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11 α β Γ γ δ ɛ ζ η Θ θ ι κ Λ λ µ ν Ξ ξ o Π π ρ Σ σ τ υ Φ φ χ Ψ ψ Ω ω

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14 (point) (line) (curved line) 3 (euclidean geometry) (definition) (axiom) (postulation )

15 A = B, B = C A = C A = B A + C = B + C A = B A C = B C (angle) (degree) (perpendicular) (intersection) (angle) L 1 L 2 A, B, A, B A A B B (opposite angle) 180

16 16 2 L 1 B A L 2 L 3 C' D D' A' C B' L 2 L 3 (parallel) L 2 L 3 L 1 L 2 A C A C (corresponding angle) A A A C A C A C (alternate angle) 2.3 (triangle) n (n 2) 180 (inner angle) (exterior angle) ABC A A B C BAC A 360 A AB DAC AC 180

17 ABC A BC LL L A L' B C B LAB C L AC A = 180 ( A + B) A + B A (congruence) BC5

18 18 2 D E m B' A n C' B C DE A DA EA DA C EA B ADE EBC BC DE (similar) ADE AB C (m : n) B C DE DE : B C = m : n DE = m n B C (circle) (radius) (diameter) (circle ratio) π d πd r 2πr π π (circumscribed circle) (inscribed circle) A B C ACB C C 1, C 2, C 3

19 (angle of circumference) O OAB, OBC, OAC OAC O OAC OBC O OBC AOB ACB C OAB AOB C ACB AOB 1 2 AB O AOB (right triangle)

20 20 2 ABC C BC (base) AC (height) AB (hypotenuse) A B c a b C (Pythagoras s theorem) a b c c 2 = a 2 + b 2 (2.1) 1 4 a + b c 2 a b a b b a b c c c 2 c a c a b b a a 2 b 2 a b c 2 a 2 + b 2 a + b A b C AB P AB Q

21 B c P a θ b 90 θ θ b A 2θ b C 90 θ Q ACP = θ ACP APC = θ QAC = 2θ ACQ AQC = ACQ PCQ = θ ACB = 90 PCB = 90 θ QBC PBC PB : CB = CB : QB a, b, c (c b) : a = a : (c + b) a 2 = (c b)(c + b) a 2 + b 2 = c 2 (2.2) 4 3m 4m 5m π n n 1 6 π 3

22 Q b O x P' P a OPQ OP P OP x a b a b (6 2 n ) OP Q ( a ) = x 2 + (2.3) 2 QP P ( a ) 2 b 2 = (1 x) 2 + (2.4) ( a 2 ) 2 b 2 = 2 2x (2.5) 2.3 x = 4 a 2 2 b a b 2 = 2 4 a 2 b = 2 4 a 2 (2.6) a = 1 b = π 6 π a 48, 96, 192, 384 π π 8

23 (natural number) 1, 2, 3,... 0 (integer number) (negative number) (finite decimal) (circulating decimal) (rational number) (real number) 1cm m 0.01m (irrational number) π e 2, 3 (imaginary number) 2.5 (coordinate) 17 xy x y x y x y x y A A(3,2) B B(-1, 0.5)

24 24 2 y A(3, 2) B(-1, 0.5) x x y xy (quadrant) x y A B d 2 (x, y) d 2 = x 2 + y 2 (2.7) AB (x a, y a ), (x b, y b ) d ab d 2 ab = (x a x b ) 2 + (y a y b ) 2 (2.8) 2.6 (algebra) (variable) x v t x = vt (2.9)

25 r A π A = πr 2 (2.10) π (constant) (unkown) (coefficient) π r A r r = (equation) (root) (solution) y = ax + by + cz (2.11) ax, by, cz (term) (identical equation) θ sin 2 θ + cos 2 θ = 1 (2.12) θ (formula) x, y a, b y = ax + b (linear equation) a = 1 2, b = 1

26 26 2 y x y = 1 2 x + 1 (2.13) (gradient) x a x y y x b (intercept) y y x v t x v t v = 50(km/h), t = 1(h) (simultaneous equation) x = vt v = 50(km/h) (2.14) t = 1(h) x 1, x 2, x a 11,, a 34 a 11 x 1 + a 12 x 2 + a 13 x 3 = a 14 (2.15) a 21 x 1 + a 22 x 2 + a 23 x 3 = a 24 (2.16) a 31 x 1 + a 32 x 2 + a 33 x 3 = a 34 (2.17) Gauss (elimination method) Gauss

27 x a21 a a31 a b 22,, b 34 a 11 x 1 + a 12 x 2 + a 13 x 3 = a 14 (2.18) b 22 x 2 + b 23 x 3 = b 24 (2.19) b 32 x 2 + b 33 x 3 = b 34 (2.20) x b32 b c 33, c 34 a 11 x 1 + a 12 x 2 + a 13 x 3 = a 14 (2.21) b 22 x 2 + b 23 x 3 = b 24 (2.22) c 33 x 3 = c 34 (2.23) 2.23 x x x x 2 (quadratic equation) x 3 (cubic equation) (nonlinear equation) y = x(x + 2) (-1, -1) x 0-2 y x x

28 28 2 y 0 x y = 0 x Newton- Raphson x 2 4 = 0 (factorisation) (x + 2)(x 2) = 0 x = ±2 x 2 = 4 x = ± x = ± 4 4 = 2 2 = (irrational number) (square root) a, b, c x ax 2 + bx + c = 0 (2.24) x 2 + b a x + c a = 0 x 2 + b a x = c a x 2 + b ( ) 2 ( ) 2 b b a x + = c 2a 2a a ( x + b ) 2 = b2 4ac 2a 4a 2 x + b 2a = ± b2 4ac 4a 2 x = b 2a ± b2 4ac 4a 2 = b ± b 2 4ac 2a a c a ( ) 2 b 2a (2.25) x

29 x 1 2 x 2.25 b 2 4ac x 2 = 1 x x 2 = 1 x = 1 1 i (imaginary unit) imagenary unit 17 (imaginary number) a, b a + bi (2.26) (complex number) b 2 4ac < 0 x 2 +x+1 = 0 x = 1 ± = 1 ± 3 2 = 1 2 ± 3 2 i (2.27) 1 2 ± 3 2 i (complex conjugate number) 2.8 (function) x 17 f f(x) 18 x, y, z f(x, y, z)

30 30 2 (argument) x, y a, b y = ax + b x y f(x) = ax + b (2.28) x f(x) = ax 2 + bx + c (2.29) x y y = f(x) (trigonometric ratio) A B θ c a b C θ sin θ b c, cos θ a c, tan θ b a (2.30) sin cos tan

31 θ θ θ tan θ = sin θ cos θ (2.31) b c a c = b a b2 a 2 +b 2 c 2 sin 2 θ + cos 2 θ = 1 (2.32) c + a2 2 c = a2 +b 2 2 c a 2 + b 2 = c 2 2 = 1 r 2 Hipparchus sin, cos Euler (degree) (radian) 1 (radian) y 90 π/2 rad θ rad 90 π rad θ rad 360 2π rad x 270 3π/2 rad π(rad) 180 π(rad) π π r r θ

32 (unit circle) (x, y) x θ { x = cos θ y = sin θ (2.33) θ (trigonometric function) f(θ) sin(θ) 2π 3π/2 π π/2 π/2 π 3π/2 2π θ cos(θ) cos θ, sin θ -1 1 x 2π (period) cos y (even function) sin (odd function) cos sin x θ π 2 cos θ + π 2 = sin θ sin (sine wave) t y = a sin(ωt φ) (2.34) a (amplitude) ω (angular frequency) φ (phase) 2π 1 1 sin cos (additional theorem) sin(α + β) sin(α + β) OAB AOB = α OB OB OBC BOC = β α + β

33 O α β 1 cos β sin β cos α C sin β α R Q cos β sin α P A cos β cos α sin β sin α B sin(α + β) OC 1 C OA OA P OB Q B CP R OAB OPQ OPQ BCQ BCR = α sin(α + β) CP sin(α + β) = AB + RC OC 1 BC sin β CR BC cos α = sin β cos α OB cos β AB OB sin α = cos β sin α sin(α + β) = sin α cos β + cos α sin β (2.35) cos(α + β) cos(α + β) OP cos(α + β) = OA P A OB cos β OA OB cos α = cos β cos α BC sin β BR BC sin α = sin β sin α cos(α + β) = cos α cos β sin α sin β (2.36) { sin(α + β) = sin α cos β ± cos α sin β cos(α + β) = cos α cos β sin α sin β (2.37) (sine formula) ABC A, B, C a, b, c

34 34 2 C b a A c D B C AB C D A B sin A = CD b, sin B = CD a (2.38) b sin A = a sin B = CD a sin A = b sin B B C A BC b sin B = a sin A = b sin B = c sin C c sin C (2.39) (cosine formula) BCD a 2 = CD 2 + BD 2 CD = b sin A, BD = c b cos A a 2 = b 2 sin 2 A + (c b cos A) 2 = b 2 sin 2 A + c 2 2bc cos A + b 2 cos 2 A = b 2 (sin 2 A + cos 2 A) + c 2 2bc cos A = b 2 + c 2 2bc cos A (2.40) b, c A a cos A = b2 +c 2 a 2 2bc a 2 = a 2 = b 2 + c 2 2bc cos A b 2 = c 2 + a 2 2ca cos B c 2 = a 2 + b 2 2ab cos C (2.41)

35 ABC A AP ABP ACB APB ABP AP ABP ABC BC sin A = CA sin B = AB sin C ACB = APB sin C = sin P ABP sin P = AB AB AP sin P = AP AB sin C = AP R BC sin A = CA sin B = AB = 2R (2.42) sin C x f() y y = f(x) f() y x (inverse function) x = f 1 (y) y = f(x) = ax + b x = y b a f 1 (y) = y b a A (2.43) B θ c a b C

36 36 2 θ θ = sin 1 b c, θ = cos 1 a c, θ = tan 1 b a (2.44) (-1) sin 1 cos 1 tan (exponential function) a x f(x) = a x (2.45) a (base) x (expornent) a x a y = a x+y (a x ) y = a xy a x = 0 y = y y=3 x y=2 x y=1.5 x y=0.5 x x (logarithmic function) y = a x x log a f(x) = log a x (2.46) 10 (common logarithm x = 8 f(8) = 10 8 f(8)

37 log log 10 x + log 10 y = log 10 xy log 10 x log 10 y = log 10 x log 10 y log 10 x y = y log 10 x log a x = log b x log b a log = 8 log = 8 a y y = log e x y = log 5 x y = log 10 x x x = 1 y = 0 x = 0 y = y = f(x) x 0 m x x y y m = f(x 0 + x) f(x 0 ) x (2.47) x 0 x

38 38 2 y x y x 0 x x 0 x 1 0 x y x 1 x 0 x x 0 x 0 m = f(x 0 + x) f(x 0 ) lim x 0 0 x (2.48) lim x 0 m (differential coefficient) (derived function) (differentiation) dy dx

39 dy dx = lim y x 0 x = lim f(x + x) f(x) x 0 x (2.49) y x dy dx 17 y, f (x) 18 y = x 2 + 2x dy dx = lim (x + x) 2 + 2(x + x) (x 2 + 2x) x 0 x (2x + 2) + ( x) 2 = lim x 0 x = lim 2x x x 0 = 2x + 2 (2.50) x x 0 (extremal value) x 0 x 2x + 2 = 0 x = y = a y = ax y = ax 2 y = x a y = sin x y = cos x y = tan x y = sin 1 x y = cos 1 x y = tan 1 x y = e x y = a x y = ln x dy dx = 0 dy dx = a dy dx = 2ax dy dx = axa 1 dy dx = cos x dy dx = sin x dy dx = 1 cos 2 x dy dx = 1 1 x 2 dy dx = 1 1 x 2 dy dx = 1 1+x 2 dy dx = ex dy dx = ax ln a dy dx = 1 x

40 40 2 y = a x 2.49 dy dx = lim a x+ x a x x 0 x = lim x 0 a x 1 x ax = Ca x lim x 0 a x 1 x = C (2.51) C = 1 a e y = e x e e (natural logarithm) log e ln 2.49 y = f(x) ± g(x) y = f(x)g(x) y = f(x) g(x) y = f(u), u = g(x) dy dx = df dx ± dg dx dy dx = df dx g ( + f dg dx dy dx = 1 df g 2 dy dx = dy du du dx ) dx g f dg dx (composite function) y = (x + a) 2 y = x 2 + 2ax + a 2 dy dx = 2x + 2a u = x + a y = u 2 u = x + a dy du = 2u du dx = 1 dy dx = dy du du dx = 2u = 2(x + a) (2.52)

41 f(x) x x, y z z = f(x, y) a, b, c z = ax 2 + bxy + cy 2 (2.53) z 0 x, y (partial differentiation) z x dz z dx d x x y y z = 2ax + by (2.54) x z = bx + 2cy (2.55) y z 0 x, y { 2ax + by = 0 (2.56) bx + 2cy = (integration) x x f(x) = x 2 0 x x S(x) x S(x + x)

42 42 2 y f(x) x S(x) x x S(x + x) S(x) x f(x) S(x + x) S(x) xf(x) S(x + x) S(x) x x f(x) (2.57) x 0 S(x + x) S(x) lim = f(x) (2.58) x 0 x f(x) S(x) S(x) (primitive function) S(x) = f(x)dx (2.59) f(x) = x 2 S(x) S(x) = 1 3 x3 + C C 0 C f(x) = x 2 x 0 x n x S S = xn x 0 f(x)dx (2.60) x 0 x n (definite integral) S(x) S(x n ) S(x 0 ) C

43 S = = xn x 0 x 2 dx [ 1 3 x3 + C ] xn x 0 = 1 3 x3 n 1 3 x3 0 (2.61) (vector) 19 (scalar) A(x a, y a ) B(x b, y b ) A B AB AB = (xb x a, y b y a ) y x AB = CD AB AB a a a

44 44 2 ( ) xb x a = a y b y a (2.62) m a (a 1, a 2,, a m ) a = a 1 a 2. a m (2.63) a k k m a = ( xa y a ) ( kxa, ka = ky a ) (2.64) a b a b a b c y b a c x ( ) ( ) ( ) xa xb xa + x a =, b =, a + b = b (2.65) y a y b y a + y b

45 c a b m a a a = x 2 a + y 2 a (2.66) m a (a 1, a 2,, a m ) a = a a a2 m (2.67) 1 (unit vector) (inner product) a b θ a b a b cos θ (2.68) a b θ a b cos θ a b b cos θ a b a θ = 90 0 θ = 0 a b a b cos θ a, b θ a b

46 46 2 a-b b θ a a b a b 2 = a 2 + b 2 2 a b cos θ (2.69) a b 2 (a b) (a b) a b 2 = (a b) (a b) = a 2 + b 2 2a b (2.70) a b = a b cos θ a = (a x, a y, a z ) b = (b x, b y, b z ) x e x = (1, 0, 0) y e y = (0, 1, 0) z e z = (0, 0, 1) a a = a x e x + a y e y + a z e z (2.71) a b = (a x e x + a y e y + a z e z ) (b x e x + b y e y + b z e z ) = a x e x b x e x + a x e x b y e y + a x e x b z e z + a y e y b x e x + a y e y b y e y + a y e y b z e z + a z e z b x e x + a z e z b y e y + a z e z b z e z (2.72) cos θ 0 1 { e x e y = e x e z = e y e z = 0 e x e x = e y e y = e z e z = 1 (2.73) a b = a x b x + a y b y + a z b z (2.74)

47 cos θ = a b a b (2.75) ( ) ax a y a z b x b y = a x b x + a y b y + a z b z (2.76) b z (outer product) a b θ a b a b sin θk (2.77) k b θ b sin θ a k a b a b (normal vector) a b b sin θ a = (a x, a y, a z ) b = (b x, b y, b z ) x e x = (1, 0, 0) y e y = (0, 1, 0) z e z = (0, 0, 1)

48 48 2 a b a b = (a x e x + a y e y + a z e z ) (b x e x + b y e y + b z e z ) = a x e x b x e x + a x e x b y e y + a x e x b z e z + a y e y b x e x + a y e y b y e y + a y e y b z e z + a z e z b x e x + a z e z b y e y + a z e z b z e z (2.78) sin θ 1 0 e x e y = e z e y e z = e x e z e x = e y e y e x = e z (2.79) e z e y = e x e x e z = e y e x e x = e y e y = e z e z = 0 a b = a x b y e z a x b z e y a y b x e z + a y b z e x + a z b x e y a z b y e x (2.80) = (a y b z a z b y )e x + (a z b x a x b z )e y + (a x b y a y b x )e z a b = a y b z a z b y a z b x a x b z a x b y a y b x (2.81) xy a = (a x, a y ) b = (b x, b y ) a b = (a x b y a y b x )e z (2.82) (a x b y a y b x ) 2.13 (matrix) 19

49 A Ã A A = a 11 a 12 a 1n a 21 a 22 a 2n... a m1 a m2 a mn (2.83) (column) (row) a ij i, j i j (a 11, a 12,, a 1n ). (a m1, a m2,, a mn ) (2.84) a 11 a 1n a 21. a 2n. a m1 a mn (2.85) (square matrix) A k k ( ) ( ) a b ka kb A =, ka = c d kc kd (2.86) ( ) ( ) ( ) a b e f a + e b + f + = c d g h c + g d + h (2.87) A, B A B A B

50 A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 B = b 11 b 12 b 13 b 21 b 22 b 23 b 31 b 32 b 33 AB = a 11b 11 + a 12 b 21 + a 13 b 31 a 11 b 12 + a 12 b 22 + a 13 b 32 a 11 b 13 + a 12 b 23 + a 13 b 33 a 21 b 11 + a 22 b 21 + a 23 b 31 a 21 b 12 + a 22 b 22 + a 23 b 32 a 21 b 13 + a 22 b 23 + a 23 b 33 a 31 b 11 + a 32 b 21 + a 33 b 31 a 31 b 12 + a 32 b 22 + a 33 b 32 a 31 b 13 + a 32 b 23 + a 33 b 33 (2.88) i j i j m n A n k B m k i j c ij Σ c ij = n a il b lj (2.89) l=1 2 3 A 3 2 B ( ) b a11 a 12 a 11 b b a 21 a 22 a 21 b 22 (2.90) 23 b 31 b 32 B c 11 a 11, a 12, a 13 b 11, b 21, b 31 c 11 = a 11 b 11 + a 12 b 21 + a 13 b 31 (2.91) c 12 a 11, a 12, a 13 b 12, b 22, b 32 c 12 = a 11 b 12 + a 12 b 22 + a 13 b 32 (2.92)

