4 14 4 14 4 1

1 4 1.1................................................ 4 1............................................. 4 1.3................................................ 5 1.4 1............................................ 7 1.5............................................ 8 1.6........................................ 9 1.7................................................ 11 1.8.................................................. 11 1.8.1 1 k, k 1................................. 11 1.8. u f(u) u........................ 1 1.8.3 3 3............................... 1 1.8.4 4 4............................... 13 1.8.5 5 5............................... 14 1.8.6 6 Gram-Schmidt........................... 15 1.8.7 7 w.................................... 16 1.8.8 8.................................... 16 1.8.9 9 (1.6.).................................... 16 1.8.1 1................................ 17 1.8.11 11...................... 18 1.8.1 1 (1.8.14).................................. Bessel Kepler 3.1................................................ 3....................................... 4..1.................................. 4......................................... 4.3.............................................. 5.3.1........................................ 5.3. Kepler..................................... 6.4............................................ 7.4.1 Bessel...................................... 7.4. Kepler..................................... 8.4.3 Bessel............................... 3.5........................................ 31.5.1 e 1................................. 31.5. e 1..................................... 33.6................................................ 34

35

3 Contents 1. Tadashi YANO: Quaternions and Spherical Linear Interpolation. Kenji SETO: Bessel Function and Kepler Equation 3. Tadashi YANO: Editorial Comments

4 Quaternions and Spherical Linear Interpolation 1 Tadashi Yano 1.1 1. 1.3 1.4 1.5 1.6 1. 3 4 1 yanotad@earth.ocn.ne.jp 3 interpolation 4 extrapolation

5 Shoemake 1985 [1] 1.3 1.1 x y B y y E (1 t)s C z k 1 y z ts x O k x D x A 1.1: 1 1.: z x y z = k (t)x + k 1 (t)y, t 1 (1.3.1) 5 t k (t), k 1 (t) x y z k (t), k 1 (t) t 1 k (t) = at + b, (1.3.) k 1 (t) = ct + d (1.3.3) a, b, c, d t = z = x t = 1 k () = 1, k 1 () = (1.3.4) z = y k (1) =, k 1 (1) = 1 (1.3.5) 5 t 1

6 k (t), k 1 (t) a, b, c, d k () = 1, k (1) = k 1 () =, k 1 (1) = 1 (1.3.6),(1.3.7),(1.3.8),(1.3.9) 6 (1.3.1) b = 1, (1.3.6) a + b = (1.3.7) d = (1.3.8) c + d = 1 (1.3.9) a = 1, b = 1, c = 1, d = k (t) = 1 t, k 1 (t) = t (1.3.1) z = (1 t)x + ty (1.3.11) k (t) = k, k 1 (t) = k 1 t (1.3.1) 1. z O, A, B, C, D, E 1. DC = OE = k 1 y y EC = OD = k x x 1. z = k x + k 1 y (1.3.1) 1. AB = s AC = ts, CB = (1 t)s EC OA ECB OAB k k x (1 t)s = x s k = (1 t)s s DC OB DAC OAB k 1 k 1 y ts = y s k 1 = ts s (1.3.1) (1.3.13) (1.3.1),(1.3.13) (1.3.1) (1 t)s k =, k 1 = ts (1.3.14) s s (1.3.1) (1.3.14) 6 k (t), k 1 (t) 1

7 1.4 1 1.3 y k 1 y z k x x O B (1 t)θ I y C E H z tθ A F D x G r r 1.3: 1 1.4: x = y = z = r (1.4.1) z = k x + k 1 y (1.4.) k, k 1 7 (1.4.) (1.3.1) (1.4.1) x y θ θ k, k 1 (1.3.14) 8 k, k 1 (1.3.14) s θ θ k = (1 t)θ, k 1 = tθ θ θ (1.4.3) θ (1.4.3) θ t k, k 1 sin (1.4.3) (1 t)θ sin (1 t)θ tθ sin tθ θ sin θ k = sin (1 t)θ sin tθ, k 1 = sin θ sin θ (1.4.4) 7 1.4 k, k 1 1.3 8 x y s x y θ

