The painter of the Lascaux Cave (B.C.15,000) knew the geometry of apparent contours. ohmoto/class.html 25 ( ) 2 / 5
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1 1 / ohmoto/class.html
2 The painter of the Lascaux Cave (B.C.15,000) knew the geometry of apparent contours. ohmoto/class.html 25 ( ) 2 / 52
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4 4 / Desargues:
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11 11 / 52 :
12 11 / 52 :
13 11 / 52 :
14 12 / 52 : l l H H
15 13 / 52 : P 2
16 13 / 52 : P 2 K = R C K P 2 RP 2 CP 2 P 2 := K 3 {0}/ x y λ K {0} s.t. x = λy
17 13 / 52 : P 2 K = R C K P 2 RP 2 CP 2 P 2 := K 3 {0}/ x y λ K {0} s.t. x = λy P 2 = U 0 U 1 U 2 U 0 = {[1 : u 0 : v 0 ] P 2 }, U 1 = {[u 1 : 1 : v 1 ] P 2 }, U 2 = {[u 2 : v 2 : 1] P 2 }.
18 14 / 52 : 0.1 Pappus 300 l, l l A, B, C l A, B, C AB A B P BC B C Q CA. C A R P, Q, R.
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21 17 / 52 : P D : RE = P D : DB = F R : RC = AR : RG = BE : EQ = DR : EQ P DR = REQ DRP EQR P, Q, R q.e.d.
22 18 / 52 =
23 18 / 52 = d d
24 18 / 52 = d d =
25 18 / 52 = d d = =
26 19 / 52
27 20 / 52 Gauss K P M elliptic point hyperbolic point parabolic point
28 ohmoto/class.html 25 ( ) 21 / 52
29 22 / 52 M R 3 P 3 p R 3 M M p ϕ : M P 2, x px contour generator ϕ apparent contour ϕ
30 0.2 f, g : R m, 0 R n, 0 A- source target σ, τ R m, 0 f R n, 0 σ R m, 0 g τ R n, ohmoto/class.html 25 ( ) 23 / 52
31 ohmoto/class.html 25 ( ) 23 / f, g : R m, 0 R n, 0 A- source target σ, τ R m, 0 f R n, 0 σ R m, 0 g τ R n, f : R m, 0 R n, 0 Jacobi = f (m n) (m > n) A- f : R m, 0 R, 0, df(0) = 0, det Hess f (0) 0 = f ±x 2 1 ± ± x2 m A-.
32 0.4 ( (1979, 84, 86)) M p ϕ x M ϕ : M, x P 2, ϕ(x) f R 2, 0 R 2, 0, (x, y) (y, f(x, y)) A- type codim. f(x, y) type codim. f(x, y) 1(regular) 0 x 7(gulls) 2 x 4 + x 2 y + xy 2 2(fold) 0 x 2 8(butterfly) 2 x 5 ± x 2 y + xy 3(cusp) 0 x 3 + xy 9 ± 3 x 3 ± xy 4 4 ± (lips/beaks) 1 x 3 ± xy x. 4 + x 2 y + xy 3 5(goose) 2 x 3 + xy x 5 + xy 6(swallowtail) 1 x 4 + xy ohmoto/class.html 25 ( ) 24 / 52