51 AB BA m = k m = k (unit matrix)e(unit matrix) E = (2.93) A E 2 2 A E ( ) ( ) ( ) a11 a a11 a = 12 a 21 a a 21 a 22 (2.94) (transposed matrix) A A T ( ) a b c a b d e f c d e (2.95) f a d 2 a T a d 2 = a T a = ( x a y a z a ) x a y a z a = x 2 a + y 2 a + z 2 a (2.96) (inverse matrix) A A 1 E A 1 A = E 2 2

52 52 2 A X ( ) ( ) ( x11 x 12 a11 a = x 21 x 22 a 21 a a 11 x 11 + a 21 x 12 = 1 a 12 x 11 + a 22 x 12 = 0 a 11 x 21 + a 21 x 22 = 0 a 12 x 21 + a 22 x 22 = 1 x 11,, x 22 ( ) ( x11 x 12 1 a22 a = 12 x 21 x 22 a 11 a 22 a 12 a 21 a 21 a 11 ) ) (2.97) (2.98) (2.99) 2 2 A ( ) A 1 1 a22 a = 12 a 11 a 22 a 12 a 21 a 21 a 11 1 a 11 a 22 a 12 a 21 ( ) ( ) a22 a 12 a11 a 12 = a 21 a 11 a 21 a 22 ( ) (2.100) (2.101) Gauss { a 1 x + a 2 y = a 3 b 1 x + b 2 y = b 3 (2.102) ( a1 a 2 b 1 b 2 ) ( x y ) = ( a3 b 3 ) (2.103)

53 ( ) 1 ( ) ( ) ( ) 1 ( ) a1 a 2 a1 a 2 x a1 a = 2 a3 b 1 b 2 b 1 b 2 y b 1 b 2 b 3 ( ) ( ) ( ) 1 ( ) 1 0 x a1 a = 2 a3 0 1 y b 1 b 2 b 3 ( ) ( ) ( ) x 1 b2 a = 2 a3 y a 1 b 2 a 2 b 1 b 1 a 1 b 3 (2.104) r x 2 + y 2 = r 2 (2.105) r y r P -r θ r x -r P x θ { x = r cos θ (2.106) y = r sin θ θ (parameter)

54 a b y = ax + b y θ x (x 0, y 0 ) a = (x a, y a ) t { x = x a t + x 0 y = y a t + y 0 (2.107) t a (x 0, y 0 ) (x 1, y 1 ) (x 0, y 0 ) a = (x 1 x 2, y 1 y 2 ) { x = (x 1 x 0 )t + x 0 y = (y 1 y 0 )t + y 0 (2.108) a, b t = 0 (x 0, y 0 ) t = 1 (x 1, y 1 ) t t (x 0, y 0 ) (x 1, y 1 ) 1 0 < t < 1 θ a = tan θ (cos θ, sin θ) (x 0, y 0 ) (x 1, y 1 ) (x 2, y 2 ) (x 3, y 3 ) (x 2, y 2 ) (x 3, y 3 ) { x = (x 3 x 2 )s + x 2 y = (y 3 y 2 )s + y 2 (2.109)

55 s x y { (x 1 x 0 )t + x 0 = (x 3 x 2 )s + x 2 (y 1 y 0 )t + y 0 = (y 3 y 2 )s + y 2 (2.110) ( x1 x 0 x 2 x 3 y 1 y 0 y 2 y 3 ) ( t s ) = ( ) x2 x 0 y 2 y 0 (2.111) t, s t ( ) ( t 1 y2 y = 3 x 3 x 2 s (x 1 x 0 )(y 2 y 3 ) (x 2 x 3 )(y 1 y 0 ) y 0 y 1 x 1 x 0 ) ( x2 x 0 y 2 y 0 ) (2.112) t = (x 2 x 0 )(y 2 y 3 ) (x 2 x 3 )(y 2 y 0 ) (x 1 x 0 )(y 2 y 3 ) (x 2 x 3 )(y 1 y 0 ) x = (x 1 x 0 ) (x 2 x 0 )(y 2 y 3 ) (x 2 x 3 )(y 2 y 0 ) (x 1 x 0 )(y 2 y 3 ) (x 2 x 3 )(y 1 y 0 ) + x 0 y = (y 1 y 0 ) (x 2 x 0 )(y 2 y 3 ) (x 2 x 3 )(y 2 y 0 ) (x 1 x 0 )(y 2 y 3 ) (x 2 x 3 )(y 1 y 0 ) + y 0 (2.113) (2.114) (x 0, y 0 ) a = (x a, y a ) P(x p, y p )

56 56 2 P t D 2 = (x a t + x 0 x p ) 2 + (y a t + y 0 y p ) 2 (2.115) t t t 0 t dd 2 = 0 dt 2x a (x a t + x 0 x p ) + 2y a (y a t + y 0 y p ) = 0 (x 2 a + y 2 a)t = x a (x p x 0 ) + y a (y p y 0 ) t = x a(x p x 0 ) + y a (y p y 0 ) x 2 a + y 2 a (2.116) t (x q, y q ) (x p, y p ) P(x p, y p ) (x a t + x 0 x p, y a t + y 0 y p ) 0 x a (x a t + x 0 x p ) + y a (y a t + y 0 y p ) = 0 (x 2 a + y 2 a)t = x a (x p x 0 ) + y a (y p y 0 ) t = x a(x p x 0 ) + y a (y p y 0 ) x 2 a + y 2 a (2.117) t (x p, y p ) (x 0, y 0, z 0 ) (x 1, y 1, z 1 )

57 (v x, v y, v z ) (x 1 x 0, y 1 y 0, z 1 z 0 ) x x 0 v x = y y 0 v y = z z 0 v z (2.118) (v x, v y, v z ) (direction vector) = t x = x 0 + v x t y = y 0 + v y t z = z 0 + v z t (2.119) t t x, y, z z n t = 0 (x 0, y 0, z 0 ) t = 1 (x 0 + v x, y 0 + v y, z 0 + v z ) (v x, v y, v z ) 1 v x x v y y v z z (direction cosine) z c v a b y x

58 58 2 v x v a y b y c v x = cos a v y = cos b (2.120) v z = cos c n n A(x a, y a, z a ) B(x b, y b, z b ) m : n (x, y, z) A x = x a + (x b x a )t y = y a + (y b y a )t z = z a + (z b z a )t (2.121) t = 0 A t = 1 B A, B m : n t = m m+n m x = x a + (x b x a ) m + n = nx a + mx b m + n m y = y a + (y b y a ) m + n = ny a + my b (2.122) m + n m z = z a + (z b z a ) m + n = nz a + mz b m + n (x a, y a, z a ) (v x, v y, v z ) B(x b, y b, z b ) xy B B ((x a + v x t) x b, (y a + v y t) y b, (z a + v z t) z b ) (2.123)

59 (v x, v y, v z ) 0 t t v x {(x a + v x t) x b } + v y {(y a + v y t) y b } + v z {(z a + v z t) z b )} = 0 (v 2 x + v 2 y + v 2 z)t = v x (x b x a ) + v y (y b y a ) + v z (z b z a ) t = v x(x b x a ) + v y (y b y a ) + v z (z b z a ) v 2 x + v 2 y + v 2 z (2.124) ax + by + cz = 1 (2.125) x, y z ax + by + cz = d d a d x + b d y + c d z = a, b, c z k(a, b, c) 1/a 1/c 1/b y x (a, b, c) (normal vector) 1 a x 1 b y 1 c z s, t x = x 0 + v x1 s + v x2 t y = y 0 + v y1 s + v y2 t z = z 0 + v z1 s + v z2 t (2.126) (x 0, y 0, z 0 ) v 1 (v x1, v y1, v z1 ), v 2 (v x2, v y2, v z2 )

60 60 2 k k(a, b, c) v 2 (v x2, v y2, v z2 ) v 1 (v x1, v y1, v z1 ) (0 < s + t < 1) (0 < s < 1) (0 < t < 1) (a, b, c) x y z x y = 0, z = 0 s, t x x a b, c 2.82 v 1, v 2 v 1 v 2 = (v y1 v z2 v z1 v y2, v z1 v x2 v x1 v z2, v x1 v y2 v y1 v x2 ) (2.127) (a, b, c) (x 0, y 0, z 0 ) x, y, z t a(x 0 + v x t) + b(y 0 + v y t) + c(z 0 + v z t) = 1 (av x + bv y + cv z )t = 1 (ax 0 + by 0 + cz 0 ) t = 1 ax 0 by 0 cz 0 av x + bv y + cv z (2.128) t

61 (a, b, c) (x 0, y 0, z 0 ) x = x 0 + at y = y 0 + bt z = z 0 + ct (2.129) t a(x 0 + at) + b(y 0 + bt) + c(z 0 + ct) = 1 (a 2 + b 2 + c 2 )t = 1 (ax 0 + by 0 + cz 0 ) t = 1 ax 0 by 0 cz 0 a 2 + b 2 + c 2 (2.130) t 2.16 (conic section)

62 (circle) r y r P -r θ r x -r P 2.1 x 2 + y 2 = r 2 (2.131) P x θ { x = r cos θ y = r sin θ (2.132) P (x, y) rectangular coordinates: (r, θ) (polar coordinate) r L L = 2πr dθ rdθ 0 2π L = 2π 0 rdθ = [rθ] 2π 0 = 2πr (2.133) S 0 r 2πr r S = 2πrdr 0 [ = 2π 1 ] r 2 r2 0 = πr 2 (2.134)

63 XYZ r Z r dθ r cosθ θ Y r X Z XY θ r cos θ 2πr cos θ rdθ 0 π 2 S 2 π 2 S = 2 2πr cos θrdθ 0 = 4πr 2 [sin θ] π 2 0 = 4πr 2 (2.135) V 0 r V = = r 0 [ 4 3 πr3 4πr 2 dr ] r 0 = 4 3 πr3 (2.136) (ellipsoid) (focal point)f F X X y (semi-major axsis) a (semi-minor axis) b

64 64 2 P P F P F 2a P x P F = a c P F = a + c P F + P F = 2a y P F = P F a b 2 + c 2 = a 2 b P -a F' -c F c a -b P F P F P (x, y) a, c P F = (x c) 2 + y 2 (2.137) P F = (x + c) 2 + y 2 (2.138) P F + P F = 2a (x c)2 + y 2 + (x + c) 2 + y 2 = 2a (x c)2 + y 2 = 2a (x + c) 2 + y 2 (2.139) a (x + c) 2 + y 2 = a 2 + cx (2.140) (a 2 c 2 )x 2 + a 2 y 2 = a 2 (a 2 c 2 ) (2.141) a 2 (a 2 c 2 ) b 2 + c 2 = a 2 x 2 a 2 + y2 a 2 c 2 = 1 (2.142) x 2 a 2 + y2 b 2 = 1 (2.143)

65 a a P x θ x, y { x = a cos θ y = a sin θ (2.144) a b P' P -a F' -c θ F c a -b -a P x P x y P b a { x = a cos θ y = b sin θ (2.145) x 2 a 2 = cos2 θ y 2 b 2 = sin2 θ (2.146) sin 2 θ + cos 2 θ = 1 x2 a + y2 2 b = y y = ± a b a2 x 2 (2.147)

66 a 1/4 S a b S = 4 a2 x 0 a 2 dx = 4 b a a2 x a 2 dx 0 = 4 b πa 2 a2 x a 4 2 a = πab (2.148) a b a b (oblateness) f f = a b a (2.149) a c (eccentricity) e e = c a = (a2 b 2 ) a (2.150) e 0 1 e 1 (parabola) 1 (hyperbola) P (x p, y p ) P T T P S

67 T' y b P T -a F' -c S F c a x -b (N x, N y ) f(x, y) f(x, y) = x2 a 2 + y2 b 2 1 (2.151) x, y f(x, y) x f(x, y) y = 2x a 2 = 2y b 2 (2.152) P (a cos θ, b sin θ) (N x, N y ) N x = 2 cos θ a N y = 2 sin θ b (2.153) t x = 2 cos θ t + a cos θ a y = 2 sin θ t + b sin θ b (2.154)

68 68 2 S x s y = 0 t t = b2 2 x s x s = b2 cos θ + a cos θ a ( a 2 b 2 ) = cos θ a cos θ = x p a = a2 b 2 a 2 x p e = e 2 x p (2.155) F S F S F S = ae + e 2 x p = e(a + ex p ) (2.156) F S = ae e 2 x p = e(a ex p ) (2.157) F P F P c c = ae a (x + c) 2 + y 2 = a 2 + aex (x + c)2 + y 2 = a + ex (2.158) F P F P = a + ex p F P F P = a + ex p (2.159) F P = a ex p (2.160) F S = ef P F S = ef P P S F P F 5 F F a (hyperbola) a e 1 x P P F P F 2a

69 y P F' -c -a a c F x P F P F P (x, y) a, c P F = (x c) 2 + y 2 (2.161) P F = (x + c) 2 + y 2 (2.162) P F P F = ±2a (x c)2 + y 2 (x + c) 2 + y 2 = ±2a (x c)2 + y 2 = ±2a + (x + c) 2 + y 2 (2.163) a (x + c) 2 + y 2 = a 2 + cx (2.164) (c 2 a 2 )x 2 a 2 y 2 = a 2 (c 2 a 2 ) (2.165) b 2 = c 2 a 2 b 2 x 2 a 2 y 2 = a 2 b 2 x 2 a 2 y2 b 2 = 1 (2.166) a a P P x Q

70 70 2 y P P' θ O a Q x P θ x y x = x = a cos θ (2.167) a cos θ cos 2 θ y2 b 2 = 1 y 2 = b 2 ( 1 1 cos 2 ) = b 2 ( cos 2 θ 1 cos 2 = b 2 tan 2 θ ) y = b tan θ (2.168) θ { x = a sec θ y = b tan θ (2.169) (parabola) e 0 P P F P Q

71 y Q P -c c F x P F P Q P Q = P F (x c)2 + y 2 = x + c P Q = x + c (2.170) P F = (x c) 2 + y 2 (2.171) (x c) 2 + y 2 = (x + c) 2 y 2 = 4cx (2.172) P (x p, y p ) P T T x QQ

72 72 2 y T' Q P Q' T c F x y 2 = 4cx y x x = 1 4c y2 (2.173) x y dx dy = 1 2c y (2.174) P (x p, y p ) 1 2c y p x b x = 1 2c y py + b P (x p, y p ) b x p = 1 2c y2 p + b b = x p 1 2c y2 p = x p 2x p = x p (2.175) T F x p + c P Q x p + c P F T F P T P Q P T F T P Q = T P F x

73 (effective digit) mm 0.1mm

74 (cancellation of significant digits) = = = = (3.1) π C float double float double

75 (histogram) n x (x 1, x 2,, x n ) (3.2) n x i (rank) a 0 a 1 1cm (frequency) n (a 0 a 1, a 1 a 2,, a n 1 a n ) (3.3) f (f 1, f 2,, f n ) (3.4) n = f 1 + f f n

76 76 3 n (relative frequency) 1 ( f1 n, f 2 n,, f ) n n 100 (3.5) (mean) x x = x 1 + x 2 + x x n n n = i=1 n x i (3.6) (m 1, m 2,, m n ) x = m 1f 1 + m 2 f 2 + m 3 f 3 + m n f n n n m i f i = i=1 n (3.7) (median) x i n/2 n

77 (mode) (deviation) d i d 1 = x 1 x d 2 = x 2 x. (3.8) d n = x n x 0 n (variance)s 2 xx S 2 xx = n (x i x) 2 i=1 n (3.9) n (degree of freedom) (independent variable) n n x x n 1 (population variance) (sample variance) s 2 xx s 2 xx = n (x i x) 2 i=1 n 1 (3.10) n = + (3.11)

78 (standard deviation) x, y x i y i Syy 2 Syy 2 = n (y i ȳ) 2 (covariance)s 2 xy i=1 (3.12) n S 2 xy = n (x i x)(y i ȳ) i=1 n (3.13) x y x, y x, y x y (correlation) x y (independent) x y x y (skew) S s = n (x i x) 3 i=1 ns 3 (3.14) 0

79 (kurtosis) S k = n (x i x) 4 i=1 ns 4 (3.15) (permutation) = 13! (3.16)! (factorial) n n! = 13! 8! (3.17) n r n! (n r)! np r P Permutation np r = n! (n r)! (3.18) (combination) r! n P r r! nc r C Combination nc r = n! r!(n r)! (3.19)

80 p 1 p ( 1 2 ) ( 1 2 ) C 2 5 C 2 ( 1 2 )5 1 2 n r f(r) f(r) = n C r p r (1 p) (n r) (3.20) (binomial distribution) (normal distribution) µ σ f(x) f(x) = 1 σ 2π e 1 2 ( x µ σ )2 (3.21) x = 0 f(x) = Ce h2 x 2 18 (De Moivre) (Gauss) (Hagen) p 1 2 x ɛ n ɛ x nɛ y ( 1 2 )n n n r x

81 x = (n r)ɛ rɛ y y = n C r ( 1 2) n = = (n 2r)ɛ (3.22) n! r!(n r)! ( ) n 1 (3.23) 2 r + 1 x 1 y 1 x 1 = (n r 1)ɛ (r + 1)ɛ y 1 = n C r+1 ( 1 2) n = = (n 2r 2)ɛ (3.24) n! (r + 1)!(n r 1)! x x = (n 2r)ɛ (n 2r 2)ɛ ( ) n 1 (3.25) 2 = 2ɛ (3.26) y ( n ( ) n 1 1 y = n C r n C r+1 2) 2 ( ) ( ) n n(n 1) (n r + 1) n(n 1) (n r) 1 = r! (r + 1)! 2 ( ) ( ) n n(n 1) (n r + 1) n(n 1) (n r)/(r + 1) 1 = r! (r + 1)!/(r + 1) 2 ( = 1 n r ) ( ) ( ) n n(n 1) (n r + 1) 1 r + 1 r! 2 ( ) ( ) n 2r n + r 1 = nc r r ( ) 2r n + r = y r + 1 ( ) 2x + 2ɛ = y 3.22 r = nɛ x (n + 2)ɛ x 2ɛ 2xy n ɛ (3.27) nɛ