8 u f(u) u sin tan u f(u) u sin u f(u) u f(u) 1.5 1.5 [] 9 k, k 1 z C x A OB I H 1.4 1 1.4 tθ (1 t)θ AOC COB r 1 tθ (1 t)θ θ = AOB 1.6-1.9 EC OA ECI OAH EC r sin(1 t)θ = x r sin θ EC = OD = k x 11 k = sin(1 t)θ sin θ (1.5.1) z C y B OA G F 1.4 DC OB DGC OFB DC r sin tθ = y r sin θ 1 DC = OE = k 1 y k 1 = sin tθ sin θ 9 [] [3] 1 1.4 k x k 1 y 1.3 11 CI OCI CI = r sin(1 t)θ 1 CG OGC CG = r sin tθ (1.5.)

9 (1.5.1),(1.5.) k = z = sin(1 t)θ, k 1 = sin θ sin tθ (1.4.4) sin θ sin(1 t)θ sin tθ x + sin θ sin θ y (1.5.3) (1.4.4) θ (1.4.3) (1.5.3) (1.3.11) (1.5.3) z z z z = r (1.5.3) 3, 4, 5 1.6 [8] 3 1. a = a 1 a. t a, ( t 1, t : ) 3. a = a + t a x, y 1 1. x, y x y z z = yx 1 13. z ( z) t 14 3. x z(t) = ( z) t x 13 8 [8] 14 (fraction)

1 x, y z x = y = z = 1 15 x, y x = 1, y = cos θ + n sin θ x, y n = in x + jn y + kn z 1 z x z ( z)x = y x 1 z z = yx 1 (1.6.1) ( z) t = (yx 1 ) t = cos tθ + n sin tθ (1.6.) 16 3 x ( z) t x = 1 z = ( z) t x = (cos tθ + n sin tθ) 1 = cos tθ + n sin tθ (1.6.3) z z = Ax + By (1.6.4) A, B x = 1, y = cos θ + n sin θ (1.6.3),(1.6.5) (1.6.6),(1.6.7) A, B z = A + B cos θ + Bn sin θ (1.6.5) cos tθ = A + B cos θ, (1.6.6) sin tθ = B sin θ (1.6.7) sin(1 t)θ A = sin θ sin tθ B = sin θ 15 r r = 1 16 ( z) t z 9

11 x y z z = z (1.5.3) sin(1 t)θ sin tθ x + sin θ sin θ y (1.6.8) 1.7 1 [9] 1.8 1.8.1 1 k, k 1 1.5 x y x y y x P(t) P(t) O z z t z = x + t(y x) 17 z = (1 t)x + ty (1.3.11) k = 1 t, k 1 = t t = y z y x P(t) t(y x) O x 1.5: k, k 1 t = 1 z = x z = y 17 t x y z

1 z = k x + k 1 y (1.3.1) k (t) = 1 t, k 1 (t) = t k () = 1, k 1 () = k (1) =, k 1 (1) = 1 1.8. u f(u) u u f(u) u sin tan u f(u) u [4] u F (u) u sin u, tan u, arcsin u, arctan u, sinh u, tanh u, sinh 1 u, tanh 1 u log(1 + u), e u sin u log(1 + u) 1 + u z z x, y sin tan 1.8.3 3 3 3 3 [5] [6] x y 1.6 z x y 1.7 z = (cos tθ)x + (sin tθ)y (1.8.1) x y x Gram-Schmidt 18 x v x y v w 19 w z x w z = (cos tθ)x + (sin tθ)w (1.8.) w x y (1.5.3) 3 18 6 19 1 r 7

13 w v y (1 t)θ y (1 t)θ z tθ z tθ x x r r 1.6: 3.1 1.7: 3. 1.8.4 4 4 4 [7] y k 1 y z k x (1 t)θ x tθ O y E B (1 t)θ C z tθ A F D x G r r 1.8: 4 1.9: 5 x y z z = k x + k 1 y (1.4.) z x y z x z y 1.8

