Matlab - Calcula del area de la ventana de viviani

syms t r;

x=r*cos(t) ;

y=r*sin(t) ;

z=sqrt (1-r^2) ;

f= [x y z ] ;

a=diff(f,r) ;

b=diff (f, t) ;

c=cross (a, b) ;

d=norm (c) ;

area=4*int (int (d, r,0,cos(t), t, 0,pi/2)

clear all

syms t   positive;

syms r

assume(r,'positive')

assumeAlso(r<1)

assumptions(r)

x=r*cos(t)

y=r*sin(t)

z=sqrt (1-r^2)

f= [x y z ]

a=diff(f,r)

b=diff (f, t)

c=cross (a, b)

d=norm (c)

d=simplify(d)

I1=int(d, 0,cos(t))

I2=int(I1,0,pi/2)

I=double(I2)

>> area_viani

ans =

[ r < 1, 0 < r]

x =

r*cos(t)

y =

r*sin(t)

z =

(1 - r^2)^(1/2)

f =

[ r*cos(t), r*sin(t), (1 - r^2)^(1/2)]

a =

[ cos(t), sin(t), -r/(1 - r^2)^(1/2)]

b =

[ -r*sin(t), r*cos(t), 0]

c =

[ (r^2*cos(t))/(1 - r^2)^(1/2), (r^2*sin(t))/(1 - r^2)^(1/2), r*cos(t)^2 + r*sin(t)^2]

d =

((r*cos(t)^2 + r*sin(t)^2)^2 - (r^4*abs(cos(t))^2)/(r^2 - 1) - (r^4*abs(sin(t))^2)/(r^2 - 1))^(1/2)

d =

r/(1 - r^2)^(1/2)

I1 =

1 - (sin(t)^2)^(1/2)

I2 =

pi/2 - 1

I =

    0.5708