1/2)/2

41^(1/2)/2 + 7/2

12. >> clear, syms x y; >> eq = 'exp(2*x) = 3*y';

>> [x] = solve(eq,x)

log(3*y)/2

13. 2x+3y-4z=5;

y+4z+x=10;

-2z+3x+4y=0;

- 57 -

>> clear, syms x y z;

>> eq1 = '2*x-3*y+4*z = 5'

2*x-3*y+4*z = 5

>> eq2 = 'y+4*z+x = 10'

y+4*z+x = 10

>> eq3 = '-2*z+3*x+4*y = 0'

-2*z+3*x+4*y = 0

>> [x,y,z] = solve(eq1,eq2,eq3,x,y,z)

165/74

>> syms x;

>> f=x^2;

>> limit(f,2)

- 58 -

>> limit(f,inf)

>> limit(1/x,inf)

>> limit(log(abs(x)),0)

Exemplu personal

15. >> syms x; >> f=sin(x)./x

sin(x)/x

>> limit(f,Inf)

>> limit(f,0)

- 59 -

Capitolul II

GEOMETRIA DIFERENȚIALĂ A CURBELOR

ÎN PLAN CU APLICAȚII ÎN MATLAB

Reprezentarea explicită a unei curbe plane

Exemplu personal 1. >> syms y;

>> y=sin(x);

>> ezplot(y)

>> title('Graficul functiei sin(x)');

>> legend('- sin(x)')

>> xlabel 'x'

>> ylabel ('f(x)=sin(x)')

Exemplu personal 2. >> x=0:.1:2*pi;

>> y=sin(x);

>> plot(y)

>> title('Graficul functiei sin(x)', 'FontSize', 14);

>> plot(y, '*r')

>> legend('functia sin(x)')

- 60 -

3. >> x=[-pi:.3:pi]; >> y=sin(x);

>> z=cos(x);

>> plot(x,y,'-r');

>> hold on

>> plot(x,z,'--g');

>> hold on

>> plot(x,0,'*-b')

>> legend ('-sin(x)','-cos(x)','y=0');

- 61 -

Reprezentarea implicită a unei curbe plane

1. >> x=linspace(-2,2);

>> plot(x,sqrt(4-x.^2))

>> hold on

>> plot(x,-sqrt(4-x.^2))

>> axis equal

2. >> t=-3:.01:3; >> x=t.^3-4*t;

>> y=t.^2-4;

>> plot(x,y)

- 62 -

3. >> t=0:.01:2*pi; >> x=((sin(t)).^2+1).*cos(t);

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

>> plot(x,y)

>> title('Curba lui Talbot')

Curbe date în coordonate polare

1. >> t=linspace(0,2*pi,200); >> r=1+sin(t);

>> polar(t,r);

>> title('Cardioida')

- 63 -

2. >> th=0:2*pi/100:2*pi; >> ro=1+.2*cos(th);

>> polar(th,ro)

>> [x,y] = pol2cart(th,ro) ;

>> plot(x,y) ;

>> axis equal

>> [x,y] = pol2cart(th,ro)

- 64 -

3. >> th=0:2*pi/100:2*pi; >> ro=3*th;

>> polar(th,ro)

Tangenta și normala într-un punct al unei curbe plane

Tangenta și normala într-un punct al unei curbe plane date explicit

1. >> syms x; >> y=sin(x);

>> ezplot(y,[0,2*pi])

>> dydx=diff(y)

dydx =

cos(x)

>> m=subs(dydx,x,2)

-0.4161

>> y0=subs(y,x,2);

>> hold on;

>> ezplot(m*(x-2)+y0,[0,2*pi]);

>> title('tangenta dusa la o curba data explicit')

- 65 -

- 66 -

Capitolul III

GEOMETRIA DIFERENȚIALĂ A CURBELOR ÎN

SPAȚIU. REPREZENTǍRI ÎN MATLAB

1. >> t = -2*pi:.001:2*pi;

>> x=cos(t); y=sin(t);z=t;

>> plot3(x,y,z)

2. >> a=2; >> b=0.1;

>> w=2;

>> t=linspace(0,12*pi,500);

>> x=a*cos(w*t);

>> y=a*sin(w*t);

>> z=b*t;

>> plot3(x,y,z)

- 67 -

3. >> alpha=0.2; >> t=linspace(-12*pi,12*pi,500);

>> x=cos(t)./sqrt(1+alpha^2*t.^2);

>> y=sin(t)./sqrt(1+alpha^2*t.^2);

>> z=alpha*t./sqrt(1+alpha^2*t.^2);

>> plot3(x,y,z)

- 68 -

4. >> r=1

>> phi=linspace(0,pi,30);

>> theta=linspace(0,2*pi,40);

>> [phi,theta]=meshgrid(phi,theta);

>> x=r*sin(phi).*cos(theta);

>> y=r*sin(phi).*sin(theta);

>> z=r*cos(phi);

>> mhndl=mesh(x,y,z)

mhndl =

171.0072

5. >> r=1; >> phi=linspace(0,pi,30);

>> x=2*r*sin(phi).*cos(theta);

>> y=2*r*sin(phi).*sin(theta);

>> z=2*r*cos(phi);

>> mhndl1=mesh(x,y,z)

- 69 -

mhndl1 =

171.0117

>> set(mhndl1,...