14 z x tθ z x = r cos tθ (1.8.3) z y (1 t)θ z y = r cos(1 t)θ (1.8.4) (1.4.) z x, y z x = r (k + k 1 cos θ) (1.8.5) z y = r (k cos θ + k 1 ) (1.8.6) (1.8.3),(1.8.5) (1.8.4),(1.8.6) k, k 1 k + k 1 cos θ = cos tθ k cos θ + k 1 = cos(1 t)θ 1 k, k 1 (1.4.4) 1.8.5 5 5 5 [8] 1.8 z = k x + k 1 y (1.4.) 1.9 DC OB DGC OFB OB BF = DC CG (1.8.7) DGC DC = k 1 y CG = r sin tθ OFB OB = y BF = r sin θ 1 y r sin θ = k 1 y r sin tθ DC OB COB = OCD = (1 t)θ k x, k 1 y, z (1.8.8) k x sin(1 t)θ = k 1 y sin tθ (1.8.9) 1 [8] CG 1.9 DGC CG = k 1 y sin θ OGC CG = r sin tθ (1.8.8) r r = y

15 (1.8.8) (1.8.9),(1.8.1) k = k 1 = sin tθ sin θ sin(1 t)θ sin θ (1.8.1) (1.8.11) 1.8.6 6 Gram-Schmidt x y x x v y v = y ax a a v x v x = x y a x = a = x y x θ x y = cos θ v r v r w N w = r N w = Nv N v = w = r N > N = r v = 1 sin θ N = r v = 1 sin θ v = y (cos θ)x v = r sin θ r w w = 1 [y (cos θ)x] (1.8.1) sin θ [8]

16 1.8.7 7 w (1.8.1) (1.8.) y (cos θ)x z = (cos tθ)x + (sin tθ) sin θ = = ( cos tθ sin tθ cos θ sin θ ) x + sin tθ sin θ y sin(1 t)θ sin tθ x + sin θ sin θ y (1.5.3) 1.8.8 8 [1] x y x θ y x y z ( z)x = y (1.8.13) z (1.8.13) x 1 z = yx 1 (1.6.1) z y x 1.8.9 9 (1.6.) (1.6.) z 11 x = 1 x 1 = x x x x = 1, y = cos θ + n sin θ y x = cos θ + n sin θ = e nθ e nθ = cos θ + n sin θ (1.8.14) 3 (1.8.14) ( z) t = (yx 1 ) t = exp[t log(y x)] = exp (tnθ) = exp (ntθ) = cos tθ + n sin tθ (1.6.) 3 (1.8.14) 1

17 1.8.1 1 z = x + iy z = x + iy = re iθ = r(cos θ + i sin θ) (polar form) [11] r = z : z θ = arg(z) : z e iθ = 1 : e iθ 1 1. 1. (quaternion) p p = w + ix + jy + kz p = p p = (w + ix + jy + kz)(w ix jy kz) = w + x + y + z m = p m = p = w + x + y + z p = m p m u = p = W + ix + jy + kz m uū = 1 u = W + X + Y + Z = 1 W = cos θ, X = n x sin θ, Y = n y sin θ, Z = n z sin θ n = n x + n y + n z = 1, n = in x + jn y + kn z

18 u = 1 u = W + ix + jy + kz = cos θ + n sin θ = e nθ p = mu = p e nθ = w + x + y + z e nθ 1.8.11 11 [1] p = w + ix + jy + kz = w + r exp p = exp (w + ix + jy + kz) w ix + jy + kz 1 p exp p = e w exp (ix + jy + kz) exp (ix + jy + kz) = e nθ = cos θ + n sin θ exp p = e w exp nθ (1.8.15) p = q, w = a, ix + jy + zk = v, θ = r = v, n = ix+jy+zk r exp q ( exp q = e a cos v + v ) sin v v w = θ = α, ix+jy+kz r = e w (cos θ + n sin θ) (1.8.15) = v v, [1] p.13 = in x + jn y + kn z = n [13] p.169 9-16 exp p = cos α + n sin α