33 A 0 fold cusp 0.5 type normal form 0(regular) (x, y) (x, y) 1(fold) (x, y) (x 2, y) 2(cusp) (x, y) (x 3 + xy, y) ohmoto/class.html 25 ( ) 25 / 52
34 26 / 52 2: P T M P A P A ϕ : M R 2
35 27 / 52 A P ϕ P ϕ(p ) P
36 28 / 52 3: P M P ξ 2 + η 3 = 0 3/2-
37 29 / 52 P M 3 2
38 30 / 52 P
39 31 / 52 P P
40 32 / 52
41 33 / 52
42 1 : 1 A t A t type miniversal unfolding 3, 4(lips/beaks) (x, y) (x 3 ± xy 2 + ax, y) 6(swallowtail) (x, y) (x 4 + xy + ax 2, y).. ohmoto/class.html 25 ( ) 34 / 52
43 35 / 52 1 : 4: Lips/Beaks P 3
44 36 / 52 1 : 6: Swallowtail P P 4
45 2 : 2 A s,t A (s0,t 0 ) type miniversal unfolding 5(goose) (x, y) (x 3 + xy 3 + axy + bx, y) 7(seagull) (x, y) (x 4 + x 2 y + xy 2 + axy + bx, y) 8, 9(butterfly) (x, y) (x 5 ± x 3 y + xy + ax 3 + bx 2, y). ohmoto/class.html 25 ( ) 37 / 52
46 38 / 52 2 : 5: Goose
47 39 / 52 2 : 7: Gulls
48 40 / 52 2 : 8 Butterfly
49 41 / ( ) K > 0 (Fold) K = 0 (Fold) { (Lips/Beaks) 1 3 (Goose) 2 4 (Gulls) 2 K < 0 (Fold) 3 (Cusp) 4 (Swallowtail) 1. 5 (Butterfly) type 9, 10, 11
50 42 / 52 M S = {x 1, x 2,, x s } ϕ 1 (y) ϕ ϕ : M, S P 2, y A-
51 43 / 52 1
52 44 / 52 2
53 45 / 52 3 P 3 (= CP 3 ) d ( 4) M := { p = [x 0 : x 1 : x 2 : x 3 ] P 3 f(x 0, x 1, x 2, x 3 ) = 0 }, f d )
54 46 / 52 2 Q:
55 47 / 52 Q. butterfly
56 47 / 52 Q. butterfly Butterfly = 5d(d 4)(7d 12)
57 47 / 52 Q. butterfly Butterfly = 5d(d 4)(7d 12) Q.
58 47 / 52 Q. butterfly Butterfly = 5d(d 4)(7d 12) Q. Seagull = 2d(d 2)(11d 24) (V. Kulikov, 1983)
59 48 / 52 X n ξ = ξ 1 + c 1 (ξ) + + c n (ξ) H (X; Z), ( c k (ξ) H 2k (X) )
60 48 / 52 X n ξ = ξ 1 + c 1 (ξ) + + c n (ξ) H (X; Z), ( c k (ξ) H 2k (X) ) k ɛ k ξ = c j (ξ) = 0 (j n k + 1) CP 1 ξ 1 ɛ 1
61 49 / 52 X n ξ = ξ 1 + c 1 (ξ) + + c n (ξ) H (X; Z), ( c k (ξ) H 2k (X) ) k ɛ k ξ = c j (ξ) = 0 (j n k + 1) CP 1 ξ 1 ɛ 1 c 1 (ξ 1 ) = 1, c 1 (ɛ 1 ) = 0 H 2 (CP 1 ; Z) Z
62 0.9 Classification of map-germs = Thom polynomials η : singularity type T p(η) Z[c 1, c 2, ] f : N n P p n η T p(η)(c(f)) = f η- H 2n (M). = Z c i = c i (f) := c i (f T P T N) A 4 - Tp T p(a 4 ) = c c 2 1c 2 + 2c c 1 c 3 + 6c 4 ohmoto/class.html 25 ( ) 50 / 52.