'EdgeColor',[0.6,0.6,0.6])

>> axis equal

>> axis off

>> title('Curba lui Viviani')

Exemplu personal

6. >> t=linspace(0,2*pi,40); >> z=linspace(-2*r,2*r,20);

>> [t,z]=meshgrid(t,z);

>> x=r+r*cos(t);

>> y=r*sin(t);

>> z=z;

>> hold on

mhndl2 =

171.0129

>> set(mhndl2,...

'EdgeColor',[0.8,0,0])

>> view(50,20)

- 70 -

7. >> r=1; >> phi=linspace(0,pi,30);

>> z=2*r*cos(phi);

mhndl1 =

171.0137

>> set(mhndl1,...

'EdgeColor',[0.6,0.6,0.6])

>> axis equal

>> axis off

>> z=linspace(-2*r,2*r,20);

>> x=r+r*cos(t);

>> y=r*sin(t);

>> z=z;

>> hold on

mhndl2 =

173.0132

- 71 -

>> set(mhndl2,...

>> view(50,20)

8. >> theta=linspace(0,2*pi,40); >> r=linspace(-1,1,30);

>> [theta,r]=meshgrid(theta,r);

>> x=r.*cos(theta);

>> y=r.*sin(theta);

>> z=r;

mhndl =

171.0144

>> set(mhndl,...

'EdgeColor',[.6,.6,.6])

>> hold on

>> [x,y]=meshgrid(-1:0.2:1);

>> z=0.5*ones(size(x));

>> phndl=mesh(x,y,z);

>> set(phndl,...

'EdgeColor',[0.625,0,0])

>> view(116,38)

>> axis equal

>> axis off

- 72 -

9. >> theta=linspace(0,2*pi,40); >> r=linspace(-1,1,30);

>> [theta,r]=meshgrid(theta,r);

>> x=r.*cos(theta);

>> y=r.*sin(theta);

>> z=r;

>> mhndl=mesh(x,y,z);

>> set(mhndl,...

'EdgeColor',[0.625,0,0])

>> set(mhndl,...

'EdgeColor',[.6,.6,.6])

>> hold on

>> [x,y]=meshgrid(-1:0.1:1);

>> z=y+0.25;

>> phndl=mesh(x,y,z);

>> set(phndl,...

'EdgeColor',[0.625,0,0])

>> view(70,55)

>> axis equal

- 73 -

10. >> r=linspace(0,1,30); >> theta=linspace(0,2*pi,30);

>> [r,theta]=meshgrid(r,theta);

>> x=r.*cos(theta);

>> y=r.*sin(theta);

>> z=r;

>> mesh(x,y,z)

>> axis tight

>> box on

>> xlabel('x-axis')

>> ylabel('y-axis')

>> zlabel('z-axis')

- 74 -

11. >> u=linspace(0,6*pi,60); >> v=linspace(0,2*pi,60);

>> [u,v]=meshgrid(u,v);

>> x=2*(1-exp(u/(6*pi))).*cos(u).*cos(v/2).^2;

>> y=2*(-1+exp(u/(6*pi))).*sin(u).*cos(v/2).^2;

>> z=1-exp(u/(3*pi))-sin(v)+exp(u/(6*pi)).*sin(v);

>> mesh(x,y,z)

>> view(160,10)

>> box on

>> surf(x,y,z,...

'FaceColor','interp',...

'FaceLighting','phong')

>> camlight left

>> view(160,10)

>> axis equal

>> box on

12. >> a=3; >> b=4;

>> c=5;

>> u=linspace(0,2*pi,30);

>> x=a*cos(u).*sin(v);

>> y=b*sin(u).*sin(v);

>> z=c*cos(v);

>> mesh(x,y,z)

>> box on

- 75 -

Exemplu personal 13. >> a=9;

>> b=12;

>> c=6;

>> u=linspace(0,2*pi,60);

>> v=linspace(0,pi,60);

>> x=a*cos(u).*sin(v);

>> y=b*sin(u).*sin(v);

>> z=c*cos(v);

>> mesh(x,y,z)

>> box on

>> surf(x,y,z,...

'FaceColor','interp',...

'EdgeColor','none',...

'FaceLighting','phong')

>> camlight left

>> view(160,10)

>> axis equal

>> box on

- 76 -

14. >> x = pi/100:pi/100:10*pi; >> y = sin(x)./x;

>> plot(x,y)

>> grid

- 77 -

15. >> x = pi/100:pi/100:10*pi; >> y = sin(x)./x;

>> plot(x,y)

>> grid

>> r=1;

>> phi=linspace(0,pi,30);

>> x=r*sin(phi).*cos(theta);

>> y=r*sin(phi).*sin(theta);

>> z=r*cos(phi);

mhndl =

171.0076

>> set(mhndl,...