19 1 p p = p e nθ, n = in x + jn y + kn z p = w + r, r = xi + yj + zk e nθ = w p + r p cos θ + n sin θ = w p + r p w r cos θ = w p, w = p cos θ, n sin θ = r p r = n p sin θ p log p = log p + log e nθ = log p + nθ = log p + n arccos( w p ) θ = arccos w v p. p = q, w = a, n = v [1] p.13 log q = log q + v v arccos a q exp log p = p p t p t = exp (t log p) = exp [t(log p + nθ)] = p t e ntθ p = p e nθ p t = p t e ntθ

1.8.1 1 (1.8.14) e nθ = cos θ + n sin θ (1.8.14) (4) [14] p p = w + r, r = ix + jy + kz r = ix + jy + kz Napir e e ix+jy+kz 4 e ix+jy+kz = cos x + y + z + ix + jy + kz x + y + z sin x + y + z (1.8.16) e ix+jy+kz e ix Maclaurin e ix+jy+kz = 1 + ix + jy + kz 1! + (ix + jy + kz)! + (ix + jy + kz)3 3! + (ix + jy + kz)4 4! (ix + jy + kz) n n =, 3, 4, + (1.8.17) (ix + jy + kz) = (x + y + z ) (ix + jy + kz) 3 = (x + y + z )(ix + jy + kz) (ix + jy + kz) 4 = (x + y + z ) (ix + jy + kz) 5 = (x + y + z ) (ix + jy + kz) (ix + jy + kz) 6 = (x + y + z ) 3 (ix + jy + kz) 7 = (x + y + z ) 3 (ix + jy + kz) (ix + jy + kz) 8 = (x + y + z ) 4 (ix + jy + kz) = (x + y + z ) (1.8.17) (ix + jy + kz) (ix + jy + kz) (ix + jy + kz) [ e ix+jy+kz = 1 (x + y + z ) + (x + y + z ) (x + y + z ) 3 ] +! 4! 6! [ 1 + 1! (x + y + z ) + (x + y + z ) (x + y + z ) 3 ] + (ix + jy + kz) 3! 5! 7! = cos x + y + z ix + jy + kz + x + y + z sin x + y + z r = x + y + z 4 Napir e =.7188188

1 e ix+jy+kz = cos r + ix + jy + kz r sin r (1.8.18) ix + jy + kz = in x + jn y + kn z = n r r = θ (1.8.18) (1.8.18) e nθ = cos θ + n sin θ (1.8.14) e ix+jy+kz = 1 e ix+jy+kz 1 x, y, z 3 r, ϕ, ψ x = r cos ϕ y = r sin ϕ cos ψ z = r sin ϕ sin ψ e r(i cos ϕ+j sin ϕ cos ψ+k sin ϕ sin ψ) = cos r + (i cos ϕ + j sin ϕ cos ψ + k sin ϕ sin ψ) sin r (6) [14] n x = x r = cos ϕ n y = y r n z = z r = sin ϕ cos ψ = sin ϕ sin ψ n x + n y + n z = 1 (14. 4. 1) [1] K. Shoemake, Animating Rotation with Quaternion Curves, Computer Graphics, 19(3), (1985) 45-54 []

[3] http://marupeke96.com/dxg_no57_shearelinearinterwithoutqu.html [4] M. R. Spiegel, 1995 111-11 [5] S. R. Buss, 3D Computer Graphics: A Mathematical Introduction with Open GL (Cambridge University Press, 3) 1-15 [6] http://en.wikipedia.org/wiki/slerp [7] Y. S. Kim, http://www.purose.net/~y-kim/ [8] F. Dunn and I. Parberry, 3D, 8 17-176 [9] 7 [1] F. Dunn and I. Parberry, 3D, 8 167-168 [11] http://rip9455.wordpress.com/1/8//quaternions [1] http://en.wikipedia.org/wiki/quaternion [13] F. Dunn and I. Parberry, 3D, 8 169-17 [14] (1.3) 14-4

