︠5ac13f93-6807-49f7-ba01-9cbc5bf75748i︠ %md ## Section 8.1, n. 26 Juste le graphique pour voir, même si ce n'était pas demandé. ︡35721c16-5424-4784-a98a-5aeda2265b4a︡{"done":true,"md":"## Section 8.1, n. 26\n\nJuste le graphique pour voir, même si ce n'était pas demandé."} ︠91d88093-5000-4fa2-956f-bf7b330e18c3︠ var('t,s') C = parametric_plot3d([sin(t), cos(t), (sin(t))^2], (t,0,2*pi), color= "blue", thickness = 4) # La courbe donnée Cyl = parametric_plot3d([cos(t), sin(t), s], (t,0,2*pi), (s, -1, 2), color="green", opacity = 0.25) # Le cylindre circulaire Cyl2 = parametric_plot3d([t,s,t^2],(t,-1,1),(s,-1.5,1.5), color="red", opacity = 0.25)#Le cylindre parabolique show(C+Cyl + Cyl2) ︡272d88a8-fd89-4f9d-bf96-5525a9c9b17f︡{"html":"
($\\displaystyle t$, $\\displaystyle s$)
"}︡{"file":{"filename":"398adfa5-d7d3-4870-b5a2-4ff0f009575f.sage3d","uuid":"398adfa5-d7d3-4870-b5a2-4ff0f009575f"}}︡{"done":true}︡ ︠3d279bb6-3e92-4fff-bffd-41e6676cb9d3i︠ %md ## Section 8.3, n. 38 ︡18dddd8c-e8f6-47e9-b7cb-18a0639224e5︡{"done":true,"md":"## Section 8.3, n. 38"} ︠e2e22e41-75ed-4cfd-b1a5-ffbe3fd9fdefs︠ var('t') typeset_mode(True) x(t) = sin(3*t) y(t) = sin(2*t) z(t) = sin(3*t) Courbe = parametric_plot3d([x(t),y(t),z(t)],(t,0,2*pi), color= "blue", thickness= 3) show(Courbe) ︡7d3ba8aa-474f-44dc-910e-0232996ea4c0︡{"stdout":"t\n"}︡{"file":{"filename":"69a36811-1cb4-4f69-8567-ac4e17a713ca.sage3d","uuid":"69a36811-1cb4-4f69-8567-ac4e17a713ca"}}︡{"done":true}︡ ︠360ee494-2b72-4d11-b2ce-2ac9b987a378s︠ r = vector([x(t),y(t),z(t)]) v = diff(r,t) a = diff(r,t,2) ︡64e1ede8-2c95-4c93-a3ae-c7139e0ccb08︡{"done":true}︡ ︠aa3e28e4-ac97-48b0-9d24-323b1ae1c1bci︠ %md On calcule la fonction de courbure avec la formule $$\kappa(t) =\frac{|\mathbf{r}'(t) \times \mathbf{r}''(t)|}{|\mathbf{r}'(t)|^3}$$ ︡eeeb9c42-f390-4e75-a1c6-cc0b60b81cf7︡{"done":true,"md":"On calcule la fonction de courbure avec la formule\n$$\\kappa(t) =\\frac{|\\mathbf{r}'(t) \\times \\mathbf{r}''(t)|}{|\\mathbf{r}'(t)|^3}$$"} ︠9b7a4426-5ce1-4d51-95af-a64fbdb3d1das︠ k(t) = (v.cross_product(a)).norm()/(v.norm())^3 ︡2c4e1c25-d99a-484d-ab7b-69a9515d6d6d︡{"done":true}︡ ︠e0716b9f-daa1-4636-8cd3-04050afdf0c3s︠ Courbe = parametric_plot((t,k(t)),(t,0,2*pi),title='Fonction de courbure', axes_labels=['$t$','$\kappa (t)$'], ticks=pi/3,tick_formatter=pi,plot_points=600) show(Courbe, aspect_ratio = 0.1, figsize=12) ︡24a8e3f7-85ed-4a4f-b00e-fe742f871684︡{"file":{"filename":"/home/user/.sage/temp/project-4aacee0b-64bd-4d37-99a1-d87b7a2c4cd6/492/tmp_ZGgCcN.svg","show":true,"text":null,"uuid":"5ef20e40-caba-428d-9f43-fa7db2f1dee0"},"once":false}︡{"done":true}︡ ︠fd41a12c-01e9-4958-bb55-4f373f6f7ce7︠