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2019-06-16 16:51:19, Á¶È¸¼ö : 952 |
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4. The First Fundamental Form <- Figure I-24 ÂüÁ¶
∘ If 𝛂(t) = 𝐗 (u(t), v(t)), a¡Â t ¡Âb, <- a curve on a surface; s = s(t): arc length, total length L = s(b) = ¡òba¡«𝛂'(t)¡«dt,
(ds/dt)2 = ¡«𝛂'¡«2 = 𝛂' ∙ 𝛂' = (u'𝐗1 + v'𝐗2) ∙ (u'𝐗1 + v'𝐗2) = u'2(𝐗1 ∙ 𝐗1) + 2u'v'(𝐗1 ∙ 𝐗2) + v'2(𝐗2 ∙ 𝐗2) <- by eq.(3)
𝐸 = 𝐗1 ∙ 𝐗1, 𝐹 = 𝐗1 ∙ 𝐗2, 𝐺 = 𝐗2 ∙ 𝐗2 <- following Gauss [4-4]
(ds/dt)2 = 𝐸(du/dt)2 + 2𝐹[(du/dt)(dv/dt)] + 𝐺(dv/dt)2 [4-5]
ds2 = 𝐸 du2 + 2𝐹 dudv + 𝐺 dv2 <- the first fundamental form ð¯ìéÐñÜâû¡ãÒ or metric form ͪÕáû¡ãÒ [4-6]
∘ If 𝐯 = a𝐗1 + b𝐗2, 𝐰 = c𝐗1 + d𝐗2, <- (a,b,c,d´Â ½Ç¼ö) 𝐌ÀÇ ÇÑÁ¡¿¡¼ÀÇ tangent vectors
then 𝐯 ∙ 𝐰 = (a𝐗1 + b𝐗2) ∙ (c𝐗1 + d𝐗2) = 𝐸ac + 𝐹(ad + bc) + 𝐺bd.
⌈ 𝐸 𝐹 ⌉ ⌈ c ⌉
𝐯 ∙ 𝐰 = (a,b) ⌊ 𝐹 𝐺 ⌋ ⌊ d ⌋ <- matrix product Ç¥±â ¹æ½Ä [4-8]
∘ 𝐔 = 𝐗1 ⨯ 𝐗2 /¡«𝐗1 ⨯ 𝐗2¡« <- unit normal vector, ¡ñ 𝐔¡Ñ𝐗1 and 𝐗2, ¡«𝐔¡«= 1, 𝐗1 ⨯ 𝐗2 ¡Á 0
¡«𝐗1 ⨯ 𝐗2¡«2 = (𝐗1 ∙ 𝐗1) (𝐗2 ∙ 𝐗2) - (𝐗1 ∙ 𝐗2)2 = 𝐸𝐺 - 𝐹2 <- the identity of Lagrange ¶ó±×¶ûÁÖ ùöÔõãÒ [4-9]
¡ñ 𝐗1 ⨯ 𝐗2 = 𝐗1 𝐗2 sin𝜃, 𝐗1 ∙ 𝐗2 = 𝐗1 𝐗2 cos𝜃, ∣𝐗1∣2 ∣𝐗2∣2 sin2𝜃 = ∣𝐗1∣2 ∣𝐗2∣2 - ∣𝐗1∣2 ∣𝐗2∣2 cos2𝜃
∘ 𝑔𝑖𝑗 = 𝐗𝑖 ∙ 𝐗𝑗 [¡Ø notaional change] ¡æ 𝑔11 = 𝐸, 𝑔21 = 𝐹, 𝑔22 = 𝐺
⌈ 𝑔11 𝑔12 ⌉ ⌈ 𝐸 𝐹 ⌉
⌊ 𝑔21 𝑔22 ⌋ = ⌊ 𝐹 𝐺 ⌋ <- the matrix of the geometric form
∘ 𝑔 = ¡« 𝐗1 ⨯ 𝐗2¡«2 <- let 𝑔 = det (𝑔𝑖𝑗) = 𝐸𝐺 - 𝐹2 [¡Ø notaional change] [4-10]
( the Einstein Summation Convention ¾ÆÀν´Å¸ÀÎ ùê Ð®å³ À» Àû¿ëÇϱ⠽ÃÀÛÇÕ´Ï´Ù .) <- Tensor Çؼ® I-1 ÂüÁ¶
∘ ds2 = 𝑔𝑖𝑗du𝑖du𝑗 [notaional change: ¢² 𝑖,𝑗 𝑔𝑖𝑗du𝑖du𝑗] <- the first fundamental form or metric form by eq.(4-6)
∘ 𝐯 ∙ 𝐰 = 𝑔𝑖𝑗v𝑖w𝑗 [notaional change: ¢² 𝑖,𝑗 𝑔𝑖𝑗v𝑖] <- by eq.(8); 𝐯¡Ñ𝐰, if and only if * 𝐯 ∙ 𝐰 = 0. [4-11]
∘ 𝛂' = u𝑖'𝐗𝑖 [notaional change: ¢² 𝑖 u𝑖'𝐗𝑖] <- by eq.(3)
∘ 𝑔𝑖𝑗: the components of the matrix inverse of (𝑔𝑖𝑗) ↦ (𝑔𝑖𝑗)(𝑔𝑖𝑗) = 𝐈 <- 𝐈: identity matrix
𝑔𝑖𝑗𝑔𝑗𝑘 = 𝛿𝑘𝑖, 𝛿𝑘𝑖 = {(1 if 𝑖 = 𝑘), (0 if 𝑖 ¡Á 𝑘)} [notaional change: ¢² 𝑖,𝑗 𝑔𝑖𝑗𝑔𝑗𝑘] [4-12]
𝑔11 = 𝑔22/𝑔, 𝑔12 = 𝑔21 = -𝑔12/𝑔, 𝑔22 = 𝑔11/𝑔 <- inverse matrix**, 𝑔 = det (𝑔𝑖𝑗); if and only if 𝑔 ¡Á 0
∘ If a surface 𝐗;𝐃 ¡æ E3, a region 𝛺 ¡ø 𝐃 on which 𝐗 is one-to-one, ¾î¶»°Ô 𝐗(𝛺) ¸éÀûÀ» ¹ß°ßÇÒ ¼ö ÀÖÀ»±î¿ä?
