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1.6 TORSION ET MOUVEMENT DANS L’ESPACE

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Texte intégral

(1)

cours 6

1.6 TORSION ET

MOUVEMENT DANS

L’ESPACE

(2)

Au dernier cours, nous avons vu

Courbure

Le plan et le cercle osculateur Formule pour la courbure

(3)

Aujourd’hui, nous allons voir

Torsion

Formules de Frenet-Serret Formule pour la torsion

Cinématique

(4)

Les mots vitesse, courbé et tordu sont souvent employé dans divers contextes.

Personnellement, j’ai tendance à les voir comme une même idée appliquée en différentes dimensions.

La torsion mesure à quel point on n’est pas planaire (dimension 2) La vitesse mesure à quel point on n’est pas constant (dimension 0)

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La courbure mesure à quel point on n’est pas linéaire (dimension 1) k(s) = dT~

ds

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(5)

Une courbe sera planaire si son plan osculateur ne change pas.

L’équation vectorielle du plan osculateur à la courbe paramétrée par en est

~r(t)

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t = a

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(x, y, z<latexit sha1_base64="14ZG1Pb+Iu9A/wfkLVouLYQ6GRU=">AAAC73icjVHLSsNAFD3GV31XXboZLELVUlIRdCOIblxJBWsFK5JMpzU0TeJkUqziP7hzJ279Abf6F+If6F94Z0zBB6ITkpw5954zc+91I9+LlW2/9Fn9A4NDw5mR0bHxicmp7PTMQRwmkosKD/1QHrpOLHwvEBXlKV8cRlI4bdcXVbe1rePVjpCxFwb7qhuJ47bTDLyGxx1F1El2KX9eYN0Cu1jcYLWO4EzmncXlxMB9DTsG7hI8yebsom0W+wlKKcghXeUw+4wa6gjBkaANgQCKsA8HMT1HKMFGRNwxLomThDwTF7jCKGkTyhKU4RDbom+TdkcpG9Bee8ZGzekUn15JSoYF0oSUJwnr05iJJ8ZZs795XxpPfbcu/d3Uq02swimxf+l6mf/V6VoUGlg3NXhUU2QYXR1PXRLTFX1z9qkqRQ4RcRrXKS4Jc6Ps9ZkZTWxq1711TPzVZGpW73mam+BN35IGXPo+zp/gYKVYIry3mtvcSkedwRzmkad5rmETOyijQt7XeMAjnqwz68a6te4+Uq2+VDOLL8u6fwcfPZ0a</latexit><latexit sha1_base64="14ZG1Pb+Iu9A/wfkLVouLYQ6GRU=">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</latexit><latexit sha1_base64="14ZG1Pb+Iu9A/wfkLVouLYQ6GRU=">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</latexit><latexit sha1_base64="14ZG1Pb+Iu9A/wfkLVouLYQ6GRU=">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</latexit> ) = ~r(a) + uT~ (a) + vN~ (a)

Or ces vecteurs directeurs peuvent changer sans que le plan change.

Par contre, si la direction du vecteur normal au plan B~ (a) = T~ (a) ^ N~ (a)

<latexit sha1_base64="iecug87aZjXWjrMoP7lYM/9fLmQ=">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</latexit><latexit sha1_base64="iecug87aZjXWjrMoP7lYM/9fLmQ=">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</latexit><latexit sha1_base64="iecug87aZjXWjrMoP7lYM/9fLmQ=">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</latexit><latexit sha1_base64="iecug87aZjXWjrMoP7lYM/9fLmQ=">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</latexit>

change, alors le plan change.

(6)

La torsion d’une courbe est

= dB~ ds

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et donne une mesure de la non-planarité d’une courbe

(7)

Exemple Essayons de trouver dB~ ds

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pour la fonction

N~ = T~0(t) kT~0(t)k

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B~ (t) = T~(t) ^ N~ (t)

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dB~

ds = dB~ dt

ds dt

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=<latexit sha1_base64="M6GXJLJlv9N97EwtiZjzUUiwOZI=">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</latexit><latexit sha1_base64="M6GXJLJlv9N97EwtiZjzUUiwOZI=">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</latexit><latexit sha1_base64="M6GXJLJlv9N97EwtiZjzUUiwOZI=">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</latexit><latexit sha1_base64="M6GXJLJlv9N97EwtiZjzUUiwOZI=">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</latexit> B~ 0(t)k~r 0(t)k