3 Bessel Kepler Bessel Function and Kepler Equation 5 Kenji Seto 6.1 Bessel 1),),3) Bessel 3 Bessel Bessel Friedrich Wilhelm Bessel (1784-1846) Bessel Kepler Bessel Bessel 174 Daniel Bernoulli (17-178) Riccati Bernoulli Bessel 4) Euler, Lagrange, Fourier, Poisson Bessel 184 Bessel Kepler Bessel Hankel, Lommel, Neumann, Schläfli Bessel Kepler Kepler..3 Kepler.4 Bessel Kepler Bessel.5 Kepler 5 6 seto@pony.ocn.ne.jp

4...1 (x, y) r (r, θ) r r θ e r, e θ e r, e θ x, y i, j t e r = cos θ i + sin θ j, e θ = sin θ i + cos θ j (..1) de r dt = dθ dt e θ, de θ dt = dθ dt e r (..) dr/dt r = re r (..) dr dt = dr dt e r + r dθ dt e θ (..3) d r dt = d r dt e r + dr dθ dt dt e θ + r d θ ( dθ ) er dt e θ r (..4) dt d r dt [ d = r ( dθ ) ] dt r e r + 1 dt r d ( dt r dθ dt ) e θ (..5)...1 F 1, F P a, ae ( e < 1).1 P F 1 P r, F 1 P θ F P a r P D F DP (ae + r cos θ) + (r sin θ) = (a r) (..6) r = l 1 + e cos θ, l = a(1 e ) (..7)

5 e l θ = π/ r a b b = a 1 e l l = b /a e 1 C DP a Q CQ ϕ ϕ θ ϕ (..7) r cos ϕ = ae + r cos θ a (..8) (..7) cos ϕ = e + cos θ 1 + e cos θ cos θ = cos ϕ e 1 e cos ϕ (..9) (..1) r cos ϕ (..1) cos θ sin θ r = a(1 e cos ϕ) (..11) sin θ = 1 cos θ = 1 e sin ϕ 1 e cos ϕ (..1) sine n θ = nπ ϕ nπ θ, ϕ.3.3.1 M m G m d r dt (..5) r θ (.3.3) = GMm r e r (.3.1) d r ( dθ ) dt r GM = dt r (.3.) d ( r dθ ) dt dt r dθ dt = (.3.3) = Const. h (.3.4) h h/

6 r, θ t r θ θ d r dt = d dt (.3.) dr dt = dθ dr dt dθ = h dr r dθ ( h dr ) r = dθ dθ dt h r d dθ d dθ ( h dr ) r = h d dθ r dθ ( h dr ) r dθ (.3.5) (.3.6) ( h dr ) ( h ) GM r r = dθ r r (.3.7) h /GM l h GM (.3.8) d ( l dr ) ( l ) dθ r dθ r 1 = (.3.9) l r 1 = ζ (.3.1) l dr = dζ (.3.11) r ζ (.3.9) d ζ dθ + ζ = (.3.1) e, δ r r = ζ = e cos(θ δ ) (.3.13) l 1 + e cos(θ δ ) (.3.14) (..7) F 1 δ θ.1 A δ.3. Kepler t (.3.4) t ht = θ r dθ (.3.15) t θ δ = (.3.14) r ht = θ l ( 1 + e cos θ ) dθ (.3.16)

7 (.3.15) θ ϕ (..1) sin θdθ = (1 e ) sin ϕ dϕ (.3.17) (1 e cos ϕ) (..1) sin θ 1 e dθ = dϕ (.3.18) 1 e cos ϕ r ϕ (..11) (.3.15) ht = a ϕ 1 e (1 e cos ϕ )dϕ (.3.19) ht = a 1 e (ϕ e sin ϕ) (.3.) a b b = a 1 e S S = πab = πa 1 e, h/ T (.3.) T T = πa 1 e h ϕ e sin ϕ = πt T (.3.1) (.3.) Kepler Kepler Johannes Kepler(1571-163) Sir Isaac Newton(164-177) Kepler Tycho Brahe (1546-161) Kepler Kepler Kepler Newton Joseph-Louis Lagrange(1736-1813) Kepler 14 177.4 (.3.) t ϕ ϕ Bessel 1817.4.1 Bessel Bessel Bessel exp[(z/)(t 1/t)] Taylor [ z exp t ( 1 )] [ = t k= 1 k! ( zt ) k ][ l= 1 ( z ) l ] l! t t Laurent = [ n= m= ( 1) m z ) n+m ] m!(n + m)!( t n + [ n=1 m= (.4.1) ( 1) n+m z ) n+m ] t m!(n + m)!( n (.4.)

