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The painter of the Lascaux Cave (B.C.15,000) knew the geometry of apparent contours. http://www.math.sci.hokudai.ac.jp/ ohmoto/class.html 25 ( ) 2 / 52
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4 / 52 16 17 Desargues: 1593-1662
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11 / 52 :
11 / 52 :
11 / 52 :
12 / 52 : l l H H
13 / 52 : P 2
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
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 }.
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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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.
18 / 52 =
18 / 52 = d d
18 / 52 = d d =
18 / 52 = d d = =
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20 / 52 Gauss K P M elliptic point hyperbolic point parabolic point
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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
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, 0.. http://www.math.sci.hokudai.ac.jp/ ohmoto/class.html 25 ( ) 23 / 52
http://www.math.sci.hokudai.ac.jp/ ohmoto/class.html 25 ( ) 23 / 52 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, 0. 0.3 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-.
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 2 10 3 x. 4 + x 2 y + xy 3 5(goose) 2 x 3 + xy 3 11 3 x 5 + xy 6(swallowtail) 1 x 4 + xy http://www.math.sci.hokudai.ac.jp/ ohmoto/class.html 25 ( ) 24 / 52
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) http://www.math.sci.hokudai.ac.jp/ ohmoto/class.html 25 ( ) 25 / 52
26 / 52 2: P T M P A P A ϕ : M R 2
27 / 52 A P ϕ P ϕ(p ) P
28 / 52 3: P M P ξ 2 + η 3 = 0 3/2-
29 / 52 P M 3 2
30 / 52 P
31 / 52 P P
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1 : 1 A t A t0 1 0.6 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).. http://www.math.sci.hokudai.ac.jp/ ohmoto/class.html 25 ( ) 34 / 52
35 / 52 1 : 4: Lips/Beaks P 3
36 / 52 1 : 6: Swallowtail P P 4
2 : 2 A s,t A (s0,t 0 ) 2 0.7 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). http://www.math.sci.hokudai.ac.jp/ ohmoto/class.html 25 ( ) 37 / 52
38 / 52 2 : 5: Goose
39 / 52 2 : 7: Gulls
40 / 52 2 : 8 Butterfly
41 / 52 0.8 ( ) 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) 2-1 - 3 type 9, 10, 11
42 / 52 M S = {x 1, x 2,, x s } ϕ 1 (y) ϕ ϕ : M, S P 2, y A-
43 / 52 1
44 / 52 2
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 )
46 / 52 2 Q:
47 / 52 Q. butterfly
47 / 52 Q. butterfly Butterfly = 5d(d 4)(7d 12)
47 / 52 Q. butterfly Butterfly = 5d(d 4)(7d 12) Q.
47 / 52 Q. butterfly Butterfly = 5d(d 4)(7d 12) Q. Seagull = 2d(d 2)(11d 24) (V. Kulikov, 1983)
48 / 52 X n ξ = ξ 1 + c 1 (ξ) + + c n (ξ) H (X; Z), ( c k (ξ) H 2k (X) )
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
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
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). 0.10 A 4 - Tp T p(a 4 ) = c 4 1 + 6c 2 1c 2 + 2c 2 2 + 9c 1 c 3 + 6c 4 http://www.math.sci.hokudai.ac.jp/ ohmoto/class.html 25 ( ) 50 / 52.
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). 0.10 A 4 - Tp T p(a 4 ) = c 4 1 + 6c 2 1c 2 + 2c 2 2 + 9c 1 c 3 + 6c 4 Butterfly = 5d(d 4)(7d 12). http://www.math.sci.hokudai.ac.jp/ ohmoto/class.html 25 ( ) 50 / 52
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51 / 52 H (P n ; Z) = Z[a]/(a n+1 )
51 / 52 H (P n ; Z) = Z[a]/(a n+1 )
51 / 52 H (P n ; Z) = Z[a]/(a n+1 )
51 / 52 H (P n ; Z) = Z[a]/(a n+1 )
51 / 52 H (P n ; Z) = Z[a]/(a n+1 )
Modern Art integrates multiple views. http://www.math.sci.hokudai.ac.jp/ ohmoto/class.html 25 ( ) 52 / 52