63 0.9 Classification of map-germs = Thom polynomials η : singularity type T p(η) Z[c 1, c 2, ] f : N n P p n η T p(η)(c(f)) = f η- H 2n (M). = Z c i = c i (f) := c i (f T P T N) A 4 - Tp T p(a 4 ) = c c 2 1c 2 + 2c c 1 c 3 + 6c 4 Butterfly = 5d(d 4)(7d 12). ohmoto/class.html 25 ( ) 50 / 52
64 51 / 52
65 51 / 52 H (P n ; Z) = Z[a]/(a n+1 )
66 51 / 52 H (P n ; Z) = Z[a]/(a n+1 )
67 51 / 52 H (P n ; Z) = Z[a]/(a n+1 )
68 51 / 52 H (P n ; Z) = Z[a]/(a n+1 )
69 51 / 52 H (P n ; Z) = Z[a]/(a n+1 )
70 Modern Art integrates multiple views. ohmoto/class.html 25 ( ) 52 / 52
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467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 B =(1+R ) B +G τ C C G τ R B C = a R +a W W ρ W =(1+R ) B +(1+R +δ ) (1 ρ) L B L δ B = λ B + μ (W C λ B )
23 14 24 14 25 15 26 15 27 15 28 15 29 16 30 N = Q 17 31 R = R 2 18 32 N < R 18 33 19 34 20 35 20 36 21 37 21 38 21 39 22 40 22 2
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1W II K200 : October 6, 2004 Version : 1.2, kawahira@math.nagoa-u.ac.jp, http://www.math.nagoa-u.ac.jp/~kawahira/courses.htm TA M1, m0418c@math.nagoa-u.ac.jp TA Talor Jacobian 4 45 25 30 20 K2-1W04-00
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x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x
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No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
VI VI.21 W 1,..., W r V W 1,..., W r W W r = {v v r v i W i (1 i r)} V = W W r V W 1,..., W r V W 1,..., W r V = W 1 W
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1 16 10 5 1 2 2.1 a a a 1 1 1 2.2 h h l L h L = l cot h (1) (1) L l L l l = L tan h (2) (2) L l 2 l 3 h 2.3 a h a h (a, h) 4 2 3 4 2 5 2.4 x y (x,y) l a x = l cot h cos a, (3) y = l cot h sin a (4) h a
> > <., vs. > x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D > 0 x (2) D = 0 x (3
13 2 13.0 2 ( ) ( ) 2 13.1 ( ) ax 2 + bx + c > 0 ( a, b, c ) ( ) 275 > > 2 2 13.3 x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D >
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I n n A AX = I, YA = I () n XY A () X = IX = (YA)X = Y(AX) = YI = Y X Y () XY A A AB AB BA (AB)(B A ) = A(BB )A = AA = I (BA)(A B ) = B(AA )B = BB = I (AB) = B A (BA) = A B A B A = B = 5 5 A B AB BA A
A A = a 41 a 42 a 43 a 44 A (7) 1 (3) A = M 12 = = a 41 (8) a 41 a 43 a 44 (3) n n A, B a i AB = A B ii aa
1 2 21 2 2 [ ] a 11 a 12 A = a 21 a 22 (1) A = a 11 a 22 a 12 a 21 (2) 3 3 n n A A = n ( 1) i+j a ij M ij i =1 n (3) j=1 M ij A i j (n 1) (n 1) 2-1 3 3 A A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33
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yoshi@image.med.osaka-u.ac.jp http://www.image.med.osaka-u.ac.jp/member/yoshi/ II Excel, Mathematica Mathematica Osaka Electro-Communication University (2007 Apr) 09849-31503-64015-30704-18799-390 http://www.image.med.osaka-u.ac.jp/member/yoshi/
x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y)
x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 1 1977 x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) ( x 2 y + xy 2 x 2 2xy y 2) = 15 (x y) (x + y) (xy
0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9
1-1. 1, 2, 3, 4, 5, 6, 7,, 100,, 1000, n, m m m n n 0 n, m m n 1-2. 0 m n m n 0 2 = 1.41421356 π = 3.141516 1-3. 1 0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c),
2S III IV K A4 12:00-13:30 Cafe David 1 2 TA 1 appointment Cafe David K2-2S04-00 : C
2S III IV K200 : April 16, 2004 Version : 1.1 TA M2 TA 1 10 2 n 1 ɛ-δ 5 15 20 20 45 K2-2S04-00 : C 2S III IV K200 60 60 74 75 89 90 1 email 3 4 30 A4 12:00-13:30 Cafe David 1 2 TA 1 email appointment Cafe
卓球の試合への興味度に関する確率論的分析
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II (10 4 ) 1. p (x, y) (a, b) ε(x, y; a, b) 0 f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a 2 x a 0 A = f x (
II (1 4 ) 1. p.13 1 (x, y) (a, b) ε(x, y; a, b) f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a x a A = f x (a, b) y x 3 3y 3 (x, y) (, ) f (x, y) = x + y (x, y) = (, )