82 82 3 y x = 2xy 2nɛ 2 (3.28) n ɛ nɛ 2 1 h 2 dy dx = 2h2 xy dy y = 2h2 xdx (3.29) 1 y dy = 2h 2 xdx ln y = h 2 x 2 + C y = Ce h2 x 2 (3.30) f(x) = e x2 y x x = 0 C, h 1 C = y = h π h π e h2 x 2 (3.31) dy dx = 2h3 e h2 x 2 x π d 2 y dx 2 = 2h3 e h2 x 2 + 4h5 e h2 x 2 x 2 π π = 2h3 π e h2 x 2 (1 2h 2 x 2 ) (3.32)

83 h 2 x 2 = 0 x x = ± 1 2h (σ 2 ) 3.31 x 2 σ 2 = h π = 1 2h 2 x 2 e h2 x 2 dx σ = ± 1 2h (3.33) σ σ = 1 2h h = 1 2σ 3.31 f(x) = 1 σ 2π e 1 2 ( x σ )2 (3.34) 0 µ y x σ +σ σ +2σ σ +3σ

84 (error) 1. (gross error) 2. (systematic error) 3. (accident error) (calibration) (most probable value) (residual) XY n (x 1, y 1 ), (x 2, y 2 ),, (x n, y n ) ( x, ȳ) (X, Y )

85 Y (x, y) (x i, y i ) (X, Y) X X σ x σ x = n (x i x)/n (3.35) i=1 RMSE(Root Mean Square Error) RMSE = n (x i X)/n (3.36) i=1 3.5 (propagation law of errors)

86 Lm k x x = L/k n x (x 1, x 2,, x n ) X e i e 1 = x 1 X e 2 = x 2 X. (3.37) e n = x n X σ 2 x σ 2 x = n (x i X) 2 i=1 n (3.38) y a y y = ax e 1 = a(x 1 X) e 2 = a(x 2 X). (3.39) e n = a(x n X) σ y n a 2 (x i X) 2 i=1 σy 2 = n = a 2 σx 2 σ y = aσ x (3.40) 2cm 100 2m 68%

87 x i σ x y i σ y z i z i = x i + y i X, Y e 1 = x 1 X + y 1 Y e 2 = x 2 X + y 2 Y. (3.41) e n = x n X + y n Y σ z n (x i X + y i Y ) 2 σz 2 = = i=1 n n (x i X) 2 + (y i Y ) 2 + 2(x i X)(y i Y ) i=1 = σ 2 x + σ 2 y + n n 2 (x i X)(y i Y ) i=1 n (3.42) P n i=1 (x i X)(y i Y ) n x y x i X y i Y 0 σ z σz 2 = σx 2 + σy 2 σ z = σx 2 + σy 2 (3.43) 1cm 1.41cm 1cm 0.1cm cm 1cm 3.5.3

88 88 3 x, y z z = xy e 1 = (x 1 X)(y 1 Y ) e 2 = (x 2 X)(y 2 Y ). (3.44) e n = (x n X)(y n Y ) σ z σ 2 z = = n ((x i X)(y i Y )) 2 i=1 n n yi 2 (x i X) 2 + x 2 i (y i Y ) 2 + 2XY (x i X)(y i Y ) (x i y i XY ) 2 i=1 n (3.45) P n P i=1 (xiyi XY n )2 i=1 n (xi X)(yi Y ) n x y x i X y i Y 0 σ z P n i=1 y2 i n P n i=1 x2 i n σ 2 z = n i=1 y 2 i n σ2 x + n i=1 n x 2 i σ2 y x, y σz 2 = y 2 σx 2 + x 2 σy 2 σ z = y 2 σx 2 + x 2 σy 2 (3.46) x i σ xi y y f y = f(x 1, x 2,, x n ) (3.47)

89 y σ y ( ) 2 ( ) 2 ( ) 2 f f f σy 2 = σx1 2 + σx σxn 2 (3.48) x 1 x 2 x n (x 1, x 2,, x n ) x y = ax σ y x σ x σ 2 y = ( ) 2 y σx 2 x = a 2 σ 2 x σ y = aσ x (3.49) x, y z = x + y σ z x, y σ x, σ y σ 2 z = ( ) 2 z σx 2 + x ( ) 2 z σy 2 y = σx 2 + σy 2 (3.50) σ z = σx 2 + σy 2 (3.51) x, y z = xy σ z x, y σ x, σ y σ 2 z = ( ) 2 z σx 2 + x ( ) 2 z σy 2 y = y 2 σx 2 + x 2 σy 2 σ z = y 2 σx 2 + x 2 σy 2 (3.52) n (x 1,, x n ) σ x x

90 90 3 x = n i=1 x i n (3.53) x σ 2 x = = n ( x i=1 n i=1 x i ) 2 σ 2 x ( ) 2 1 σx 2 n = 1 n σ2 x (3.54) σ x = 1 n σ x (3.55) 1 n Gauss X n (x 1, x 2, x n ) X x i (least square method) X Φ n Φ = (x i X) 2 (3.56) i=1 Φ X 2 Φ X 0 X

91 dφ n dx = 2 (x i X) = 0 i=1 nx n x i = 0 i=1 X = n x i /n (3.57) i= AB BC AB x 1, x 2 BC y AC z AB BC A B C x 1 x 2 y z AB BC X, Y Φ = (X x 1 ) 2 + (X x 2 ) 2 + (Y y) 2 + (X + Y z) 2 (3.58) Φ X, Y Φ X Y 0 X, Y Φ X = 2(X x 1) + 2(X x 2 ) + 2(X + Y z) = 0 (3.59) Φ = 2(Y y) + 2(X + Y z) = 0 Y { 3X + Y X + 2Y = x 1 + x 2 + z = y + z (3.60) X, Y

92 (x, y) x, y y = ax + b (regression analysis) (regression function) y y=ax+b (x 1, y 1 ) (x n, y n ) (x 2, y 2 ) x y = ax + b n (x 1, y 1 ), (x 2, y 2 ), (x n, y n ) ax i + b y i a, b Φ Φ = n (ax i + b y i ) 2 (3.61) i=1 Φ X, Y Φ X Y 0 X, Y Φ n a = 2 {x i (ax i + b y i )} = 0 i=1 Φ n b = 2 (ax i + b y i ) = 0 i=1 (3.62) [] ( [ ] x 2 i [x i ] [x i ] n ) ( a b ) = ( [xi y i ] [y i ] ) (3.63)

93 a, b ( ) ( a 1 = b n [x 2 i ] [x [ n [xi ] y i ] [x i ] [y i ] i] [x i ] x 2 i [yi ] [x i ] [x i y i ] ) (3.64) (correlation coefficient) (variance) (co-variance) x ( x) (v x ) y (ȳ) (v y ) x, y (v xy ) v x = v y = v xy = n (x i x) 2 /n (3.65) i=1 n (y i ȳ) 2 /n (3.66) i=1 n (x i x)(y i ȳ)/n (3.67) i=1 x, y (r) r = v xy vx vy (3.68) 1 r y-y i y-y i x-x i x-x i (x i x) (y i ȳ)

94 94 3 (x i x) (y i ȳ) (x i x) a i (y i ȳ) b i n n a(a 1, a 2, a 3,, a n ) b(b 1, b 2, b 3,, b n ) a 1 b 1 +a 2 b 2 +a 3 b 3 + +a n b n n a 2 = (a 2 1 +a 2 2 +a a 2 n) r θ 1 r r = a b = cos θ (3.69) a b 3.8 (coordinate transformation) (u, v) (x, y) (x, y) (u, v) f x, f y { x = f x (u, v) y = f y (u, v) (3.70) uv uv xy uv 1 xy m

95 uv (x 0, y 0 ) m uv xy { x = mu + x 0 y = mv + y 0 (3.71) u, v { u = x x 0 m v = y y 0 m (3.72) z r B C

96 96 3 y C(x 2, y 2 ) B(x 1, y 1 ) β α A(r, 0) x r x A(r, 0) α B(x 1, y 1 ) α r x 1 = r cos α (3.73) y 1 = r sin α (3.74) B(x 1, y 1 ) β C(x 2, y 2 ) α, β, r x 1, y 1 x 2 = r cos(α + β) = r cos α cos β r sin α sin β = x 1 cos β y 1 sin β (3.75) y 2 = r sin(α + β) = r sin α cos β + r cos α sin β = x 1 sin β + y 1 cos β (3.76) B(x 1, y 1 ) C(x 2, y 2 ) ( ) ( ) ( ) x2 cos β sin β x1 = sin β cos β y 2 y 1 (3.77) (rotation matrix) { x = cos θu sin θv + x 0 y = sin θu + cos θv + y 0 (3.78) (u, v) (x, y) θ (x 0, y 0 ) (Helmart transformation) a, b, c, d { x = au bv + c y = bu + av + d (3.79) c, d a 2 + b

97 cos θ a sin θ b cos sin -1 1 a, b V Y U X (Affine transformation) (skew) V Y U X a, b, c, d, e, f { x = au + bv + c y = du + ev + f (3.80) x y z xy z z x x y y

98 98 3 X Y Z cos θ sin θ 0 sin θ cos θ cos θ 0 sin θ sin θ 0 cos θ cos θ sin θ 0 sin θ cos θ x y z x y z x y z (3.81) (3.82) (3.83) (two dimensional projective transformation) a 1, a 2,, a 8 u = a 1x + a 2 y + a 3 a 7 x + a 8 y + 1 v = a 4x + a 5 y + a 6 a 7 x + a 8 y + 1 (3.84) V U X Y

99 (three dimensional projective transformation) a 1, a 2,, a 11 u = a 1x + a 2 y + a 3 z + a 4 a 9 x + a 10 y + a 11 z + 1 v = a (3.85) 5x + a 6 y + a 7 z + a 8 a 9 x + a 10 y + a 11 z + 1 Z V U X Y (ground control point data) (x i, y i ) (u i, v i ) (Newton-Raphson method) f(x) = 0 x x = x 0 x = x 0 f(x) f(x) x

100 100 3 x = x 1 x = x 1 f(x) f(x) x 2 f(x i ) f(x i+1 ) y f(x) x 2 x 1 x 0 x x = x 0 x x = x 1 f (x 0 ) = f(x 0) x 0 x 1 (3.86) x 1 x 1 = x 0 f(x 0) f (x 0 ) x 2 = x 1 f(x 1) f (x 1 ) x 3 = x 2 f(x 2) f (x 2 ) (3.87) f(x) = x 2 5 f(x) = 0 5 f(x) = x 2 5 f (x) = 2x

101 x 0 = 3 x 1 = = x 2 = = x 3 = = x 4 = = (3.88) 5 = (Taylor) (sequence) (series) f(x) (Taylor series) f(x) = f(a) + f (1) (a) 1! (x a) + f (2) (a) 2! (x a) 2 + f (3) (a) (x a) 3 + (3.89) 3! f (n) (a) f(a) n f(x) = f(a) + f (1) (a)(x a) f(x) f(a) = f (1) (a)(x a) f (1) (a) = f(x) f(a) x a a (3.90)

102 (Maclaurin) a = 0 (Maclaurin series) f(x) = f(0) + f (1) (0) 1! x + f (2) (0) 2! x 2 + f (3) (0) x 3 + (3.91) 3! sin(x) = ! x + 0 2! x ! x ! x ! x5 + = 1 1! x 1 3! x ! x5 1 7! x7 + (3.92) cos(x) = ! x + 1 2! x ! x ! x ! x5 + = 1 1 2! x ! x4 1 6! x6 + (3.93) x(radian) sin ( ) π 4 7 ( π ) sin 1 ( π ) 1 ( π ) 3 1 ( π ) 5 1 ( π ) 7 + (3.94) 4 1! 4 3! 4 5! 4 7! (3.95) sin ( π 4 ) = π e x e x = ! x + 1 2! x ! x ! x ! x5 + (3.96) x = 1 e e =

103 (Euler s formula) sin, cos e ix (function of complex variable) e ix cos sin e ix = ! (ix) + 1 2! (ix) ! (ix) ! (ix) ! (ix)5 + = 1 1 2! x ! x4 1 ( 1 6! x6 + + i 1! x 1 3! x ! x5 1 ) 7! x7 + = cos x + i sin x (3.97) e ix e ix = cos x i sin x x = π e iπ = 1 e ±ix = cos x ± i sin x (3.98) e sin, cos (e ix ) n = e inx (de Moivre s theorem) (cos x + i sin x) n = cos nx + i sin nx (3.99) e i(α+β) = cos(α + β) + i sin(α + β) e i(α+β) = e iα e iβ = (cos α + i sin α)(cos β + i sin β) = cos α cos β sin α sin β + i(sin α cos β + cos α sin β) (3.100) sin x, cos x e ix + e ix = (cos x + i sin x) + (cos x i sin x) = 2 cos x (3.101)

104 104 3 e ix e ix = (cos x + i sin x) (cos x i sin x) = 2i sin x (3.102) cos x = eix + e ix 2 sin x = eix e ix 2i (3.103) (3.104) e x θ e iθ = cos θ + i sin θ sinθ θ cosθ (complex plane) e iθ f(x) 2π (Fourier series) f(x) = a (a n cos nx + b n sin nx) (3.105) n=1 a n, b n (Fourier coefficient) n 2π n a n, b n

105 a n, b n a n = 1 π b n = 1 π π π π π f(x) cos nxdx (n = 0, 1, 2, ) (3.106) f(x) sin nxdx (n = 1, 2, 3, ) (3.107) e ix + e ix a n cos nx + b n sin nx = a n 2 e ix + e ix = a n 2 = a n ib n 2 + b n e ix e ix 2i ib n e ix e ix 2 e ix + a n + ib n e ix (3.108) 2 C 0 = a0 2, C n = an ibn 2, C n = an+ibn f(x) = C 0 + n= C n e inx (3.109) C n = a n ib n 2 C n = 1 2 a n i 1 2 b n = 1 2π = 1 2π = 1 2π π π π π π π f(x) cos nxdx i 1 2π π π f(x)(cos nx i sin nx)dx f(x) sin nxdx f(x)e inx dx (3.110) C n C n 2π n (power spectrum) a n, b n C n C n = a n ib n 2 = a2 n + b 2 n 4 an + ib n 2 (3.111) a n, b n θ n (phase spectrum) θ n = tan 1 ( bn a n ) (3.112)

106 π (Fourier transformation) f(x) g(s) g(s) = 1 2π f(x)e isx dx (3.113) g(s) f(t) (inverse Fourier transformation) f(x) = 1 2π g(s)e isx ds (3.114) 3.13 P (x p, y p, z p ) P (x p, y p, z p ) (v x, v y, v z ) (gradient) xyz xyz φ ( φ x, φ y, φ ) z (3.115)

107 ( φ φ = x, φ y, φ ) z (3.116) φ gradφ φ ( = x, y, ) (3.117) z (divergence) A(A x, A y, A z ) A = A x x + A y y + A z z (3.118) A diva φ (A x, A y, A z ) (rotation) A(A x, A y, A z ) ( Az A = y A y z, A x z A z x, A y x A ) x y (3.119) A rota φ (A x, A y, A z ) (Laplacian) φ φ φ

108 108 3 ( φ = x, y, z = ) ( 2 φ x φ y φ z 2 ( φ x, φ y, φ ) z ) (3.120) = = 2 φ (3.121) 2

109 SI (International System of Units) MKSA

110 110 4 M K S A 4 1 (the meter standard) 1m (µm) (nm) (atomic clock) m 1m 1/299,792, ,792,458(m/s) 1m 1 0 2π = = (total station) (transit) (theodolite)

111 (laser scanner) LiDAR(light detecting and ranging) GPS GPS(global positioning system) (positioning) GPS

112 112 4 GPS GPS m cm mm GPS GPS GPS GPS VRS GPS INS INS(inertia navigation system) GPS GPS INS IMU(inertia measurement system) 4.3

113 (levering plate) x y x x y y y x x

114 x A h Z + P X r h A + Y P r P X r (r, 0, 0) Y (h) Z (A) P A, h x y z = = cos( A) sin( A) 0 sin( A) cos( A) r cos( h) cos( A) r cos( h) cos( A) r sin( h) cos( h) 0 sin( h) sin( h) 0 cos( h) r 0 0 (4.1)

115 X P cm 2cm

116 (forward intersection) (trilateration) C (x c, y c ) A, B x AB y C (x c, y c ) b a A (x a, 0) θ B (x b, 0) c x 1 x AC b BC a CAB cos θ c = x b x a cos θ = b2 + c 2 a 2 2bc (4.2) C x x c x c = x a + x 1 = x a + b cos θ = x a + b b2 + c 2 a 2 2bc = x a + b2 + c 2 a 2 2c (4.3)

117 C y y c = b 2 x 2 1 (4.4) x 1 = b2 +c 2 a 2 σ x1 σ a, σ b, σ c 2c ( σx1 2 x1 = a = ) 2 σ 2 a + ( a c ) 2 σ 2 a + ( x1 b ( b c ) 2 σ 2 b + ) 2 σ 2 b + y c σ yc σ 2 yc = ( ) 2 yc σa 2 + a ( ) 2 yc σb 2 + b ( x1 c ( c 2 b 2 + a 2 2c 2 ) 2 σ 2 c ) 2 σ 2 c (4.5) ( ) 2 yc σc 2 (4.6) c y c a y c = yc 2 a (yc 2 ) a = 1) (x2 a 2y c y c a = 2x 1 x 1 a y c y c a = + c 2 a 2 b2 a 2c c y c a = a ( b 2 + c 2 a 2 ) y c 2c 2 y c b (yc 2 ) = 2b (x2 1) b b y c 2y c b = 2b 2x x 1 1 b y c y c b = b b2 + c 2 a 2 b 2c c y c b = 1 ( 2c 2 y c 2c 2 b b2 + c 2 a 2 2c 2 = b ( a 2 b 2 + c 2 ) y c 2c 2 ) b (4.7) (4.8)

118 118 4 yc c (yc 2 ) = (x2 1) c c 2y c y c c = 2x 1 x 1 c y c y c c = x 1 c2 b 2 + a 2 y c 2c 2 y c C C y C (x c, y c ) b a A x a A c B x b B x C A, B x AB c C A B a sin A = b sin B = c sin C C 180 A B sin C sin C = sin(180 (A + B)) = sin(a + B) (4.9) = sin A cos B + cos A sin B (4.10) C b b b = C c sin B sin A cos B + cos A sin B (4.11) c cos A sin B x c = b cos A + x a = sin A cos B + cos A sin B + x a (4.12) c sin A sin B y c = b sin A = sin A cos B + cos A sin B (4.13)