~r<latexit sha1_base64="gPT520ojjkO34FLcQQjqzDepidY=">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</latexit><latexit sha1_base64="gPT520ojjkO34FLcQQjqzDepidY=">AAAC3HicjVHLSsNAFD2N73fUhQs3wSJUkJJWQTeC6MZlBWsLrUoyndbQvJhMCqW4cydu/QG3+j3iH+hfeGdMQS2iEzI5c+49J/fOdWPfS6Rtv+aMsfGJyanpmdm5+YXFJXN55TyJUsF4lUV+JOquk3DfC3lVetLn9VhwJ3B9XnO7xype63GReFF4JvsxvwicTui1PeZIoq7MtWaPM0sU5NZBQW5b8rKstp2tKzNvF229rFFQykAe2apE5guaaCECQ4oAHCEkYR8OEnoaKMFGTNwFBsQJQp6Oc9xglrQpZXHKcIjt0t6hUyNjQzorz0SrGf3Fp1eQ0sImaSLKE4TV3ywdT7WzYn/zHmhPVVufvm7mFRArcU3sX7ph5n91qheJNvZ1Dx71FGtGdccyl1Tfiqrc+tKVJIeYOIVbFBeEmVYO79nSmkT3ru7W0fE3nalYdWZZbop3VSUNuPRznKPgvFwsET7dzR8eZaOexjo2UKB57uEQJ6igqut/xBOejUvj1rgz7j9TjVymWcW3ZTx8ABBblqY=</latexit><latexit sha1_base64="gPT520ojjkO34FLcQQjqzDepidY=">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</latexit><latexit sha1_base64="gPT520ojjkO34FLcQQjqzDepidY=">AAAC3HicjVHLSsNAFD2N73fUhQs3wSJUkJJWQTeC6MZlBWsLrUoyndbQvJhMCqW4cydu/QG3+j3iH+hfeGdMQS2iEzI5c+49J/fOdWPfS6Rtv+aMsfGJyanpmdm5+YXFJXN55TyJUsF4lUV+JOquk3DfC3lVetLn9VhwJ3B9XnO7xype63GReFF4JvsxvwicTui1PeZIoq7MtWaPM0sU5NZBQW5b8rKstp2tKzNvF229rFFQykAe2apE5guaaCECQ4oAHCEkYR8OEnoaKMFGTNwFBsQJQp6Oc9xglrQpZXHKcIjt0t6hUyNjQzorz0SrGf3Fp1eQ0sImaSLKE4TV3ywdT7WzYn/zHmhPVVufvm7mFRArcU3sX7ph5n91qheJNvZ1Dx71FGtGdccyl1Tfiqrc+tKVJIeYOIVbFBeEmVYO79nSmkT3ru7W0fE3nalYdWZZbop3VSUNuPRznKPgvFwsET7dzR8eZaOexjo2UKB57uEQJ6igqut/xBOejUvj1rgz7j9TjVymWcW3ZTx8ABBblqY=</latexit> (t) = (t, t2, t3) ~r<latexit sha1_base64="t41QpKAe14+xOPxMiwSrt5LKz9I=">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</latexit><latexit sha1_base64="t41QpKAe14+xOPxMiwSrt5LKz9I=">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</latexit><latexit sha1_base64="t41QpKAe14+xOPxMiwSrt5LKz9I=">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</latexit><latexit sha1_base64="t41QpKAe14+xOPxMiwSrt5LKz9I=">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</latexit> 0(t) = (1, 2t, 3t2) k~r 0(t)k = p

1 + 4t2 + 9t4

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T~ (t) =

1

p1 + 4t2 + 9t4 , 2t

p1 + 4t2 + 9t4 , 3t2

p1 + 4t2 + 9t4

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T~0(t) =

4t 18t3

(1 + 4t2 + 9t4) 32 , 2 18t4

(1 + 4t2 + 9t4) 32 , 6t + 12t3

(1 + 4t2 + 9t4) 32

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kT~0(t)k =

s✓ 4t 18t3 (1 + 4t2 + 9t4)32

2

+

2 18t4

(1 + 4t2 + 9t4)32

2

+

6t + 12t3 (1 + 4t2 + 9t4)32

2

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