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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

(3)

Au dernier cours, nous avons vu

Courbure

(4)

Au dernier cours, nous avons vu

Courbure

Le plan et le cercle osculateur

(5)

Au dernier cours, nous avons vu

Courbure

Le plan et le cercle osculateur Formule pour la courbure

(6)

Aujourd’hui, nous allons voir

(7)

Aujourd’hui, nous allons voir

Torsion

(8)

Aujourd’hui, nous allons voir

Torsion

Formules de Frenet-Serret

(9)

Aujourd’hui, nous allons voir

Torsion

Formules de Frenet-Serret Formule pour la torsion

(10)

Aujourd’hui, nous allons voir

Torsion

Formules de Frenet-Serret Formule pour la torsion

Cinématique

(11)

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

(12)

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.

(13)

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 vitesse mesure à quel point on n’est pas constant (dimension 0) v<latexit sha1_base64="f9BnFOqw3fqK86x+iQiQN9fmx6c=">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</latexit><latexit sha1_base64="f9BnFOqw3fqK86x+iQiQN9fmx6c=">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</latexit><latexit sha1_base64="f9BnFOqw3fqK86x+iQiQN9fmx6c=">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</latexit><latexit sha1_base64="f9BnFOqw3fqK86x+iQiQN9fmx6c=">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</latexit> (t) = k~r 0(t)k

(14)

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 vitesse mesure à quel point on n’est pas constant (dimension 0) v<latexit sha1_base64="f9BnFOqw3fqK86x+iQiQN9fmx6c=">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</latexit><latexit sha1_base64="f9BnFOqw3fqK86x+iQiQN9fmx6c=">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</latexit><latexit sha1_base64="f9BnFOqw3fqK86x+iQiQN9fmx6c=">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</latexit><latexit sha1_base64="f9BnFOqw3fqK86x+iQiQN9fmx6c=">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</latexit> (t) = k~r 0(t)k

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

ds

<latexit sha1_base64="zmy2ifZX1s0eAVKDVdax85wRvR0=">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</latexit><latexit sha1_base64="zmy2ifZX1s0eAVKDVdax85wRvR0=">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</latexit><latexit sha1_base64="zmy2ifZX1s0eAVKDVdax85wRvR0=">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</latexit><latexit sha1_base64="zmy2ifZX1s0eAVKDVdax85wRvR0=">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</latexit>

(15)

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)

v<latexit sha1_base64="f9BnFOqw3fqK86x+iQiQN9fmx6c=">AAAC2nicjVHLSsNAFD2Nr1pfVXHlJlhE3ZRUBN0IRTcuK9gHGJFkHHVomoTJpCDVjTtx6w+41Q8S/0D/wjvjCD4QnZDk3HPvOTN3bphGIlOe91xwhoZHRseK46WJyanpmfLsXCtLcsl4kyVRIjthkPFIxLyphIp4J5U86IURb4fdXZ1v97nMRBIfqIuUH/WCs1icChYooo7LC/1VtbbtX/p9zlzpuysU+pfH5YpX9cxyf4KaBRXY1UjKT/BxggQMOXrgiKEIRwiQ0XOIGjykxB1hQJwkJEye4wol0uZUxakiILZL3zOKDi0bU6w9M6NmtEtErySli2XSJFQnCevdXJPPjbNmf/MeGE99tgv6h9arR6zCObF/6T4q/6vTvSicYsv0IKin1DC6O2ZdcnMr+uTup64UOaTEaXxCeUmYGeXHPbtGk5ne9d0GJv9iKjWrY2Zrc7zqU9KAa9/H+RO01qs1wvsblfqOHXURi1jCKs1zE3XsoYEmeQ9wjwc8Or5z7dw4t++lTsFq5vFlOXdvGiGWug==</latexit><latexit sha1_base64="f9BnFOqw3fqK86x+iQiQN9fmx6c=">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</latexit><latexit sha1_base64="f9BnFOqw3fqK86x+iQiQN9fmx6c=">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</latexit><latexit sha1_base64="f9BnFOqw3fqK86x+iQiQN9fmx6c=">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</latexit> (t) = k~r 0(t)k

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

ds

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(16)

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

(17)

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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(18)

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=">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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><latexit sha1_base64="14ZG1Pb+Iu9A/wfkLVouLYQ6GRU=">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</latexit> ) = ~r(a) + uT~ (a) + vN~ (a)

(19)

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=">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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><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.

(20)

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=">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</latexit><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> ) = ~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

(21)

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=">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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><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>

(22)

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=">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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><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)

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change, alors le plan change.

(23)

La torsion d’une courbe est

⌧ = dB~ ds

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

(24)

Exemple

Essayons de trouver dB~ ds

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pour la fonction

~r<latexit sha1_base64="gPT520ojjkO34FLcQQjqzDepidY=">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</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><latexit sha1_base64="gPT520ojjkO34FLcQQjqzDepidY=">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</latexit> (t) = (t, t2, t3)

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