119 x c y c σ xc σ yc A B σ a, σ b ( σxc 2 xc = A ( σyc 2 yc = A ) 2 σa 2 + ) 2 σa 2 + ( xc B ( yc B ) 2 σb 2 (4.14) ) 2 σb 2 (4.15) x c, y c x c A c sin A sin B(sin A cos B + cos A sin B) c cos A sin B(cos A cos B sin A sin B) = (sin A cos B + cos A sin B) 2 c sin A sin B sin(a + B) c cos A sin B cos(a + B) = sin 2 (4.16) (A + B) x c B c cos A cos B(sin A cos B + cos A sin B) c cos A sin B( sin A sin B + cos A cos B) = (sin A cos B + cos A sin B) 2 c cos A cos B sin(a + B) c cos A sin B cos(a + B) = sin 2 (4.17) (A + B) y c A c cos A sin B(sin A cos B + cos A sin B) c sin A sin B(cos A cos B sin A sin B) = (sin A cos B + cos A sin B) 2 c cos A sin B sin(a + B) c sin A sin B cos(a + B) = sin 2 (4.18) (A + B) y c B c sin A cos B(sin A cos B + cos A sin B) c sin A sin B( sin A sin B + cos A cos B) = (sin A cos B + cos A sin B) 2 c sin A cos B sin(a + B) c sin A sin B cos(a + B) = sin 2 (4.19) (A + B) (σ a, σ b ) (backward intersection)

120 120 4 A B θ ab θ bc C P 1 θ ab θ bc P 2 A B θ ab B C θ bc P P A, B, C AP B θ ab AB AB ABP ABP O R o R o = 1 2 (xb x a ) 2 + (y b y a ) 2 sin θ ab (4.20) R o AB α ab BAO AB α = tan 1 y b y a x b x a BAO AOB AOB = 360 2θ ab 90 θ ab O (x o, y o ) { x o = R o cos(α ab (θ ab 90 )) + x a y o = R o sin(α ab (θ ab 90 )) + y a (4.21) α ab (θ ab 90 ) AB

121 BCP Q (x q, y q ) R q = 1 2 (xb x c ) 2 + (y b y c ) 2 sin θ cb (4.22) CBQ 90 θ cb Q (x q, y q ) { x q = R q cos(α cb + (90 θ cb )) + x c y q = R q sin(α cb + (90 θ cb )) + y c (4.23) α cb + (90 θ cb ) CB P P B P OQP OP O R 0 QP Q R q

122 122 4 OQ φ = QOP, s = OQ cos φ = R2 q + s 2 R 2 o 2R q s (4.24) OQ β β = tan 1 y q y o x q x o P { x p = R q cos(φ + β) + x o (4.25) y p = R q sin(φ + β) + y o φ + β OQ 4.6 (control point) (point of triangulation) (control point surveying) x y

123 x (latitude) y (departure) X x r y Y x y θ r { x = r cos θ y = r sin θ (4.26) x y y x x θ Y r X

124 (traversing) A(x a, y a ) B B C B Y B D θ 2 r 3 r 1 r 2 C r 4 θ 4 A θ 1 X θ 3 E A B A ( x 1, y 1 ) B (x b, y b ) { x 1 = r 1 cos θ 1 y 1 = r 1 sin θ 1 { x b = x a + x 1 y b = y b + y 1 (4.27) B B C ( x 2, y 2 ) C (x c, y c ) C θ 1 + θ 2 π { { x 2 = r 2 cos(θ 1 + θ 2 π) x c = x b + x 2 (4.28) y 2 = r 2 sin(θ 1 + θ 2 π) y c = y b + y 2 θ 2 C B C B A C

125 Y B r 2 r 1 θ 2 C A θ 1 X θ3 r 3 r 6 θ 6 θ 4 D F r 5 θ 5 r 4 E n (n 2)π δ n δ = (n 2)π θ i (4.29) δ (closing error) v i v i = 1 n δ (4.30) i=1

126 126 4 n x i = 0 (4.31) i=1 n y i = 0 (4.32) i=1 0 x ɛ x y ɛ y ɛ x = ɛ y = n x i (4.33) i=1 n y i (4.34) i=1 ɛ ɛ = ɛ 2 x + ɛ 2 y (4.35) ɛ r (closing ratio) ɛ r = ɛ ri (4.36) (compass rule) x y v xi, v yi x v xi = r i ri ɛ x (4.37) (transit rule) x y v xi = x i x i ɛ x (4.38)

127 π (triangulation) 2 2 (base line) 17 (Snell) E C F A B D AB A, B, C A, B, C π A, B, C AC BC BCD BC (trilateration)

128 128 4 GPS E C F A B D A B A(X a, Y a ) B(X b, Y b ) A Q r α X Q θ φ α S B(x b, y b ) A(x a, y a ) Y A B φ φ = tan 1 x a x b y a y b Q θ = φ α Q (x q, y q ) { x q = r sin(φ α) (4.39) y q = r cos(φ α) 4.6.6

129 r u A B A(x a, y a ) B(x b, y b ) A Q r α Y V Q(x q, y q ) U v B(x b, y b ) A(x a, y a ) X Q (x q, y q ) A B U-V Q UV { u = r cos α v = r sin α (4.40) U-V X-Y X U θ A (x q, y q ) ( xq y q θ ) ( cos θ sin θ = sin θ cos θ ) ( u v ) ( xa + y a ) (4.41) θ = tan 1 y a y b x a x b (4.42) θ θ Q tan 1 atan() ( π 2 π 2 ) π 2 θ θ π

130 130 4 atan2() ( π π) (level) 2 (staff) 2 (direct leveling) H = H a H b (4.43) (bench mark) BM Ha Hb H H BM1 BM2 n 1 h i H n = H h i (4.44) i=1

131 s n h n s 1 s 2 h 1 h 2 h n-1 H BM1 BM2 s i δ i δ i = i j=1 s j (4.45) n s j j= (indirect leveling) θ B A H H = B tan θ (4.46) H = A sin θ (4.47) H A B θ B B

132 (level point) 21km (earth ellipsoid) 3 (Eratosthenes) 17 19

133 (geodetic datum) WGS Bessel m GRS80 WGS84 cm WGS (m) Bessel , GRS , WGS , GRS80 2 1:25000 GRS80 1: m 0m (geoid) 150m

134 m

135 CCD(charge coupled device: ) CMOS(complementary metal oxide semiconductor: ) (image sensor)

136 θ 1 (incidence angle) θ 2 (reflection angle) θ 1 = θ 2 (law of reflection) A θ 1 θ 2 P O θ 3 B (refraction) (refracting angle) θ 3 (Snell) 17 (Snell s law) n 1 n 3 θ 1 θ 3 sin θ 1 sin θ 3 = n 3 n 1 (5.1) (Fermat) (Huygens) 299,792,458(m/s) 1661 (Fermat s principle) A B B PB 1 O B

137 P 1,, P 5 P 1 A P 5 B P 1 A P 5 C A BC P 2 P 4 P 5 C BC DA AE ABC CEA θ 1 θ 2 P 1 P 2 P 3 P 4 P 5 B E θ 1 θ 2 D A C P 1,, P 5 P 1 A P 5 B P 1 P 5 C A AE P 2 P 4 P 5 C θ 3 P 1 P 2 P 3 P 4 P 5 B D A θ 1 C θ 3 E

138 (focal point) (axis of lens) A F C F B A F' B C F a b a b PAC P BC AC a = BC b FC (focal length) f ACF ABP AC f AC + BC = b BC = AC b a (5.2) (5.3) 1 f = 1 a + 1 b (5.4) A F' B C F a b 1 a = 0 1 f = 1 b

139 A F' a B C F b b (resolution) (aperture) (visible light) 0.4 µm 0.8 µm D x P C t 1 D θ P x C t 2 f P P C P t 1, t 2 Ct 1, Ct 2 λ Ct 2 Ct 1 > λ (5.5) f x x f = λ D (Rayleigh) 19 Bessel

140 140 5 x f = 1.22 λ D (5.6) MTF(modulated transfer function) MTF F (F number) D f F f D (5.7) 5.2.4

141 ɛ d 1 D 2 D 1 ε d 2 f ɛ (depth of focus) D 1 d 1 D 2 d 1 D ε d 2 f (depth of field)

142 (exposure) 1 1/2, 1/4, 1/16, 1/30, 1/60, 1/125, 1/250, 1/500, 1/1000,... 1/2 F (aperture) 1.4, 2.8, 4, 5.6, 8, 11, 16, 22, 32 2 F 1/2 1/4 2 F (speed) ISO ISO , 400, 800, 1600 ISO200 ISO100 ISO

143 ISO (white balance) f W H Column Raw

144 144 5 U, V (interior orientation) (radial distortion) θ d r r d a 1, a 3, a 5, a 7 d = a 1 r + a 3 r 3 + a 5 r 5 + a 7 r 7 (5.8) (tangental distortion) (parallel projection) (central projection)

145 u (X 0, Y 0, Z 0 ) (ω, φ, κ) u v w ω φ κ X Y Z w w v v u ω φ κ (u, v, w)

146 146 5 (x, y, z) u v = cos ϕ 0 sin ϕ cos κ sin κ 0 0 cos ω sin ω sin κ cos κ 0 x y w 0 sin ω cos ω sin ϕ 0 cos ϕ z (5.9) Z Y X (x, y, z) (u, v, w) x cos( κ) sin( κ) 0 y = sin( κ) cos( κ) 0 z cos( ω) sin( ω) 0 sin( ω) cos( ω) cos( ϕ) 0 sin( ϕ) sin( ϕ) 0 cos( ϕ) u v w (5.10) P(X p, Y p, Z p ) (u p, v p, w p ) c (u, v, c) w v O(X0,Y0,Z0) O(0,0,0) u c P'(u,v,-c) Z (0,0,-c) κ X ω φ Y P(Xp,Yp,Zp) P(up,vp,wp) (X p, Y p, Z p ) (u p, v p, w p ) O(X 0, Y 0, Z 0 ) (ω, ϕ, κ) (u p, v p, w p ) (X p X 0, Y p Y 0, Z p Z 0 )

147 u p v p = cos ω sin ω w p 0 sin ω cos ω cos ϕ 0 sin ϕ sin ϕ 0 cos ϕ cos κ sin κ 0 sin κ cos κ x p x 0 y p y 0 z p z 0 a 11 a 33 u p v p = a 11 a 12 a 13 a 21 a 22 a 23 w p a 31 a 32 a 33 x p x 0 y p y 0 z p z 0 (5.11) c P P (u, v) OP c w p w c u = c u p = c a 11(x p x 0 ) + a 12 (y p y 0 ) + a 13 (z p z 0 ) w p a 31 (x p x 0 ) + a 32 (y p y 0 ) + a 33 (z p z 0 ) v = c v p = c a 21(x p x 0 ) + a 22 (y p y 0 ) + a 23 (z p z 0 ) w p a 31 (x p x 0 ) + a 32 (y p y 0 ) + a 33 (z p z 0 ) (5.12) (5.13) (co-linearity equation) (x 0, y 0, z 0 ) (ω, ϕ, κ) (x p, y p, z p ) (u, v) (exterior orientation) (u, v, c) (x p, y p, z p ) F u (x 0, y 0, z 0, ω, ϕ, κ) = c a 11(x p x 0 ) + a 12 (y p y 0 ) + a 13 (z p z 0 ) a 31 (x p x 0 ) + a 32 (y p y 0 ) + a 33 (z p z 0 ) u (5.14) F v (x 0, y 0, z 0, ω, ϕ, κ) = c a 21(x p x 0 ) + a 22 (y p y 0 ) + a 23 (z p z 0 ) a 31 (x p x 0 ) + a 32 (y p y 0 ) + a 33 (z p z 0 ) v (5.15)

148 148 5 x 00, y 00, z 00, ω 0, ϕ 0, κ 0 x 0, y 0, z 0, ω, ϕ, κ x 0 = x 00 x 0 (5.16) y 0 = y 00 y 0 (5.17) z 0 = z 00 z 0 (5.18) ω = ω 0 ω (5.19) ϕ = ϕ 0 ϕ (5.20) κ = κ 0 κ (5.21) F u, F v 3.89 x = x 0, a = x 00 x 0 x 00 = x 0 y 0, z 0, ω, ϕ, κ F u (x 0, y 0, z 0, ω, ϕ, κ) F u (x 00, y 00, z 00, ω 0, ϕ 0, κ 0 ) F u x 0 F u y 0 F u z 0 F u x 0 y 0 z 0 ω ω F u ϕ ϕ F u κ (5.22) κ F v (x 0, y 0, z 0, ω, ϕ, κ) F v (x 00, y 00, z 00, ω 0, ϕ 0, κ 0 ) F v x 0 F v y 0 F v z 0 F v x 0 y 0 z 0 ω ω F v ϕ ϕ F v κ (5.23) κ

149 F u ca 11 + ua 31 = x 0 a 31 (x p x 0 ) + a 32 (y p y 0 ) + a 33 (z p z 0 ) (5.24) F u ca 12 + ua 32 = y 0 a 31 (x p x 0 ) + a 32 (y p y 0 ) + a 33 (z p z 0 ) (5.25) F u ca 13 + ua 33 = z 0 a 31 (x p x 0 ) + a 32 (y p y 0 ) + a 33 (z p z 0 ) (5.26) F u ω = uv c (5.27) F u ϕ = u2 cos ω u sin ω c cos ω c (5.28) F u κ = F u (y p y 0 ) F u (x p x 0 ) x 0 y 0 (5.29) F v ca 21 + va 31 = x 0 a 31 (x p x 0 ) + a 32 (y p y 0 ) + a 33 (z p z 0 ) (5.30) F v ca 22 + va 32 = y 0 a 31 (x p x 0 ) + a 32 (y p y 0 ) + a 33 (z p z 0 ) (5.31) F v ca 23 + va 33 = z 0 a 31 (x p x 0 ) + a 32 (y p y 0 ) + a 33 (z p z 0 ) (5.32) F v ω = c + v2 c (5.33) F v uv = u sin ω cos ω ϕ c (5.34) F v κ = F v (y p y 0 ) F v (x p x 0 ) x 0 y 0 (5.35) x 00, y 00, z 00, ω 0, ϕ 0, κ F u () = 0, F v () = 0 x 0, y 0, z 0, ω, ϕ, κ u, v 5.12, F u (), F v () 0 (single photograph orientation)

150 150 5 (relative orientation) (bundle adjustment) 5.12, 5.13 c (x 0, y 0, z 0 ) (ω, φ, κ) u = b 1x + b 2 y + b 3 z + b 4 b 9 x + b 10 y + b 11 z + 1 v = b (5.36) 5x + b 6 y + b 7 z + b 8 b 9 x + b 10 y + b 11 z + 1 b 1, b 2,, b b 1, b 2,, b , 5.13 b 1, b 2,, b 11 (x 0, y 0, z 0 ) (ω, φ, κ) 6 z = 0 u = b 1x + b 2 y + b 3 b 7 x + b 8 y + 1 v = b (5.37) 4x + b 5 y + b 6 b 7 x + b 8 y

151 z b 1, b 2,, b 8 c Z m ω = tan 1 (cb 8 ) (5.38) φ = tan 1 ( cb 7 cos ω) (5.39) κ = tan 1 ( b 4 /b 1 ) φ = 0 (5.40) κ = tan 1 (b 2 /b 5 ) φ 0, ω = 0 (5.41) κ = tan 1 {(A 1 A 3 A 2 A 4 )/(A 1 A 2 A 3 A 4 )} φ 0, ω 0 (5.42) z 0 = c cos ω (A A2 3 )/(A2 1 + A2 4 ) + Z m (5.43) x 0 = b 3 (tan ω sin κ/ cos φ tan φ cos κ)(z m z 0 ) (5.44) y 0 = b 6 (tan ω cos κ/ cos φ + tan φ sin κ)(z m z 0 ) (5.45) A 1, A 2,, A 4 A 1 = 1 + tan 2 φ A 2 = b 1 + b 2 tan φ/ sin ω A 3 = b 4 + b 5 tan φ/ sin ω A 4 = tan φ/(cos φ tan ω) (5.46) (stereoscopic viewing) B A θ a θ b θa θb

152 P P (X p, Y p, Z p ) (X L0, Y L0, Z L0 ) (ω L0, φ L0, κ L0 ) (X R0, Y R0, Z R0 ) (ω R0, φ R0, κ R0 ) P (u L0, v L0, c)(u R0, v R0, c) P P P (corresponding point) P (image matching) (stereo matching) w v w v (X L0,Y L0,Z L0 ) u (X R0,Y R0,Z R0 ) u c c (u L1,v L1,-f) (X L1,Y L1,Z L1 ) (u R1,v R1,-f) (X R1,Y R1,Z R1 ) P(Xp,Yp,Zp)

153 P P P (u L0, v L0, c) (X L1, Y L1, Z L1 ) (X L0, Y L0, Z L0 ) u L1 v L1 = a 11 a 12 a 13 a 21 a 22 a 23 X L1 X L0 Y L1 Y L0 (5.47) c a 31 a 32 a 33 Z L1 Z L0 P X L1 X L0 Y L1 Y L0 Z L1 Z L0 = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 1 u L1 v L1 c (5.48) t X = (X L1 X L0 )t + X L0 Y = (Y L1 Y L0 )t + Y L0 (5.49) Z = (Z L1 Z L0 )t + Z L0 s X = (X R1 X R0 )s + X R0 Y = (Y R1 Y R0 )s + Y R0 (5.50) Z = (Z R1 Z R0 )s + Z R0 L 2 L 2 = {(X L1 X L0 )t + X L0 (X R1 X R0 )s X R0 } 2 = {(Y L1 Y L0 )t + Y L0 (Y R1 Y R0 )s Y R0 } 2 = {(Z L1 Z L0 )t + Z L0 (Z R1 Z R0 )s Z R0 } 2 (5.51) L s, t L s, t 0 s, t L2 L2 s = 0, t = 0 s, t

154

155 155 6 (Eratosthenes)

156 (zenith) (prime meridian circle) 12 (prime vertical circle) Z P N z E h W Α S (azimuth) A (elevation) h (zenith angle) z (daily rotation) (diurnal motion) (meridian passage)

157 (solar time) (day) (standard time) (local time) 12 (year) (revolution) /4=

158 /4-1/100= /4-1/100+1/400= / (second) 1 =1/(24 *60 *60 ) (crystal oscillation) 1 133Cs 87Rb 1 133Cs 2 9,192,631, (sidereal time)

159 (ecliptic) 6.2 (Astronomical Unit) AU 1AU 17 28AU 280m 1mm

160 160 6 (Aries first point)

161 δ(celestial declination) α(right ascension) δ α α φ(latitude) λ(longitude) P λ φ