'EdgeColor',[0.6,0.6,0.6],...

'EdgeAlpha',0.5,...

'FaceAlpha',0.5)

>> axis equal

>> axis off

>> alpha=0.2;

>> t=linspace(-12*pi,12*pi,500);

>> x=cos(t)./sqrt(1+alpha^2*t.^2);

>> y=sin(t)./sqrt(1+alpha^2*t.^2);

>> z=alpha*t./sqrt(1+alpha^2*t.^2);

>> lhndl=line(x,y,z)

lhndl =

173.0071

>> set(lhndl,...

'Color',[0.625,0,0],...

'LineWidth',2)

- 78 -

16. >> t=linspace(0,4*pi,200);

>> x=r+r*cos(t);

>> y=r*sin(t);

>> z=2*r*sin(t/2);

>> vhndl=line(x,y,z)

vhndl =

0.0079

>> set(vhndl,...

'Color',[0,0,0],...

'LineWidth',2)

>> view(50,20)

- 79 -

17. >> r=1; >> phi=linspace(0,pi,30);

>> z=2*r*cos(phi);

mhndl1 =

171.0088

>> set(mhndl1,...

'EdgeColor',[0.6,0.6,0.6])

>> axis equal

>> axis off

>> z=linspace(-2*r,2*r,20);

>> x=r+r*cos(t);

>> y=r*sin(t);

>> z=z;

>> hold on

mhndl2 =

173.0083

>> set(mhndl2,...

>> view(50,20)

>> x=r+r*cos(t);

>> y=r*sin(t);

>> z=2*r*sin(t/2);

>> vhndl=line(x,y,z)

vhndl =

174.0083

>> set(vhndl,...

'Color',[0,0,0],...

'LineWidth',2)

>> view(50,20)

- 80 -

18. >> t=linspace(0,1,200); >> x=3*sin(4*pi*t);

>> y=4*sin(6*pi*t);

>> z=5*sin(8*pi*t);

>> plot3(x,y,z)

>> box on

>> view(135,30)

- 81 -

19. >> t=linspace(0,6*pi,200); >> x=cos(t).*(2-cos(2*t/3));

>> y=sin(t).*(2-cos(2*t/3));

>> z=-sin(2*t/3);

>> plot3(x,y,z)

>> box on

>> view(135,30)

20. >> t=linspace(0,2*pi,200); >> x=(6.25+3*cos(5*t)).*cos(2*t);

>> y=(6.25+3\cos(5*t)).*sin(2*t);

>> z=3.25*sin(5*t);

>> plot3(x,y,z)

>> box on

>> view(135,30)

- 82 -

21. >> a=7;b=2;c=3; >> u=linspace(0,2*pi,20);

>> v=linspace(0,2*pi,40);

>> x=(a+b*cos(u)).*cos(v);

>> y=(a+b*cos(u)).*sin(v);

>> z=c*sin(u);

>> set(mhndl,...

'EdgeColor',[.6,.6,.6],...

'FaceAlpha',0.5,...

'EdgeAlpha',0.5);

>> axis equal

>> x=(a+b*cos(5*t)).*cos(2*t);

>> y=(a+b*cos(5*t)).*sin(2*t);

>> z=c*sin(5*t);

>> lhndl=line(x,y,z);

>> set(lhndl,...

'Color',[.625,0,0],...

'LineWidth',2)

>> view(135,30)

- 83 -

22. >> r=linspace(0,1,20); >> theta=linspace(0,2*pi,40);

>> [r,theta]=meshgrid(r,theta);

>> x=r.*cos(theta);

>> y=r.*sin(theta);

>> z=r;

>> set(mhndl,...

'EdgeColor',[.6,.6,.6],...

'FaceAlpha',0.5,...

'EdgeAlpha',0.5);

>> axis equal

>> t=linspace(0,1,200);

>> x=t.*cos(20*t);

>> y=t.*sin(20*t);

>> z=t;

>> lhndl=line(x,y,z);

>> set(lhndl,...

'Color',[.625,0,0],...

'LineWidth',2)

>> view(135,30)

- 84 -

Exemple personale.

1. >> [X,Y] = meshgrid(-8:.5:8); >> R = sqrt(X.^2 + Y.^2) + eps;

>> Z = sin(R)./R;

>> mesh(X,Y,Z,'EdgeColor','black')

>> axis off

2. >> t = 0:pi/20:2*pi;

>> y = exp(sin(t));

>> plotyy(t,y,t,y,'plot','stem')

>> xlabel('X Axis')

>> ylabel('Plot Y Axis')

>> title('Two Y Axes')

- 85 -

Bibliografie

1. VALER NIMINEȚ - Geometrie computațională cu aplicații în Matlab

2. CARMEN MURARU – Geometrie computațională

3. INTERNET – E-learn.ro

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