¸ÕÀú 𝛺¸¦ u1, u2ÃàÀ» µû¶ó °¢º¯ÀÌ 𝛥u1, 𝛥u2ÀÎ ÀÛÀº Á÷»ç°¢ÇüÀ¸·Î ³ª´« ÈÄ¿¡ À̸¦ 𝐗(𝛺)·Î mappingÇØ º¾´Ï´Ù.
𝐗(𝛺)¿¡¼ 𝛺¿¡¼ÀÇ ÀÛÀº »ç°¢ÇüµéÀÌ À¯»ç ÆòÇà»çº¯ÇüÀ¸·Î µÇ¸ç °¢º¯Àº ¡«𝐗1¡«𝛥u1,¡«𝐗2¡«𝛥u2°¡ µË´Ï´Ù.
¡Å 𝛥𝐴 ≈¡«𝐗1¡«¡«𝐗2¡«sin 𝜃 𝛥u1𝛥u2 = ¡«𝐗1 ⨯ 𝐗2¡«𝛥u1𝛥u2 = (¡î 𝑔) 𝛥u1𝛥u2
𝐀 = ¡ó𝛺¡«𝐗1 ⨯ 𝐗2¡«du1du2 = ¡ó𝛺 (¡î 𝑔) du1du2 <- function 𝐗 to 𝛺: one-to-one and regular [4-13]
5. The Second Fundamental Form <- Figure I-25 ÂüÁ¶
∘ ÀÌÈķδ the Einstein Summation Convention ¾ÆÀν´Å¸ÀÎ ùê Ð®å³ À» °è¼Ó Àû¿ëÇÕ´Ï´Ù.
∘ The Gauss Formulas : a curve 𝛂(s) = 𝐗(u1(s), u2(s)) on 𝐌, s: arc length,
unit tangent vector 𝐓 = 𝛂' = u𝑖'𝐗𝑖; acceleration or curvature vector 𝐓' = 𝛂"; curvature k(s) = ¡«𝐓'¡«
𝛂" = 𝛂"tan + 𝛂"nor <- tangent and normal to the surface; in order to study the surface 𝐌 [5-14]
𝛂" = u𝑖"𝐗𝑖 + u𝑖'd𝐗𝑖/ds <- by product rule [5-15]
𝐗𝑖 ¡Õ ¡Ó𝐗/¡Óu𝑖 = ( ¡Óx/¡Óu𝑖, ¡Óy/¡Óu𝑖, ¡Óz/¡Óu𝑖), 𝑖 = 1, 2
𝐗𝑖𝑗 ¡Õ ¡Ó2𝐗/¡Óu𝑗¡Óu𝑖 = ¡Ó𝐗𝑖/¡Óu𝑗 = ( ¡Ó2x/¡Óu𝑗¡Óu𝑖, ¡Ó2y/¡Óu𝑗¡Óu𝑖, ¡Ó2z/¡Óu𝑗¡Óu𝑖), 𝑖,𝑗 = 1, 2
𝐗𝑖 = 𝐗𝑖(u1(s), u2(s)) ¡æ d𝐗𝑖/ds = (¡Ó𝐗𝑖/¡Óu1) u1' + (¡Ó𝐗𝑖/¡Óu2) u2' = (¡Ó𝐗𝑖/¡Óu𝑗)u𝑗' = u𝑗'𝐗𝑖𝑗 <- by chain rule
𝛂" = u𝑟"𝐗𝑟 + u𝑖'u𝑗'𝐗𝑖𝑗 [5-16]
𝐗𝑖𝑗 = 𝛤𝑟𝑖𝑗𝐗𝑟 + 𝐿𝑖𝑗𝐔, 𝑖,𝑗 = 1,2 <- the Gauss Formulas ; cf) Tensor Çؼ® II-4 b) Christoffel ±âÈ£ [5-17]
𝛂" = 𝛂"tan + 𝛂"nor = (u𝑟" + 𝛤𝑟𝑖𝑗 u𝑖'u𝑗')𝐗𝑟 + (𝐿𝑖𝑗 u𝑖'u𝑗')𝐔 [5-18]
𝛂" = [u1" + 𝛤111(u1')2 + 2𝛤112u1'u2' + 𝛤122u2')2]𝐗1 + [u2" + 𝛤211(u1')2 + 2𝛤212u1'u2' + 𝛤222(u2')2]𝐗2
+ [𝐿11(u1')2 + 2𝐿12u1'u2' + 𝐿22(u2')2]𝐔 <- full equation by the Einstein summation convention
∘ 𝐿𝑖𝑗 u𝑖'u𝑗' <- how does it curve relative that space? extrinsic(èâî¤îÜ) geometry [5-19]
𝐿𝑖𝑗 = 𝐗𝑖𝑗 ∙ 𝐔, 𝑖,𝑗 = 1,2 <- the second fundamental form of the surfaces [5-20]
ex) Let 𝐗(u,v) = (u, v, f(u,v)), (u,v) ∊ D, then 𝐗1 = (1, 0, fu), 𝐗2 = (0, 1, fv).