162 δ δ P φ P θ φ, δ, θ φ = δ + πrad 2 θ (6.1) P P P P φ (geographic latitude) φ φ (geocentric latitude)

163 P O φ' φ = = = GPS GPS

164 XYZ (geocentric rectangular coordinate system) m X 0 Y 90 Z Z P r O φ λ Y X (heriocentric rectangular coordinate system) λ XY

165 λ λ = tan 1 y x (6.2) φ XY x 2 + y 2 φ φ = tan 1 z x2 + y 2 (6.3) r Z z = r sin φ (6.4) X Y XY r cos φ x = r cos φ cos λ y = r cos φ sin λ (6.5)

166 166 6 P Z P r O φ λ Y X r XYZ (r, 0, 0) Y φ Z λ P x y z = cos λ sin λ 0 sin λ cos λ x y z = cos φ 0 sin φ sin φ 0 cos φ r cos φ cos λ r cos φ sin λ r sin φ r 0 0 (6.6) (6.7) Z X Y P Y φ

167 x y z φ -180 sin φ z 6.5 P φ P φ Zenith b P -a -c O φ' θ c a G φ -b P (tangential plane) P G φ P (N x, N y ) f(x, y) f(x, y) = x2 a 2 + y2 b 2 1 (6.8) f(x, y) = 0

168 D Surface x*x/3+y*y/1-1 Z axis Y axis X axis x, y f(x, y) x f(x, y) y = 2x a 2 = 2y b 2 (6.9) P (a cos θ, b sin θ) (N x, N y ) N x = 2 cos θ a (6.10) N y = 2 sin θ b t x = 2 cos θ t + a cos θ a y = 2 sin θ t + b sin θ b (6.11) φ P P y G P N (radius of prime vertical circle) (radius of curvature) N

169 Zenith b P -a -c O T S Q N c a G φ -b N G G x 0 x = 0 t y y y = (b2 a 2 ) sin θ b (6.12) G P x a cos θ y b sin θ (b2 a 2 ) sin θ b N ( N 2 = (a cos θ) 2 + b sin θ (b2 a 2 ) 2 ) sin θ b = a 2 cos 2 θ + a4 b 2 sin2 θ ) = (a a4 b a4 b 2 cos 2 θ (6.13) N θ φ φ θ tan φ tan φ = 2 sin θ b 2 cos θ a = a tan θ (6.14) b

170 170 6 cos 2 θ tan 2 φ = a2 b 2 tan2 θ cos 2 θ = = a2 b 2 ( 1 cos 2 θ 1 a 2 ) b 2 ( tan 2 φ + a2 b 2 ) (6.15) 6.13 cos 2 θ N ) N 2 = (a a4 b a4 a 2 b 2 b ( ) 2 tan 2 φ + a2 b ( 2 = a4 b 2 a 2 ) b b 2 tan 2 φ + a 2 e 2 = a2 b 2 a 2 = a4 b 2 ( b 2 tan 2 φ + a 2 + b 2 a 2 ) b 2 tan 2 φ + a 2 = a2 (tan 2 φ + 1) b 2 a tan 2 φ a 2 (tan 2 φ + 1) = b 2 a (tan 2 φ + 1) b2 2 a = = a 2 1 cos 2 φ b 2 1 a 2 cos 2 φ b2 a a 2 b 2 a ( ) b 2 a 2 2 a 2 cos2 φ N 2 = a 2 a 2 e 2 a 2 a 2 + e 2 cos 2 φ a 2 = 1 e 2 + e 2 cos 2 φ a 2 = 1 e 2 (1 cos 2 φ) = a 2 1 e 2 sin 2 φ (6.16) (6.17) P φ N T T P P x S PSQ OGS OG:PQ y b 2 a 2 sin θ : b sin θ = (b 2 a 2 ) : b 2 (6.18) b

171 a 2 > b 2 (a 2 b 2 ) : b 2 T (a 2 b 2 ) : b 2 = (N T ) : T (6.19) P N φ { x = N cos φ y = b2 a 2 N sin φ T = b2 a 2 N (6.20) φ φ tan φ = y x tan φ = b 2 a 2 N sin φ N cos φ (6.21) = b2 tan φ (6.22) a2 6.6 x y 90 z Z P r O φ λ Y X P φ λ P (x, y, z) P y φ x z P

172 172 6 (N cos φ, 0, b2 N sin φ) z +λ a2 x y z = cos λ sin λ 0 sin λ cos λ x y z = N cos φ cos λ N cos φ sin λ b2 a 2 N sin φ N cos φ 0 b2 a 2 N sin φ (6.23) (6.24)

173 173 7 (remote sensing) (artificial satellite) 1960 (satellite remote sensing) 1970 NOAA Landsat SPOT MOMS Terra ASTER ALOS PRISM GPS Inertial Measurement Unit: IMU GPS

174 (velocity) (acceleration) v x t α v t v = dx dt (7.1) α = dv dt = d2 x dt 2 (7.2) (law of inertia) 1687 v 0 (uniform motion) t = 0 x 0 t x t x = v 0 t + x 0 (7.3) v 0 t (uniform accelerated motion) α t = 0 v 0 t v t v = αt + v 0 (7.4)

175 t = 0 x 0 t x t x = 1 2 αt2 + v 0 t + x 0 (7.5) (grabity) v 0 x y x θ { v x = v 0 cos θ v y = v 0 sin θ x y y (gravitational acceleration) g { v x = v 0 cos θ v y = gt + v 0 sin θ t t { x = v 0 cos θt y = 1 2 gt2 + v 0 t sin θ v 0 =30 m/s 108 km/h) θ =45 g 9.8m/s 2 y (7.6) (7.7) (7.8) v y v 0 θ v x x

176 m (law of motion) m α (force) F F = mα (7.9) N 1kg 1m/s 2 1N F = mα (mass) 5kg (law of action and reaction) A B F a B A F b F a + F b = 0 (7.10) F a F b (momentum)p m v p = mv (7.11) F = mα v t α F = d (mv) (7.12) dt F = 0 (work) W F r W = F r (7.13)

177 F F θ r F r cos θ J 1J 1N 1m (power) W 1W 1 1J (potential energy) J (kinetic energy) 7.13 F = mα r W = mαdr = m dv dt dr = mvdv v = dr dt = 1 2 mv2 (7.14) v 2 = v v (mechanical energy) r P

178 178 7 y r P -r θ r x -r P r θ { x = r cos θ y = r sin θ (7.15) (angular velocity)ω t θ ω = dθ dt (7.16) ω θ = ωt { x = r cos ωt y = r sin ωt (7.17) rθ v v = d(rθ) dt = r dθ dt = rω (7.18) { v x = dx dt = rω sin ωt = ωy v y = dy dt = rω cos ωt = ωx (7.19) v = vx 2 + vy 2 = ω ( y) 2 + x 2 = rω (7.20)

179 { α x = dv x dt = rω 2 cos ωt = ω 2 x α y = dv y dt = rω 2 sin ωt = ω 2 y (7.21) r = (x, y) a = rω 2 (7.22) r (period) T ω θ ωt 2π t T = 2π ω (7.23) (frequency) 1 Hz ν T ν = 1 T (7.24) (moment) (torque) O F O r F O N O r F N = rf (7.25) r

180 180 7 r F y F y F F x F x r y r F y x r x F x y x r x F y y r y F x x r x F y y r y F x N = r x F y r y F x (7.26) 2.82 N = r F (7.27) p = mv (angular momentum) p r L L = r p (7.28) p r

181 y p = mv r O x m v p = mv v ω r v = ωr L = mvr = mr 2 ω (7.29) mr 2 mr 2 (moment of inertia) I (geometrical moment of inertia) F F m M r

182 182 7 m, M r F = G Mm r 2 (7.30) G (Cavendish) G = m 3 /kg/s M GM G s G e G M r 2 G s = m 3 /s 2 G e = m 3 /s 2 F = mα g m g = m/s 2 9.8m/s (8.75 ) (1.00 ) 1/10 (Cassini) 1672

183 AU 7.2 (orbit) (Kepler s laws) (elliptic orbit) a e b -a -c c a -b

184 (uniform circular motion) F (laws of gravity) F = G Mm r 2 (7.31) G M m r (areal velocity) O A, B, C, D

185 D D' C C' B O A A B C O OAB OBC O AB BC B C C B O OB CC OBC OBC D A P CAP P' b P E M v C A a

186 186 7 P E OAP OAP = πa 2 E 2π = 1 2 a2 E (7.32) OAP OAP b a OAP OAP = πa 2 E 2π b a = 1 abe (7.33) 2 CAP OAP OCP OC e ae b sin E CAP = 1 2 abe 1 aeb sin E 2 = 1 ab(e e sin E) (7.34) 2 CAP A P t T 1 2ab(E e sin E) = πab t T (7.35) E e sin E = 2π T t (7.36) E (eccentric anomaly) 2π T t M(mean anomaly) M a a r v T ν ω v = 2πr T = 2πrν = rω (7.37)

187 (frequency) T = 1/ν (angular velocity) (uniform circular motion) α (uniform motion) v A B A B AOB θ v v v B v θ v θ r v A θ A B t α α = v t (7.38) θ v v v v θ θ ω θ = ω t α α = v t = v θ t = vω = v2 r = 4π2 r T 2 (7.39)

188 F = mα m 4π2 r T 2 = G Mm r 2 (7.40) r 3 T 2 = GM 4π 2 (7.41) v a e Ω ω i a

189 (perigee) (apogee) (ascending node) (descending node) 7.1 ET days ω degree i degree Ω degree e M 0 degree M 1 rev/day M 2 rev/day a e P(x p, y p ) P P AOP AOP E { x p = a cos E y p = b sin E = a 1 e 2 sin E (7.42)

190 190 7 P' b P E M ae v A a E a T a a 3 T 2 = GM 4π 2 (7.43) GM GM = (m 3 /s 2 ) = (km 3 /day 2 ) (7.44) E E E e sin E = 2π T t = M (7.45) A t = 0 M E M

191 M E f(e) = M E + e sin E (7.46) f(e) = 0 E E 0 f(e) f (E) = e cos E 1 (7.47) E 1 E 0 E 0 f (E 0 ) f (E 0 ) = f(e 0) E 0 (7.48) E 0 E 1 = E 0 E 0 E E 0 = f(e 0) f (E 0 ) E 1 = f(e 0 E 0 ) f (E 0 E 0 ) E 2 = f(e 0 E 0 E 1 ) f (E 0 E 0 E 1 ) (7.49) E 0 M (U, V ) E U V { U = a cos E V = b sin E = a 1 e 2 sin E (7.50)

192 192 7 V P' P E ae U (U, V ) { U = a cos E ae V = a 1 e 2 sin E (7.51) (U, V ) E ω (Argument of perigee) UV UV W W i (inclination angle) 0 90 x Ω (right ascension of ascending node) z

193 z V P U O v i y x (U, V ) (x, y, z) (U, V ) z ω x i z Ω x y z = cos Ω sin Ω 0 sin Ω cos Ω cos i sin i 0 sin i cos i cos ω sin ω 0 sin ω cos ω U V 0 (7.52) Space Track ALOS ALOS (JST) a (UT) t

194 ALOS ET (UT) ω ω 0 = i i = Ω Ω 0 = e e = M 0 M 0 = M 1 M 1 = (rev/day) M 2 M 2 = (rev/day 2 ) a T M 1 (rev) (day) a ( GM a = 4π 2 M1 2 ) 1 3 (7.53) t M m M m = M 1 + M 2 t = = (rev/day) (7.54) (1) GM GM = (km 3 /day 2 ) (7.55) ALOS ( ) a = 4π = (km) (7.56) ALOS (km)

195 E E M 7.54 M 0 M (rev) M = M 0 + M 1 t M 2 t 2 = = (rev) (7.57) M (rev) 360 M = M e E E e sin E = M (7.58) E = E (U, V ) U = a cos E ae = (km) (7.59) V = a 1 e 2 sin E = (km) (7.60) ω i Ω x-y x z ω Ω i a r ω = ω (2 2.5 sin2 i) π( a r )3.5 t = (7.61) Ω = Ω cos i π( a r )3.5 t = (7.62) (U, V ) (x, y, z) (U, V ) z ω

196 196 7 x i z Ω x y z = = cos Ω sin Ω 0 sin Ω cos Ω cos i sin i 0 sin i cos i cos ω sin ω 0 sin ω cos ω U V 0 (7.63) x (rev/day) θ 0 (rev) T θ G (rev) θ G = θ T (7.64) θ 0 = (rev) (UT) θ G = (rev) θ G 360 θ G = x z X Y = cos( θ G) sin( θ G ) 0 sin( θ G ) cos( θ G ) 0 Z x y z = (7.65) r

197 X = r cos φ cos λ Y = r cos φ sin λ Z = r sin φ (7.66) r z Y X = tan λ φ = sin 1 Z = X2 + Y 2 sin 1 + Z = (7.67) λ = tan 1 Y X = tan = (7.68) tan 1 π 2 π 2 X Y XY = tan 1 π π atan2(x, y) ALOS x P (X, Y, Z) φ λ H X Y Z = (N + H) cos φ cos λ (N + H) cos φ sin λ ( b2 N + H) sin φ a2 (7.69) 6.24 A(X a, Y a, Z a ) A P

198 198 7 z P N A E O λ φ y x ( X, Y, Z) X Y = X X a Y Y a (7.70) Z Z Z a x (u, v, w) u v w = cos( π 2 φ) 0 sin( π 2 φ) sin( π 2 φ) 0 cos( π 2 φ) cos λ sin λ 0 sin λ cos λ X Y (7.71) Z u A A = tan 1 v u (7.72) h h = sin 1 w u2 + v 2 + w 2 ) (7.73) h w 7.5

199 (equatorial orbit) (inclined orbit) (polar orbit) km 6370km 600km 1m 40cm (geostationary orbit) 36000km BS CS 7.5.2

200 200 7 Ω M 0 Ω ω Ω ALOS i 98 Ω (sun synchronous orbit) 7.6 (image pickup device) (matrix array sensor) (linear array sensor)

201 RPC (rational polynomial coefficient model) (u, v) (x, y, z) f u, g u, f v, g v RPC { u = f u (x,y,z) g u (x,y,z) v = f v(x,y,z) g v (x,y,z) (7.74)

202 202 7 RPC IKONOS Quick Bird ALOS PRISM GPS RPC RPC RPC 30m (u, v) (x, y, z) a 1 a 8, b 1 b 6 u = a 1x + a 2 y + a 3 z + a 4 b 1 x + b 2 y + b 3 z + 1 v = a (7.75) 5x + a 6 y + a 7 z + a 8 b 4 x + b 5 y + b 6 z + 1 u, v RPC u v u = a 1x + a 2 y + a 3 z + a 4 b 1 x + b 2 y + b 3 z + 1 (7.76) v = a 5 x + a 6 y + a 7 z + a 8 Gupta-Hartley (gound control point) (u, v) (x, y, z) 500 RPC

203 SPOT O 1 O 2 P n P b P g P g O 1 P n O 2 P b P g P g O 2 Terra ASTER ALOS PRISM (u n, v n ) (u b, v b ) u n = f u (x, y, z) v n = f v (x, y, z) (7.77) u b = g u (x, y, z) v b = g v (x, y, z) (x, y, z)

204 GPS GPS GPS(Global Positioning System) GPS GPS GPS GPS T C C T GPS GPS GPS GPS 2 km GPS GPS (x 3, y 3, z 3 ) (x 2, y 2, z 2 ) r 2 r 3 (x 4, y 4, z 4 ) (x 1, y 1, z 1 ) r 4 r 1 GPS (x p, y p, z p )

205 7.7 GPS 205 GPS (x 1, y 1, z 1 ), (x n, y n, z n ) GPS r 1,, r n P(x p, y p, z p ) P GPS (x i, y i, z i ) r i P GPS r i (x i x p ) 2 + (y i y p ) 2 + (z i z p ) 2 = (r i + C t) 2 (7.78) t GPS t x p, y p, z p, t GPS GPS 7.3 GPS L1 1, MHz C/A P L2 1,227.6 MHz P GPS (standalone GPS) C/A 7.78 C/A m 10m

206 m 1m (differential GPS) GPS GPS GPS GPS FM FM DGPS 1m (relative GPS) L2 P cm GPS L1 19cm L2 24cm 1/10 2cm cm cm cm 3cm GPS (continuous GPS station) (virtual relative station)

207 (static electricity) 20

208 208 8 (electric charge) A,B q a, q b r F F = k q aq b r 2 (8.1) k q b F F r q a (Coulomb) (Coulomb s law) 8.1 N C r 1m N 1C k Nm 2 /C k q a r 2 (electric field) E F = q b E (8.2) E q a r q a q b N/C q a (electric flux line)

209 q a E r r r E 4πr 2 E 4πr 2 = k q a r 2 4πr2 = 4πkq a (8.3) 4πkq a 1835 (Gauss law) 4πk 1 ɛ 0 ɛ 0 (permittivity) q a ɛ0 4πr 2 E = q a E = ɛ 0 q a 4πr 2 ɛ 0 (8.4) q a 4πr (electric flux density)d 2 D = ɛ 0 E (8.5) ɛ 0 C 2 /Nm 2 E N/C C/m 2 q b q a W 7.13 W = F r W = F r = q b Er (8.6) = q b q a 4πrɛ 0

210 210 8 Er (electric potential) (difference of potential) (power voltage)v N S (magnetic pole) A,B p n, p s r F k F = k p np s r 2 (8.7) C Wb r 1m 107 (4π) = N 2 1Wb k Nm 2 /Wb 2 (magnetic field)h p n, p s (magnetic flux line) N S H

211 (Ørsted) (Ampère) I H (Ampere s rule) H = I 2πr (8.8) I A H A/m I r H r r s H = I s 4πr 2 (8.9) (Biot-Savart s law) E ɛ D H (magnetic permeability)µ 0 (magnetic flux density)b B = µ 0 H (8.10) Wb/m 2 µ 0 N/A 2 4π 10 7 N/A 2

212 (electrical current) 1A 1 1C ρ v S I I = ρvsq (8.11) v m α F = ma 8.2 F = qe α = qe m v T v = qe m T 8.11 I = ρ qe m Sq E I = ρq2 S m E (8.12) I V R I = V R (8.13) (Ohm s law) 8.12 E V 1 R (Lorentz force) B I F F B I

213 F = (I B)l (8.14) l F I B F B I (Faraday) 1831 (electromagnetic induction) V φ t V = φ t (8.15) φ B S (Faraday s law)

