exercice 49 p 194.sagews

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# Ellipsoïde

Ellipsoïde

var('x, y, z')
F(x,y,z) = 2*(x-2)^2+(y-1)^2+(z-3)^2
show(F)

(x, y, z)

$\displaystyle \left( x, y, z \right) \ {\mapsto} \ 2 \, {\left(x - 2\right)}^{2} + {\left(y - 1\right)}^{2} + {\left(z - 3\right)}^{2}$

r = 8
G = implicit_plot3d(F(x,y,z)==10, (x, -r, r), (y, -r, r), (z, -r, r), color='orange', mesh=1, opacity=.7, spin=1)
show(G)

3D rendering not yet implemented

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# Plan tangent

Plan tangent

gradF = F.gradient()
show(gradF)

$\displaystyle \left( x, y, z \right) \ {\mapsto} \ \left(4 \, x - 8,\,2 \, y - 2,\,2 \, z - 6\right)$

gradF0 = gradF(3, 3, 5)
show(gradF0)

$\displaystyle \left(4,\,4,\,4\right)$

PP0 = vector([x-3, y-3, z-5])
show(PP0)

$\displaystyle \left(x - 3,\,y - 3,\,z - 5\right)$

eq_plan_tangent = gradF0.dot_product(PP0)==0
show(eq_plan_tangent)

$\displaystyle 4 \, x + 4 \, y + 4 \, z - 44 = 0$

G += implicit_plot3d(eq_plan_tangent, (x, -r, r), (y, -r, r), (z, -r, r), color='blue')
show(G)

3D rendering not yet implemented

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# Droite normale

Droite normale

var('s')
s_max=3
G += parametric_plot3d([3+s, 3+s, 5+s], (s, -s_max, s_max), color='red')
show(G)

s

3D rendering not yet implemented