𝐸 = 1 + fu2, 𝐹 = fuv, 𝐺 = 1 + fv2, 𝑔 = 𝐸𝐺 - 𝐹2 = 1 + fu2 + fv2, 𝐗1 ⨯ 𝐗2 = (-fu, -fv, 1), 𝐔 = (-fu, -fv)/¡î 𝑔
𝐗11= (0, 0, fuu), 𝐗12= (0, 0, fuv), 𝐗11= (0, 0, fvv) ¡æ 𝐗𝑖𝑗 = (0, 0, f𝑖𝑗); ¡Å 𝐿𝑖𝑗 = f𝑖𝑗 / ¡î 𝑔, 𝑖,𝑗 = 1, 2
∘ Definition I-3
𝐯 = v𝑖 𝐗𝑖°¡ 𝐏¿¡¼ 𝐌¿¡ Á¢ÇÏ´Â a unit vector¶ó¸é, the normal curvature ÛöÍØáã in the v direction k´Â,
𝑘n(𝐯) = 𝐿𝑖𝑗 v𝑖v𝑗 ·Î Á¤ÀǵȴÙ. [5-21]
∘ 𝛂°¡ 𝐌¿¡ ÀÖ´Â ÀÓÀÇÀÇ °î¼±À¸·Î 𝛂(s0) = 𝐏, 𝛂(s0)' = 𝐯À̸é, 𝛂(s0)'= u𝑖'(s0)𝐗𝑖(u1(s0), u2(s0)), v𝑖 = u𝑖'(s0)ÀÌ µË´Ï´Ù.
𝑘n(𝐯) = 𝐿𝑖𝑗 u𝑖'u𝑗' = 𝛂" ∙ 𝐔 = ∓¡«𝛂"¡« <- by eq.16 ¿¡¼ ur" 𝐗r ∙ 𝐔 = 0; eq. 20 [5-22]
'0'ÀÌ¾Æ´Ñ tangent vector¿¡ ´ëÇؼ normal curvatureÀÇ °³³äÀ» ´ÙÀ½°ú °°ÀÌ È®Àå½Ãų ¼ö ÀÖ½À´Ï´Ù.
𝑘n(𝐯) = 𝐿𝑖𝑗 v𝑖v𝑗/¡«𝐯¡«2 = 𝐿𝑖𝑗 v𝑖v𝑗/ 𝑔mnvmvn <- by eq. 21; ¿ìº¯ÀÇ v¸¦ v/¡«v¡«·Î ±³Ã¼ÇÔ.
p.s. °î¸é 𝐌 À§¿¡ ÀÖ´Â °î¼± ¹æÁ¤½ÄÀÇ 1Â÷ 2Â÷ ¹ÌºÐ °á°ú¸¦ ÀÌ¿ëÇؼ °î¸é ÀÚüÀÇ curvature¸¦ ¹ß°ßÇÏ´Â °ÍÀÌ ÇÙ½ÉÀÓ.
ƯÈ÷ °î¸é À§¿¡¼ÀÇ Christoffel symbols¸¦ Æ÷ÇÔÇÑ Gauss Formulas-'Tensor Çؼ® II-2'°ú ºñ±³µÊ-¿¡ ÁÖ¸ñ ¹Ù¶÷.
* 'if and only if ' ȤÀº ÁÙ¿©¼ 'iff '¶ó°íµµ ¾²´Âµ¥ µ¿Ä¡(ÔÒö·) ¶Ç´Â ÇÊ¿äÃæºÐÁ¶°ÇÀ» °¡¸®Å´.
** inverse matrix(æ½ú¼Öª) 𝐀-1 = (Ȃ𝑖𝑗)T/det 𝐀 <- cofactor(æ®ì×í) Ȃ𝑖𝑗 = (-1)𝑖+𝑗𝑀𝑖𝑗, 𝑀𝑖𝑗: the 𝑖,𝑗 minor of 𝐀 |
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