214 214 8 IH(Induction Heating) (Maxwell) πr 2 E = qa ɛ 0 r r a qa ɛ 0 b 0 q a n E θ S E a S E E θ E cos θ n E n E n = E n cos θ n = 1

215 E nds = q a (8.16) ɛ 0 v s (electric charge density)ρ q a = ρdv ɛ 0 E nds = ρdv (8.17) s v N N S H 0 H µ 0 H nds = 0 s B nds = 0 (8.18) s 8.8 H = I 2πr B B = µ 0I 2πr 2πrB = µ 0 I Bdr = µ 0 I (8.19)

216 216 8 j S I = jds Bdr = µ 0 jds (8.20) s de/dt B S I de dt de s dt ds de ɛ 0 s dt ds (displacement current density) 8.20 Bdr = µ 0 s ( ) de j + ɛ 0 ds (8.21) dt (Ampere-Maxwell s law) Bdr = µ 0 s ( ) E j + ɛ 0 nds (8.22) t 8.15 V = φ t V 8.6 E r E r V = E r V = Edr

217 φ E r V φ B S φ = BdS s Edr = d dt s BdS (8.23) B Edr = nds (8.24) s t ɛ 0 E nds = ρdv s v ( ) E Bdr = µ 0 j + ɛ 0 nds s t B nds = 0 s Edr = s B t nds E = ρ (8.25) ɛ 0 ( ) E B = µ 0 j + ɛ 0 (8.26) t B = 0 (8.27) E = B t (8.28)

218 (electromagnetic waves) (wave length)λ c (period)t (frequency)ν T c = λ/t ν 1 ν = 1/T c = νλ Hz µm 0.1nm 10nm 1µm 100µm 10mm 1m 100m 10km γ X µm µm µm 1mm 1m 8.2.2

219 E = 0 E B = µ 0 ɛ 0 t H = 0 (8.29) E = 0 (8.30) E H = ɛ 0 t H E = µ 0 t (8.31) (8.32) B H f() z t f() v t 0 z 0 t z 1 v z 0 = z 1 v t x t f(z 1 -vt 1 ) t 1 t f(z 0 -vt 0 ) z t 0 v t z 0 z 1 z

220 220 8 f(z vt) f(z 1 vt 1 ) = f(z 1 v(t 0 + t)) = f(z 1 vt 0 v t) z 0 = z 1 v t = f(z 0 vt 0 ) (8.33) f(z 1 vt 1 ) = f(z 0 vt 0 ) f(z vt) f() f(z + vt) (sine wave) sin sin x 2π λ a u(z, t) u(z, t) = a sin 2π (z vt) (8.34) λ T v λ = vt u(z, t) = a sin 2π( z λ t T ) (8.35) 2π 2π λ (wave number) k T (angular frequency) ω u(z, t) = a sin(kz ωt) (8.36) (plane wave) k k = (k x, k y, k z ) x r k n z y k n r n r u(r, t) = a sin(k r ωt) = a sin(k x x + k y y + k z z ωt) (8.37)

221 π 2 u(r, t) = ae i(k r ωt) (8.38) i u(r, t) = a cos(k r ωt) + i sin(k r ωt) (8.39) z xz yz x H y E x z y z E x H y E x = E 0 sin(kz ωt) (8.40) H y = H 0 sin(kz ωt) (8.41) 8.31 z H y z = ɛ E x 0 t 8.32 (8.42) E x z = µ H y 0 t (8.43) 8.42 z 2 H y z 2 = ɛ 0 E x t H y z = ɛ 0 µ 0 2 H y t (8.44)

222 222 8 (wave equation) 8.43 z 2 E x z 2 = µ 0 H y t E x z = ɛ 0 µ 0 2 E x t (8.45) (wave equation) (E 0 sin(kz ωt)) z 2 = ɛ 0 µ 0 2 (E 0 sin(kz ωt)) t 2 k (E 0 cos(kz ωt)) (E 0 cos(kz ωt)) = ωɛ 0 µ 0 z t k 2 (E 0 sin(kz ωt)) = ω 2 ɛ 0 µ 0 (E 0 sin(kz ωt)) k 2 = ω 2 ɛ 0 µ 0 k 2 ω 2 = ɛ 0µ 0 (8.46) k = 2π 2π λ ω = T v = λ T v = ω k c c c = ω λ = 1 ɛ0 µ 0 (8.47) c ɛ 0 µ 0 x v 2 y x 2 = 1 2 y v 2 t 2 (8.48) E H S S = E H (8.49) S (pointing vector) E H kh 0 cos(kz ωt) = ɛ 0 ωe 0 cos(kz ωt) (8.50) ke 0 cos(kz ωt) = µ 0 ωh 0 cos(kz ωt) (8.51)

223 k ω k ω = ɛ E 0 cos(kz ωt) 0 H 0 cos(kz ωt) k ω = µ H 0 cos(kz ωt) 0 E 0 cos(kz ωt) (8.52) (8.53) µ 0 {H 0 cos(kz ωt)} 2 = ɛ 0 {E 0 cos(kz ωt)} 2 {H 0 cos(kz ωt)} 2 = ɛ 0 µ 0 {E 0 cos(kz ωt)} 2 H0 2 = ɛ 0 E0 2 µ 0 ɛ0 H 0 = E 0 (8.54) µ 0 E H ν 0 ɛ 0 H 0 E (Snell s law) zx yz ɛ 1 µ 1 ɛ 2 µ 2 E vi E vr E hi Ehr x x ε 1, µ 1 θ i θ r z ε 1, µ 1 θ i θ r z ε 2, µ 2 θ t Y ε 2, µ 2 θ t Y E ht E vt zx E vi E vt E vr θ i θ t θ r x

224 224 8 zx E hi E ht E hr E iv e i(k1n r ωt) n (sin θ i, cos θ i ) r zx (z, x) u iv (z, x, t) k 1 u iv (z, x, t) = E iv e i(k 1z sin θ i k 1 x cos θ i ωt) (8.55) z E ivz E ivz = E iv cos θ i n E y y y 8.54 y H ivy H ivy = ɛ1 µ 1 E iv y z x { Eivz = E iv cos θ i H ivy = ɛ1 (8.56) µ 1 E iv sinθ i E iv H iv cosθ i sinθ i x cosθ i H ih E ih sinθi sinθ i x cosθ i n θ i cosθ i n θ i z z y E ihy E ihy = E hi H ih n E z H iz H ihz = ɛ1 µ 1 E ih cos θ i { Eihy = E ih H ihz = ɛ1 (8.57) µ 1 E ih cos θ i u rv (z, x, t) u rv (z, x, t) = E rv e i(k 1z sin θ r k 1 x cos θ r ωt) (8.58)

225 z E rvz y H rvy { Ervz = E rv cos θ r H rvy = ɛ1 (8.59) µ 1 E rv E rv sinθ r x cosθ H r rv cosθ r n x cosθ r E rh cosθ r sinθ r H rh n θ r sinθ r θ r sinθ r z z y E rhy z H rhz { Erhy = E rh H rhz = ɛ1 (8.60) µ 1 E rh u tv (z, x, t) k 2 u tv (z, x, t) = E tv e i(k 2z sin θ r k 2 x cos θ t ωt) (8.61) z E tvz y H tvy { Etvz = E tv cos θ t H tvy = ɛ2 (8.62) µ 2 E tv x z x z sinθ t θ E tv t H tv cosθ t cosθ t H th E th sinθ t sinθ t sinθ t cosθ t n cosθ t n y E thy

226 226 8 z H thz { Ethy = E th H thz = ɛ2 (8.63) µ 2 E th z x = 0 ωt { E ivz e ik1z sin θi ik1z sin θr ik2z sin θt + E rvz e = E tvz e E ihy e ik 1z sin θ i + E rhy e ik 1z sin θ r = E thy e ik 2z sin θ t (8.64) z k 1 sin θ i = k 1 sin θ r = k 2 sin θ t (8.65) θ i = θ r k 1 sin θ r = k 2 sin θ t k 1, k 2 ɛ2 k 1 = ω ɛ1 µ 1, k 2 = ω µ { E ivz + E rvz = E tvz (8.66) E ihy + E rhy = E thy { H ivy + H rvy = H tvy H ihz + H rhz = H thz (8.67) E iv cos θ i E rv cos θ r = E tv cos θ t E ih E rh = E th ɛ1 ɛ1 ɛ1 µ 1 E iv + µ 1 E rv = µ 1 E ih cos θ i + ɛ1 µ 1 E rh = ɛ2 µ 2 E tv ɛ2 µ 2 E th (8.68) n n = ɛ2 µ 2 / E rv E iv ɛ1 µ 1 = cos θ i n cos θ t cos θ i + n cos θ t (8.69) E rh E ih = n cos θ i cos θ t n cos θ i + cos θ t (8.70)

227 E tv 2 cos θ i = (8.71) E iv cos θ i + n cos θ t E th 2 cos θ i = (8.72) E ih n cos θ i + cos θ t (Fresnel s equations) θ i 0 (Brewster s angle) (polarisation) (radiation) (radiant energy) (J) (J/s) (radiant flux) (light flux) (W) (lm) (radiant exitance) (irradiance) M e Φ S M e = dφ ds (W/m 2 ) (lx = lm/m 2 ) (8.73) (solid angle) (radiant intensity)

228 228 8 I dω Φ I e Φ Ω I e = dφ dω (8.74) I e (W/sr) (cd = lm/sr) α S α r Ω S r Ω = S r 2 (8.75) sr 4πr 2 4π(sr) ds θ ds cos θ ds cos θ θ ds (radiance) L e I e L e = di e ds cos θ (8.76)

229 I e = dφ dω L e = d 2 Φ dωds cos θ (8.77) (W/sr m 2 ) (cd/cm 2 ) (radiation) (heat radiation) (black body) (black body radiation) 1859 (Kirchhoff) λ T (Planck) 1900 T < E > < E >= 1 2kT k (Boltzmann constant) E P (E) P (E) = Ae E kt (8.78) A e 3.30 E E = nhν h (Planck constant) ν n

230 230 8 n 0, 1, 2, n = 0, 1, 2, P (0), P (1), P (2), 0hνP (0) + 1hνP (1hν) + 2hνP (2hν) + (8.79) < E > = = 0hνP (0) + 1hνP (1hν) + 2hνP (2hν) + P (0) + P (1hν) + P (2hν) + hν(0 + e hν kt e 0 + e hν kt + 2e 2hν kt + + e 2hν kt + = hν 0 + x + 2x x + x 2 + x = hν 1 x = hν x 1 hν = e hν kt 1 x = e hν kt (8.80) M e (λ, T ) c c = νλ M e (λ, T ) = 2πhc2 λ 5 1 e hc kλt 1 (8.81) T =300[K] 5000[K] (K) 1000(K) 600(K) 300(K) µm 1µm 10µm 100µm 1mm 5900[K]

231 M e L e B B(λ, T ) = 2hc2 λ 5 1 e hc kλt 1 (8.82) hν kt 1 e hν kt 1 e hν kt B(λ, T ) = 2hc2 λ 5 1 e hc kλt (8.83) (Wien) λ µm T 3200 hν kt hν 1 e kt 1 hν kt B(λ, T ) = 2c kt (8.84) λ4 (Rayleigh-Jeans) λ = 3mm 30mm (vacuum discharge) 1913 (Bohr) 3 2 E 2 E 1 1

232 (energy level) 2 E 2 1 E 1 1 E 2 E 1 (radiation) (excitation) (absorption) 1890 (Rydberg) n m λ ν 1 λ = ν ( 1 c = R n 2 1 ) m 2 R E E = hc λ (8.85) = hν (8.86) E = nhν 1905 (Einstein) (photon) µm 10µm 3µm 0.7 3µm

233 aerosol N 2 O 2 CO 2 O 3 N 2 O 2 Ar (Rayleigh scattering) (Mie scattering) I s α θ λ I i dω dω ( ) 128π 5 I s = 3λ 4 α2 /dω 3 4 (I i + cos 2 θ) dω (8.87) 4π 1/10 ρ N γ (extinction coefficient)k λ K λ = 8π3 (γ 2 1) 2 3λ 4 Nρ (8.88) λ 4 b K() ( ) 2πb K λ = πb 2 K λ, γ (8.89)

234 234 8 (extinction) (emission) (Kirchhoff s law) λ j λ k λ j λ k λ = B(λ, T ) (8.90) I λ ds ρ di λ di λ = k λ ρi λ ds (8.91) di λ = j λ ρds (8.92) 8.90 j λ = k λ B(λ, T ) J λ j λ = k λ J λ di λ = k λ ρi λ ds + j λ ρds = k λ ρi λ ds + k λ J λ ρds di λ ρk λ ds = I λ + J λ (8.93) LOWTRAN AFGL Air Force

235 Geophisics Laboratory MODTRAN 6s(Second Simulation of the Satellite Signal in the Solar Spectrum) 0 1 (albedo) µ )

236 µm θ φ = 2k h cos θ < π 2 h < λ 8 cos θ (8.94) λ h k(= 2π/λ) Φ π/2

237 φ = 2k h cos θ < π 8 λ h < 32 cos θ (8.95) σ σi = P r(4π) 3 R 4 P t G 2 λ 2 (8.96) P t λ R G P r A σ 0 = σ i /A i backscattering coefficient

238

239 mm M Photoshop ERDAS ENVI

240 (binary data) 0 1 1byte 8bit (text data) ASCII 1 1byte 16 1byte 8bit 4bit 2 4bit A F 9.1 ASCII ASCII K 4 B 16 4B ASCII NUL DLE SP P p 1 SOH DC1! 1 A Q a q 2 STX DC2 2 B R b r 3 ETX DC3 # 3 C S c s 4 EOT DC4 $ 4 D T d t 5 ENQ NAC % 5 E U e u 6 ACK SYN & 6 F V f v 7 BEL ETB 7 G W g w 8 BS CAN ( 8 H X h x 9 HT EM ) 9 I Y i y A LF/NL SUB * : J Z j z B VT ESC + ; K [ k { C FF FS, < L \ l D CR GS - = M ] m } E SO RS. > N ˆ n F SI US /? O o DEL

241 ASCII 1byte 2byte JIS Shift-JIS, EUC, Windows OS Shift-JIS UNIX OS EUC OS (2.13, 2.75) (4.50, 2.75) (2.50, 0.75) u v X Y UV V column row (pixel) (cell)

242 A/D (analog-digital conversion) (sampling) 256 (quantization) (pixel)

243 (R) (G) (B) µ m 8 R=0, G=0, B=0 R=128, G=128, B=128 R=255, G=255, B=255 (additive color mixing) RGB 8bit =16,777,216 RGB 8bit VRAM RGB8bit RGB 4bit RGB YCC YCC Y C 1, C 2

244 244 9 RGB YCC R = Y + C 1 G = Y C C 2 (9.1) B = Y + C 2 Y = 0.299R G B C 1 = R Y = 0.701R 0.587G 0.114B C 2 = B Y = 0.299R 0.587G B (9.2) RGB (C) (M) (Y) (B) CMYB (subtractive color mixing) HSI H Hue S Saturation I Intensity YCC HSI 1 C1 H = tan C 2 S = C1 2 + C2 2 I = Y (9.3) RGB RGB RGB (true color image) (B) B R G B (false color image) R RGB 3

245 BMP (Bitmap) Windows OS 256 BMP TIFF (Tag Image File Format) Aldus 256 LZW LZW TIF JPEG (Joint Photographic Expert Group) ISO (DCT) (DCT) JPG GIF (Graphic Interchange Format) Compu Serve 256 JPEG GIF HDF Hierarchical Data Format HDF HDF NCSA(National Center for Supercomputing Applications)

246 246 9 RAW JPEG TIFF RAW 10bit 12bit 8bit JPEG TIFF RAW RAW RAW RAW (Column Row ) RGB RGB BIP BIP Band Interleaved by Pixel RGB R, G, B, R, G, B, R, G, B,... BMP BMP BIL BIL Band Interleaved by Line RGB BSQ BSQ Band Sequential RGB BIP BIL BSQ

247 B B B B B G G G G G R R R R R B B B B B G G G G G R R R R R BIP B B B B B G G G G G R R R R R B B B B B G G G G G R R R R R B B B B G G G G R R R R B B B B G G G G R R R R B B B B G G G G R R R R B B B B G G G G R R R R BIL B B B B B B B B B B B B B B B B G G G G G G G G G G G G G G G G R R R R R R R R R R R R R R R R BSQ Digital Number (DN) (radiometric correction) (linear array sensor) (matrix array sensor)

248 248 9 NOAA AVHRR (limb darkening) (shading) (vignetting) cos 4 cos 4 θ cos 4 θ cos 4 (standard white board)

249 P i P max a i = Pmax P i (dark noise) (swath width) (scan angle)

250 km (path radiance) (scan angle correction) 9.2.5

251 MOD- TRAN(MODerate resolution atmospheric TRANsmission) FORTRAN Web 6S(Second Simulation of a Satellite Signal in the Solar Spectrum vector code) MODIS 9.2.6

252 252 9 L L d L r L c L = L d + L r + L c (9.4) L d, L r, L c L in L d = R d L in cos θ i R d : (0 1) (9.5) L r = L in ω(θ i ) cos γ ω(θ i ): γ: (9.6) L c = L a R d L a : (9.7)

253 9.3 画像濃度の変換手法 253 下図は 上の画像における輝度値のヒストグラムを示したものです ヒストグラムの横軸は輝度値 を示し の範囲です 255 に近いほど明るい状態であります 原画像のヒストグラムは 平均 が約 54 であり 暗い輝度値に偏っていることが解ります それに対して 濃度変換を施した画像の ヒストグラムにおいては 平均が約 129 であり 輝度が にまんべんなく分布しています 標準偏差を見ても 原画像の標準偏差が 13.5 なのに対して 濃度変換後の標準偏差は 49.9 となり 非常に広く分布していることが 数値からも読み取れます リニアストレッチ 画像の明るさやコントラストの調節は 原画像の各画素の輝度を関数によって変換することで実現 できます この輝度値を変換する関数は数々ありますが もっとも簡単な関数は 線形変換 (linear strech) です 原画像の輝度値を P 一次変換によって得られる輝度値を Q とすると 次式で表すこ とができます Q = ap + b (9.8) ここで a, b は変換係数を表しています 一般に a はゲイン (gain) b はオフセット (offset) と呼 ばれています ゲインによってコントラストが オフセットによって明るさが調節されます つまり 画像の統計量は ゲインによって分布の幅 標準偏差 を調整し オフセットによって平均値を調整 することになります Photoshop などのフォトレタッチソフトウェアでも 明るさやコントラストの調節は 目視によっ て適当な調節を実行することが出来ますが 大量の画像を一度に処理をする場合には ゲインやオフ セットの値を自動的に決定することが重要となります ゲインは統計量の分布幅を調節することか ら 原画像における輝度の最小値 Pmin と最大値 Pmax を 変換によって最小値 Qmin と最大値 Qmax

254 254 9 a = Q max Q min P max P min (9.9) b = Q min ap min = Q minp max Q max P min P max P min (9.10) Q min = 0, Q max = 255 P min, P max P min, P max P min, P max P std Q std a = Q std P std (9.11) P ave Q ave b = Q ave ap ave = Q avep std Q std P ave P std (9.12) RGB RGB RGB RGB P min, P max P min, P max

255 (histogram equalization) M N P 0 Q 0 M Q x y (spatial filter) f(x) x 1 x 2 x 3 x 4 x 5 x 3 f(x) 3 (x 1, x 2, x 3 ) 3 x 2 x 1 x 2 x 3 x 4 x 5 x f(x) x 1 x 2 x 3 x 4 x 5 x

256 256 9 (x 1 + x 2 + x 3 )/3 x 3 (x 2 + x 3 + x 4 )/3 1 (moving average) (x 1, x 2, x 3 ) (1/3, 1/3, 1/3) (convolution) (1/3, 1/3, 1/3) ( 1, 0, 1) (median filter) 3 (x 1, x 2, x 3 ) (f(x 1 ), f(x 2 ), f(x 3 )) f(x 2 ) > f(x 1 ) > f(x 3 ) x 2 f(x 1 )

257 (Laplacian filter) X f(i) 1 f (i) = f(i) f(i 1) (9.13) 1 2 f (i) = f (i + 1) f(i) (9.14) 1 1 f (i) = (f (i + 1) f(i)) (f (i) f(i 1)) = f(i 1) 2f(i) + f(i + 1) (9.15) f(x) f(x) f(x) f(x) f'(i) = f(i) - f(i-1) f''(i) = f(i+1) - f(i) x x x x 1, -2, 1 2 f(i) (f(i 1) 2f(i) + f(i + 1)) = f(i 1) + 3f(i) f(i + 1) (9.16) -1, 3, X Y = =

258 第 9 章 画像処理 258 下図は 空間フィルタを施した例で 左はオリジナル画像 中央はラプラシアンフィルタを施した もの 右はシャープ化フィルタを施したものです 9.5 フーリエ変換 デジタルデータは 連続的なデータではなく 飛び飛びの離散的なデータであり また無限大の区 間でもありません したがって フーリエ変換する場合には 工夫が必要になります その工夫され た手法が 離散フーリエ変換 (Discrete Fourier Transform: DFT) です ある区間において データが n 個 それぞれの値が f0, f1,, fn 1 であったとき 離散フーリエ 変換は 次式で与えられます Fk = n 1 1 j 2π ki fi e n n i=0 (9.17) ここで j は特別に虚数単位を表しています データ番号の i と区別するためです また k は変換後 のデータ番号を表しており k = 0, 1,, n 1 です Fk は おのおのの周波数成分を表し k が 0 か n 1 に近い部分は低周波成分の強さを表し k が n/2 に近い部分は高周波成分の強さを表してい ます 元データの高周波成分がノイズであると判断された場合には 高周波成分の強さを 0 として逆 変換をかければ ノイズが除去されます 離散フーリエ逆変換 (Inverse Discrete Fourier Transform: IDFT) は 以下の式となります fi = n 1 Fk e j 2π n ki (9.18) k=0 このように 簡単な計算によって変換が可能ですが 繰り返し計算が多くなり コンピュータのパワー が必要になります そこで アルゴリズムを高速化した高速フーリエ変換 (Fast Fourier Transform: FFT) が考案されています フーリエ変換に関する書籍はたくさんあるので それらを参考にしてく ださい 通常のデジタル信号処理においては 一次元のデータを取り扱うのですが 画像の場合は 二次元 データとなります 画像をフーリエ変換する場合 画像の濃淡を信号と見なして扱います ただ そ のままフーリエ変換をしたのでは 画像の左上の画素から横方向の変化のみを信号として処理されて しまいます そこで 画像の横方向に加えて 縦方向も考慮して積分することで対応することができ

259 9.5 フーリエ変換 259 ます m n 画素の画像において 各画素の濃度を列番号 u と行番号 v として f (u, v) とすると 二次元 フーリエ変換は 次式で計算することができます F (x, y) = n 1 m 1 2π 2π 1 f (u, v)e j m xu e j n yv mn v=0 u=0 (9.19) ここで x, y は変換後の列番号と行番号を表しています この式を用いて 画像に対してフーリエ変 換を実行すると 画像の濃度の周期性を調べることができます F (x, y) の値を画像化すると 画像 の四隅が低周波成分を表し 画像の中心が高周波成分を表していることになります また 二次元フーリエ逆変換は 次式となります f (u, v) = n 1 m 1 F (x, y)e j m xu e j 2π 2π n yv (9.20) v=0 u=0 フーリエ変換を施した後の F (x, y) を上式に代入すると 元の画像が f (x, y) となって復元されます ある周波数帯においてノイズが発生しているような場合には その周波数成分を調節し 逆変換を施 せば ノイズが軽減された画像として復元されます このフーリエ変換は データ圧縮にも利用できます 例えば 周波数成分の小さいものを省略すれ ば データの保存領域を少なくすることができます 下図は 左の原画像に対してフーリエ変換を施し 周波数成分の画像を右に示したものです 周波 数成分は 中央部が低周波成分で 四隅が高周波成分になるように配列の並びを変えてあります し たがって 変換画像の中央は 直流成分を表しています そして振幅は 明るさで表現されています 振幅の大きいものほど明るくなっています そして下図は 先の原画像に横線のノイズが周期的に入った画像とその周波数成分を示しています 上の周波数成分と異なり 中央部の縦方向に周期的に高い値となっています

260 第 9 章 画像処理 260 このノイズと見られる部分の周波数成分の値を低くすることでノイズが軽減されます 9.6 分類 農業や環境 資源探査の分野において 人工衛星画像を用いて土地被覆の分類を行うことが求めら れています 各画素の地目が 森林か草地か裸地か水域かというような分類です 目視によって判読 することもなされていますが 人工衛星は様々な電磁波の波長域を捉えているので 画像処理を利用 した自動分類がある程度可能です しかし 自動分類という処理は 元来困難なものです 例えば 地表には人間が判断しても森林と草地の区別のつき辛いものも多い状況です 水域と裸地との区別に しても一つの画素の中に水域と裸地が混在している場合もあります このように実際にも区別のつき 辛い部分があるので 自動での画像分類は困難なのです また分類するということは 基本的に人間 の都合によって行われるものなので さらに話が厄介です 画像分類に限らず 例えば惑星と小惑星 の分類でも同様で 以前まで惑星と分類されていた冥王星は 矮惑星という項目に降格されたりして います とかく自然界は 白か黒かはっきりと区別することはできないのが普通で グレーの部分が 多く存在します このようなことから 筆者自身は分類することよりも物理量を求める方が重要であ ると考えています しかし 様々な分野から求められている重要な技術でもあるので 本節において 簡単に解説します バンド間演算 衛星画像は 多数の電磁波波長帯の放射量を計測しています その波長域は 可視光の領域から熱 赤外の領域までを計測する場合が多いです そして太陽光の反射は 物質によって特徴を持ってい ます 下図は 電磁波の波長別の反射率を代表的な地目ごとにグラフ化したもので 分光反射特性 (spectral reflectance) と呼ばれています 可視光の波長帯は およそ µm の範囲であり そ れより長い波長は いわゆる赤外です 植物は人間の目には緑に見えますが それは可視光において 緑の波長帯域にピークを持っているからです さらに植物は近赤外域で非常に反射率が高いのです コンクリートの反射率をも遥かに越えるものであり 近赤外域では極めて反射率が高いことが解りま

261 Terra ASTER AVNIR ASTER Band1 Band2 Band3 60 ASTER VNIR B1 B2 B3 50 Vegetation Reflectance Percentage Soil Concrete Water Wave Length (µm) (band operation) (NDVI: normalized differential vegetation index) NDV I = IR V R IR + V R (9.21) IR V R ASTER VNIR IR Band3 V R Band2 NDV I NDVI

262 µm Digital Number(DN) DN MODTRAN NDVI NDVI NDVI NDVI NDVI NDVI NDVI

263 (level slicing) NDVI NDVI NDVI (threshold) (binarization) (pseudo color) (color table) NDVI

264 第 9 章 画像処理 264 一般に 二値化は 解析に使われ シュードカラーは 分かりやすい表現のために使われています 閾値の決定は 非常に重要です 意味のある値を閾値に選ぶことが重要であり なぜその値が閾値 となり得るのかを説明できなければなりません 特に時系列データを扱う場合は 時期によって閾値 を変化させる必要もあり システマティックな閾値決定手法を構築しなければならないのですが こ れはかなり難しい課題です ディシジョンツリー ディシジョンツリー (decision tree) は レベルスライスの発展形といえます レベルスライスは 1 バンドのデータを閾値を用いて幾つかに分類することでした ディシジョンツリーは if then の ルールにより多バンドのデータを用いて分類する手法です 下図は その例を示したものです ３つ のバンドのデータを利用し ４つのルールにより分類しています START if Band1 > 128 if Band2 > 128 Class 5 Class 4 if Band3 > 128 if Band2 > 128 Class 1 Class 2 Class 3 レベルスライスと同様に閾値の決定は重要で ディシジョンツリーではさらにルールの組み立て手法 も意味付けられるものであることが望まれます

265 (supervised data) (training data) (euclidian distance) k x x k x = x 1 x 2. x k (9.22) n Y y ij i j Y = y 11 y 12 y 1n y 21 y 22 y 2n..... y k1 y k2 y kn (9.23) µ µ = µ 1 µ 2.. µ k = n y i1 /n i=1 n y i2 /n i=1. n y ik /n i=1 (9.24)

266 266 9 x µ d 2 d 2 = (x µ) T (x µ) (9.25) t d 2 = (x 1 µ 1 ) 2 + (x 1 µ 1 ) (x k µ k ) 2 (9.26) (normalized euclidian distance) x 1 µ 1 2 x 2 µ 1 x µ 2 Y s = (s 1, s 2,, s k ) µ s = s 1 s 2.. s k = n (y i1 µ 1 ) 2 /n i=1 n (y i2 µ 2 ) 2 /n i=1. n (y ik µ k ) 2 /n i=1 (9.27)

267 S s s 2 0 S = s k (9.28) d 2 S d 2 = (x µ) T S 1 (x µ) (9.29) x 1 µ 1 2 µ 2 µ 1 x (Maharanobis distance) Y 1 2 s 12 n s 12 = (y i1 µ 1 )(y i2 µ 2 )/n (9.30) i=1 Σ s 11 s 12 s 1k s 21 s 22 s 2k Σ =... s k1 s k1 s kk (9.31)

268 268 9 d 2 Σ d 2 = (x µ) T Σ 1 (x µ) (9.32) GIS (mixel) mixed pixel

269 a w a v a s I I = c w a w + c v a v + c s a s (9.33) c w, c v, c s a w, a v, a s I, a w, a v, a s c w, c v, c s I 1 = c w1 a w + c v1 a w + c s1 a s I 2 = c w2 a w + c v2 a w + c s2 a s (9.34) I 3 = c w3 a w + c v3 a w + c s3 a s I 1, I 2, I 3 a w, a v, a s (geometric correction) (ground control point) (x i, y i ) (u i, v i ) (image control point)

270 /5 1 m 1 10m 1/5 2m 1/ n (x 1, y 1 ), (x 2, y 2 ),, (x n, y n ) (u 1, v 1 ), (u 2, v 2 ),, (u n, v n ) 3.80 x y x y x i x i = au i + bv i + c δ i δ i = au i + bv i + c x i n n Φ Φ = = n i=1 δ 2 i n (au i + bv i + c x i ) 2 (9.35) i=1

271 Φ a, b, c a, b, c 0 Φ a = Φ b = Φ c = n i=1 n i=1 n i=1 2u i (au i + bv i + c x i ) = 0 (9.36) 2v i (au i + bv i + c x i ) = 0 (9.37) 2(au i + bv i + c x i ) = 0 (9.38) [] [ ] u 2 i [u [ i v] i ] [u i ] [u i v i ] v 2 i [v i ] a b = [x iu i ] [x i v i ] (9.39) [u i ] [v i ] n c [x i ] a, b, c y 3.79 a, b x y Φ n Φ = {(au i bv i + c x i ) 2 + (bu i + av i + d y i ) 2 } (9.40) i=1 a, b, c, d 0 Φ a = Φ b = Φ c = Φ d = n i=1 n i=1 n i=1 n i=1 2{u i (au i bv i + c x i ) + v i (bu i + av i + d y i )} = 0 (9.41) 2{ v i (au i bv i + c x i ) + u i (bu i + av i + d y i )} = 0 (9.42) 2(au i bv i + c x i ) = 0 (9.43) 2(bu i + av i + d y i ) = 0 (9.44) [ ] u 2 i + vi 2 [ 0 ] [u i ] [v i ] a [x i u i + y i v i ] 0 u 2 i + vi 2 [v i ] [u i ] b [u i ] [v i ] n 0 c = [ x i v i + y i u i ] [x i ] (9.45) [v i ] [u i ] 0 n d [y i ]

272 272 9 a, b, c, d { x = u cos κ v sin κ + x 0 y = u sin κ + v cos κ + y 0 (9.46) F x (x 0, κ) = u cos κ v sin κ + x 0 x (9.47) F y (y 0, κ) = u sin κ + v cos κ + y 0 y (9.48) x 00, y 00, κ 0 x 0, y 0, κ x 0 = x 00 x 0 (9.49) y 0 = y 00 y 0 (9.50) κ = κ 0 κ (9.51) F x, F y 3.89 x = x 0, a = x 00 x 0 x 00 = x 0 y 0, κ F x (x 0, κ) F x (x 00, κ 0 ) F x x 0 F x κ (9.52) x 0 κ F y (y 0, κ) F y (y 00, κ 0 ) F y y 0 F y κ (9.53) y 0 κ

273 F x = 1 x 0 (9.54) F y = 1 y 0 (9.55) F x = u sin κ v cos κ κ (9.56) F y = u cos κ v sin κ κ (9.57) F x (x 0, κ) F x (x 00, κ 0 ) x 0 + (u sin κ + v cos κ) κ (9.58) F y (y 0, κ) F y (y 00, κ 0 ) y 0 (u cos κ v sin κ) κ (9.59) n (u 1, v 1, x 1, y 1 ),, (u n, v n, x n, y n ) x 0, y 0, κ x 00, y 00, κ , 9.59 x 00, y 00, κ 0 F x, F y n F x = 0, F y = 0 F x, F y x 0, y 0, κ E n { E = Fx (x 00, κ 0 ) x 0 + (u i sin κ 0 + v i cos κ 0 ) κ } 2 i=1 n { + Fy (y 00, κ 0 ) y 0 (u i cos κ 0 v i sin κ 0 ) κ } 2 i=1 (9.60) x 0, y 0, κ 0 x 0, y 0, κ E x 0 = 2 E y 0 = 2 n { Fx (x 00, κ 0 ) x 0 + (u i sin κ 0 + v i cos κ 0 ) κ } = 0 (9.61) i=1 n { Fy (y 00, κ 0 ) y 0 (u i cos κ 0 v i sin κ 0 ) κ } = 0 (9.62) i=1 E n κ = 2 { (ui cos κ 0 v i sin κ 0 )(F x (x 00, κ 0 ) x 0 + (u i sin κ 0 + v i cos κ 0 ) κ) } 2 i=1 n { (ui sin κ 0 + v i cos κ 0 )(F y (y 00, κ 0 ) y 0 (u i cos κ 0 v i sin κ 0 ) κ) } = 0 i=1 (9.63) x 00, y 00

274 274 9 κ x 00 = x 1 u 1 (9.64) y 00 = y 1 u v (9.65) κ 0 = tan 1 y 2 y 1 x 2 x 1 tan 1 v 2 v 1 v 2 v 1 (9.66) x 0, y 0, κ (u, v, w) (x, y, z) x y z = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 u v w + x 0 y 0 z 0 (9.67) a 11 a 33 (x 0, y 0, z 0 ) (u, v, w) (x, y, z) a 11 a 33 (u, v, w) (x, y, z) x ω y ϕ z κ a 11 a 12 a 13 a 21 a 22 a 23 = cos ϕ 0 sin ϕ 0 cos ω sin ω a 31 a 32 a 33 0 sin ω cos ω sin ϕ 0 cos ϕ = cos κ sin κ 0 sin κ cos κ cos ϕ cos κ cos ϕ sin κ sin ϕ cos ω sin κ + sin ω sin ϕ cos κ cos ω cos κ sin ω sin ϕ sin κ sin ω cos ϕ sin ω sin κ cos ω sin ϕ cos κ sin ω cos κ + cos ω sin ϕ sin κ cos ω cos ϕ (9.68) 6 2

275 F x (x 0, ω, ϕ, κ) = a 11 u + a 12 v + a 13 w + x 0 x = (cos ϕ cos κ)u (cos ϕ sin κ)v + (sin ϕ)w + x 0 x (9.69) F y (y 0, ω, ϕ, κ) = a 21 u + a 22 v + a 23 w + y 0 y = (cos ω sin κ + sin ω sin ϕ cos κ)u + (cos ω cos κ sin ω sin ϕ sin κ)v + ( sin ω cos ϕ)w + y 0 y (9.70) F z (z 0, ω, ϕ, κ) = a 31 u + a 32 v + a 33 w + z 0 z = (sin ω sin κ cos ω sin ϕ cos κ)u + (sin ω cos κ + cos ω sin ϕ sin κ)v + (cos ω cos ϕ)w + z 0 z (9.71) x 00, y 00, z 00, ω 0, ϕ 0, κ 0 x 0, y 0, z 0, ω, ϕ, κ x 0 = x 00 x 0 (9.72) y 0 = y 00 y 0 (9.73) z 0 = z 00 z 0 (9.74) ω = ω 0 ω (9.75) ϕ = ϕ 0 ϕ (9.76) κ = κ 0 κ (9.77) F x, F y, F z 3.89 x = x 0, a = x 00 x 0 x 00 = x 0 y 0, z 0, ω, ϕ, κ F x (x 0, ω, ϕ, κ) F x (x 00, ω 0, ϕ 0, κ 0 ) F x x 0 F x x 0 ω ω F x ϕ ϕ F x κ (9.78) κ F y (y 0, ω, ϕ, κ) F y (y 00, ω 0, ϕ 0, κ 0 ) F y y 0 F y y 0 ω ω F y ϕ ϕ F y κ (9.79) κ F z (z 0, ω, ϕ, κ) F z (z 00, ω 0, ϕ 0, κ 0 ) F z z 0 F z z 0 ω ω F z ϕ ϕ F z κ (9.80) κ

276 276 9 F x = 1, F y = 1, F z x 0 y 0 z 0 = 1, F x ω = 0 (9.81) F x = (sin ϕ cos κ)u + (sin ϕ sin κ)v + (cos ϕ)w ϕ (9.82) F x = (cos ϕ sin κ)u (cos ϕ cos κ)v κ (9.83) F y = ( sin ω sin κ + cos ω sin ϕ cos κ)u + ( sin ω cos κ cos ω sin ϕ sin κ)v (cos ω cos ϕ)w ω (9.84) F y = (sin ω cos ϕ cos κ)u + (sin ω cos ϕ sin κ)v + (sin ω sin ϕ)w ϕ (9.85) F y = (cos ω cos κ sin ω sin ϕ sin κ)u + ( cos ω sin κ sin ω sin ϕ cos κ)v κ (9.86) F z ω = (cos ω sin κ + sin ω sin ϕ cos κ)u + (cos ω cos κ sin ω sin ϕ sin κ)v (sin ω cos ϕ)w (9.87) F z = ( cos ω cos ϕ cos κ)u + (cos ω cos ϕ sin κ)v (cos ω sin ϕ)w ϕ (9.88) F z = (sin ω cos κ + cos ω sin ϕ sin κ)u + ( sin ω sin κ + cos ω sin ϕ cos κ)v κ (9.89) x 00, y 00, z 00, ω 0, ϕ 0, κ 0,, x 0, y 0, z 0, ω, ϕ, κ (u, v, w) (x, y, z) ω 0 = 0, ϕ 0 = 0 κ (root mean square error) (x i, y i, z i ) (f x (x i, y i, z i ), f y (x i, y i, z i )) (u i, v i ) RMSE x, RMSE y RMSE x = n (u i f x (x i, y i, z i )) 2 (9.90) i=1 RMSE y = n (v i f y (x i, y i, z i )) 2 (9.91) i=1

277 (resampling) 1 (meta data) { x = au + bv + c y = du + ev + f (9.92) a, b, c, d, e, f (u, v) (x, y) U Y V X (x, y) (u, v)

278 278 9 { u = ax + by + c v = dx + ey + f (9.93) (x, y) (u, v) (x, y) U Y V X P(u i, v j ) P(u i+1, v j ) Q(u, v) P(u i, v j+1 ) P(u i+1, v j+1 ) (Nearest Neighbor) (x, y) (u, v) P (u i, v j ), P (u i+1, v j ), P (u i, v j+1 ), P (u i+1, v j+1 ) (x, y) Q(x, y) P (u i, v j ) Q(x, y) = P (u, v) (9.94) u = int(ax + by + c + 0.5) v = int(dx + ey + f + 0.5) int() 0.5

279 (Bi-Linear) u v v i P (u i, v j ), P (u i+1, v j ) Q(u, v i ) P (u i, v j ) P (u i+1, v j ) P (u i, v j ) P (u i+1, v j ) Q(u, v i ) P P(u i, v j ) Q(u, v i ) P(u i, v j ) Q(u, v j ) P(u i+1, v j ) v - v i Q(u, v) v i+1 - v u - u i 1 u i+1 - u P(u i+1, v j ) u P(u i, v j+1 ) P(u, v j+1 ) u - u i u i+1 - u P(u i+1, v j+1 ) u, v u Q(u, v j ) Q(u, v j ) = (u i+1 u)p (u i, v j ) + (u u i )P (u i+1, v j ) (9.95) v v j+1 Q(u, v j+1 ) Q(u, v j+1 ) = (u i+1 u)p (u i, v j+1 ) + (u u i )P (u i+1, v j+1 ) (9.96) Q(u, v j ) Q(u, v j+1 ) v Q(u, v) Q(u, v) = (v j+1 v)q(u, v j ) + (v v j )Q(u, v j+1 ) = (u i+1 u)(v j+1 v)p (u i, v j ) + (u u i )(v j+1 v)p (u i+1, v j ) + (u i+1 u)(v v j )P (u i, v j+1 ) + (u u i )(v v j )P (u i+1, v j+1 ) (9.97) Bi-Linear (Cubic Convolution)

280 280 9 Nearest Neighbor Cubic Convolution (orthogonal projection) (central projection) (x, y) (u, v) (x, y, z) (u, v) (orthogonal image) (DEM: Digital Elevation Model) 50m 10m 2m (DSM: Digital

281 Surface Model) DSM LiDAR(Light Detection and Ranging) 9.8 L(u, v) R(u, v) T(i, j) (u 0, v 0 ) T(i, j) L(u, v) SSDA SSDA m n (i, j)

282 282 9 T (i, j) (u, v) (u 0, v 0 ) L(u, v) d(u 0, v 0 ) n m d(u 0, v 0 ) = T (i, j) L(u 0 + i, v 0 + j) (9.98) j=0 i= T L r(u 0, v 0 ) n m ( T T (i, j))( L L(u 0 + i, v 0 + j)) j=0 i=0 r(u 0, v 0 ) = (9.99) n m ( T n m T (i, j)) 2 ( L L(u 0 + i, v 0 + j)) 2 j=0 i=0 j=0 i=0 SSDA SSDA SSDA SSDA m n T (i, j) (i c, j c ) (u c, v c )

283 L(u, v) R(u, v) T(i, j) (u c, v c ) (i c, j c ) T (i, j) = al(u, v) + b (9.100) a, b { u = c 1 (i i c ) + c 2 (j j c ) + u c v = c 3 (i i c ) + c 4 (j j c ) + v c (9.101) c 1, c 2, c 3, c 4 a, b, c 1, c 2, c 3, c 4, u c, v c F F (a, b, c 1, c 2, c 3, c 4, u c, v c ) = T (i, j) al(u, v) b (9.102) a 0, b 0, c 10, c 20, c 30, c 40, u c0, v c0 a, b, c 1, c 2, c 3, c 4, u c, v c F (a, b, c 1, c 2, c 3, c 4, u c, v c ) F (a 0, b 0, c 10, c 20, c 30, c 40, u c0, v c0 ) F F F F a b u v (9.103) a b u v c 1, c 2, c 3, c 4, u c, v c u, v u, v F = L(u, v) a (9.104) F = 1 b (9.105) F v) L(u + 1, v) L(u 1, v) = a L(u, = a u u 2 (9.106) F v) L(u, v + 1) L(u, v 1) = a L(u, = a v v 2 (9.107)

284 L(u, v) L u = L(u+1,v) L(u 1,v) 2, L v = L(u,v+1) L(u,v 1) 2 u, v { u = c 1 (i i c ) + c 2 (j j c ) + u c v = c 3 (i i c ) + c 4 (j j c ) + v c (9.108) F (a, b, c 1, c 2, c 3, c 4, u c, v c ) F (a 0, b 0, c 10, c 20, c 30, c 40, u c0, v c0 ) + L(u, v) a + b a 0 L u { c 1 (i i c ) + c 2 (j j c ) + u c } a 0 L v { c 3 (i i c ) + c 4 (j j c ) + v c } (9.109) m n a, b, c 1, c 2, c 3, c 4, u c, v c E n m E = [F (a 0, b 0, c 10, c 20, c 30, c 40, u c0, v c0 ) + L(u, v) a + b j=0 i=0 a 0 L u { c 1 (i i c ) + c 2 (j j c ) + u c } a 0 L v { c 3 (i i c ) + c 4 (j j c ) + v c }] 2 (9.110) a = 1, b = 0, c 1 = 1, c 2 = 0, c 3 = 0, c 4 = 1 u c, v c

285 (map) (Geographic Information System) GIS

286 Google Earth Grass Arc GIS 10.1

287 (point) ID ID 10.1 Point ID x y Point 1 x 1 y 1 Point 2 x 2 y 2... Point n x n y n (line) (node) (chain) (arc) GIS CG ID

288 Line ID x y Line 1 x 11 y 11 x 12 y 12.. x 1n y 1n Line 2 x 21 y 21 x 22 y 22.. x 2n y 2n ID Node 3 Line 1 Line Node ID x y Node 1 x 1 y 1 Node 2 x 2 y 2... Node n x n y n Line ID Node Node Line 1 Node 1 Node 2 Node 2 Node 3 Node 3 Node 4 Line 2 Node 5 Node 6 Node 6 Node 3 Node 3 Node 7 Node 7 Node 8 GIS

289 (polygon) Node ID x y Node 1 x 1 y 1 Node 2 x 2 y 2... Node n x n y n 10.4 Polygon ID Node Node Polygon 1 Node 1 Node 2 Node 2 Node 3 Node 3 Node 1 Polygon 2 Node 4 Node 5 Node 6 Node 7 Node 7 Node 4 (polyline) Polyline (label point) (grid)

290 (x 0, y 0 ) (size of grid) (column) (row) (attribute data) GIS ID ID (meta data)

291 GRS80 400m GIS GIS GIS GIS OS GIS DXF GIS CAD CAD DXF AutoDesk AutoCAD CAD shape shape ESRI GIS Arc View Arc GIS GIS shape GIS shape dbaseiv

292 SIMA GIS SIMA SIMA DM SIMA DM JPGIS JPGIS 2007 GIS XML XML ISO JIS XML G-XML, GML G-XML 2000 GIS JPGIS XML JPGIS ISO JIS PDA GML Open GIS Consortium G-XML GML kml kml Google Maps Google Earth XML 10.2 (map projection)

293 (large scale) 5,000 1 (medium scale) 10,000 50,000 25,000 50,000 (small scale) (cylindrical projection) (Mercator projection) (x, y) (φ, λ) R m { x = mr λ y = mr log tan(45 + φ 2 ) (10.1) λ λ 0 λ = λ λ 0 (conical projection)

294 :1,000,000 1:25,000 (transverse Mercator projection) 1:50,000 1:25,000 (Gauss-Krüger projection) (φ, λ) (x, y) N M 6.17 N = a 1 e2 sin 2 φ (10.2) a e φ N M ds dφ ds ds = Mdφ M M = ds dφ = ds dz dz dφ (10.3)

295 ds X dx Z dz cos φ = dz ds ds dz = 1 (10.4) cos φ 6.24 Z = b2 a a sin φ φ 2 1 e 2 sin 2 φ dz dφ = b 2 cos φ a(1 e 2 sin 2 φ) 3 2 (10.5) M b 2 M = a(1 e 2 sin 2 φ) 3 2 a(1 e 2 ) = (1 e 2 sin 2 φ) 3 2 b 2 = a 2 (1 e 2 ) (10.6) M (φ, λ) N, M (x, y) x y 6 60 (UTM) (universal transverse Mercator projection) ± ± :5,000 UTM 6 19 I XIX IV

296 GIS GIS P (x p, y p ) (x 0, y 0 ) w c P (u p, v p ) { u p = int( x p x 0 w c + 0.5) v p = int( y 0 y p (10.7) w c + 0.5) int() (x 1, y 1 ) (x 2, y 2 ) t { x = (x 2 x 1 )t + x 1 y = (y 2 y 1 )t + y 1 (10.8)

297 t = 0 (x 1, y 1 ) t = 1 (x 2, y 2 ) t 0 1 t w c t t w s t (x 1, y 1 ) (x 2, y 2 ) 1 t w s w s = w c (x2 x 1 ) 2 + (y 2 y 1 ) 2 (10.9) t 0 w s x y

298 GIS GIS (Morphology) (Hough transform) ρ = x cos θ + y sin θ (10.10) θ

299 ρ x y (x, y) θ ρ 8 1*cos(x)+4*sin(x) 3*cos(x)+3*sin(x) 5*cos(x)+2*sin(x) 6*cos(x)+3*sin(x) 7*cos(x)+4*sin(x) 8*cos(x)+5*sin(x) 4 ρ θ (rad) θ ρ θ = 2.35 θ = θ ρ θ ρ θ ρ θ ρ θ ρ (interpolation) (x 0, y 0 ) (x 3, y 3 ) (x 1, y 1 ) (x 2, y 2 ) x p y p

300 n (x 0, y 0 ) (x n, y n ) i i + 1 x p y p (x i, y i ) (x i+1, y i+1 ) y = y i+1 y i x i+1 x i x + y i y i+1 y i x i+1 x i x i (10.11) y p x = x p (Spline) x i x i+1 f i (x) f i (x) = a i x 3 + b i x 2 + c i x + d i (10.12) a i, b i, c i, d i

301 x i x i+1 f i (x) x i g i x i+1 g i+1 f i (x) = g i+1 g i x i+1 x i x + g i g i+1 g i x i+1 x i x i = g i + (x x i ) g i+1 g i x i+1 x i (10.13) f i(x) = f i(x i ) + g i (x x i ) (x x i) 2 g i+1 g i x i+1 x i (10.14) f i (x) g i, g i+1 f i (x) = f i (x i ) + f i(x i )(x x i ) g i(x x i ) (x x i) 3 g i+1 g i x i+1 x i (10.15) x = x i+1 f i (x i+1 ) = f i (x i ) + f i(x i )(x i+1 x i ) g i(x i+1 x i ) (x i+1 x i ) 3 g i+1 g i x i+1 x i (10.16) f i(x i ) = f i(x i+1 ) f i (x i ) x i+1 x i 1 2 g i(x i+1 x i ) (x i+1 x i )(g i+1 g i ) = y i+1 y i x i+1 x i 1 6 (x i+1 x i )(g i+1 g i ) (10.17) f i (x i ), f i (x i+1 ) y i, y i+1

302 x = x i+1 f i(x i+1 ) = f i(x i ) + g i (x i+1 x i ) (x i+1 x i ) 2 g i+1 g i x i+1 x i = f i(x i ) (x i+1 x i )(g i+1 + g i ) f i(x i ) = y i+1 y i x i+1 x i 1 6 (x i+1 x i )(g i+1 g i ) (x i+1 x i )(g i+1 + g i ) = y i+1 y i x i+1 x i 1 6 (x i+1 x i )(2g i+1 + g i ) (10.18) f i(x i ) = y i y i 1 x i x i (x i x i 1 )(2g i + g i 1 ) (10.19) y i+1 y i 1 x i+1 x i 6 (x i+1 x i )(g i+1 g i ) = y i y i 1 1 x i x i 1 6 (x i x i 1 )(2g i + g i 1 ) ( yi+1 y i 6 y ) i y i 1 = (x i+1 x i )(g i+1 g i ) (x i x i 1 )(2g i + g i 1 ) x i+1 x i x i x i 1 = (x i x i 1 )g i 1 + 2(x i+1 x i 1 )g i + (x i+1 x i )g i+1 (10.20) g i 1, g i, g i+1 i = 1, 2, 3 g 0, g 1, g 2, g 3, g 4 g 0, g 4 g 0 = g 4 = 0 g 1, g 2, g n (x 1, y 1, z 1 ),, (x n, y n, z n ) (x p, y p, z p ) z (x p, y p )

303 z p (x p, y p, z p ) z p x-y (x p, y p ) (x 4, y 4 ) z p (x 2, y 2 ) n w i z p = n w i z i i=1 (10.21) n w i i=1 w i (x i, y i ) (x p, y p ) d i w i = 1 (xi x p ) 2 + (y i y p ) 2 (10.22)

304 (semi-varience) (x i, y i, z i ) (x j y j, z j ) Γ d ij Γ(d ij ) = (z i z j ) 2 2 (10.23) d d ij d ij = (x i x p ) 2 + (y i y p ) 2 d ij d ij Γ d Γ d Γ Γ Γ(d ij ) n n C 2 50m 1km

305 10.4 データ内挿 305 離における半分散の最大値は 距離が 500m より大きくなると ほぼ一定になっていることが解りま す 距離が長くなると 二点間の標高差は 非常に近いものもあるのですが 標高差の最大値は あ まり変化しないと言えます 180 semivariogram1.txt Semi-Variance (m^2) Distance (m) このセミバリオグラムを解りやすく表現するために ある一定距離の範囲ごとに半分散を平均化した ものが下図です 4 semivariogram2.txt 3.5 Semi-Variance (m^2) Distance (m) もとが 50m メッシュのグリッドデータなので 50m ごとに区切って平均値を求めています この平 均値を各距離における半分散の値として利用できます 重み wi に関しては ４つの点を用いて内挿 する場合 セミバリオグラムから各点間の距離における半分散の値を求めると 次式が成り立ちます Γ(d1p ) = w1 Γ(d11 ) + w2 Γ(d12 )w3 Γ(d13 ) + w4 Γ(d14 ) Γ(d2p ) = w1 Γ(d21 ) + w2 Γ(d22 )w3 Γ(d23 ) + w4 Γ(d24 ) Γ(d3p ) = w1 Γ(d31 ) + w2 Γ(d32 )w3 Γ(d33 ) + w4 Γ(d34 ) Γ(d4p ) = w1 Γ(d41 ) + w2 Γ(d42 )w3 Γ(d43 ) + w4 Γ(d44 ) (10.24) それぞれの Γ(dij ) の値は 上のグラフより求めて代入すると 未知数が w1,, w4 の連立方程式と なります この連立方程式を解いて 重み wi を求め 式 を用いれば zp の値が求まります さて 半分散を計算するとき 対象範囲全体にわたってデータが同じようなばらつきを持つもの は 対象範囲全体でセミバリオグラムを作成します しかし 対象範囲が非常に広いときには 場所 によってばらつきが異なる場合もあります そのようなときは あるウィンドウサイズを設定し 場

306 (spatial analysis) (overlay) 50m

307 10.5 空間解析 究所が発行する 250m メッシュのデータをポリゴン化したもので 地質帯ごとに色分けしています 水系データは国土地理院の空間データ基盤のもので ベクトル型のデータです このように 様々な 項目のデータをオーバーレイすることにより 地図らしくなっていることがわかります 地図らしいというだけでなく 地質ごとに地形的な特徴の違いが視覚的に捉えられます 下図は 上図の上に 地すべり防止区域のデータをオーバーレイさせたものです ピンクで表され ている部分が 地すべり防止区域です 地すべり防止区域は ある特定の地質で多く発生しており しかも水系沿いに多いことが解ります 307