8 1 t n n n Bessel J n (z) = m= ( 1) m z ) n+m (.4.3) m!(n + m)!( n J n (z) = m= ( 1) m z ) n+m = m!( n + m)!( m=n [ z ( exp t 1 )] = t ( 1) m z m!( n + m)!( = m= n= Bessel ) n+m ( 1) n+m z ) n+m = ( 1) (n + m)!m!( n J n (z) (.4.4) J n (z)t n (.4.5) (.4.5) t n+1 t 1 Cauchy πij n (z) πij n (z) = 1 [ z ( t n+1 exp t 1 )] dt (.4.6) t t ϕ t = e iϕ 7 J n (z) = 1 π e i(nϕ z sin ϕ) dϕ (.4.7) π π [ π, ] [, π] [ π, ] ϕ J n (z) = 1 π π cos(nϕ z sin ϕ)dϕ (.4.8) Bessel Bessel ϕ sin ϕ Kepler.4. Kepler Kepler (.3.) τ τ πt T (.4.9) ϕ e sin ϕ = τ (.4.1) n τ = nπ ϕ nπ ϕ, τ ϕ τ ϕ ϕ + π, τ τ + π ϕ τ π ϕ τ Fourier-sine ϕ τ = A n sin(nτ) (.4.11) n=1 7 e e, e

9 sin(mτ) τ sine A n π sin(nτ) sin(mτ)dτ = π δ n,m (.4.1) A n = π π (ϕ τ) sin(nτ)dτ (.4.13) τ =, π ϕ τ A n = π π cos(nτ) ( dϕ )dτ n dτ 1 = π π cos(nτ) n dϕ dτ (.4.14) dτ Kepler (.4.1) τ ϕ A n = πn π cos[n(ϕ e sin ϕ)]dϕ (.4.15) (.4.8) Bessel ϕ ϕ = τ + A n = n J n(ne) (.4.16) n=1 Bessel n J n(ne) sin(nτ) (.4.17) ϕ cos ϕ ϕ cos ϕ = B + B n cos(nτ) (.4.18) n=1 Fourier-cosine B τ πb = π cos ϕ dτ = τ cos ϕ π π + τ sin ϕ dϕ dτ (.4.19) dτ τ = π ϕ π Kepler τ ϕ πb = π + π (ϕ e sin ϕ) sin ϕ dϕ (.4.) B = e (.4.1) n 1 B n (.4.18) cos(mτ) cosine π cos(nτ)dτ =, π cos(nτ) cos(mτ)dτ = π δ n,m, n, m 1 (.4.) π B n = π cos(nτ) cos ϕ dτ (.4.3) Kepler τ ϕ Bessel B n = 1 n[ Jn 1 (ne) J n+1 (ne) ] (.4.4)

3 cos ϕ cos ϕ = e + n=1 1 [ Jn 1 (ne) J n+1 (ne) ] cos(nτ) (.4.5) n (..11) r ) r = a (1 + e 1 [ ae Jn 1 (ne) J n+1 (ne) ] cos(nτ) (.4.6) n θ cos θ (..1) cos θ = 1 ( 1 e ) e + 1 e 1 e cos ϕ Kepler (.4.1) τ n=1 dϕ dτ = 1 1 e cos ϕ (.4.17) τ (.4.7) (.4.7) (.4.8) 1 1 e cos ϕ = 1 + J n (ne) cos(nτ) (.4.9) cos θ = e + (1 e ) e (.4.17) (.4.5) n=1 J n (ne) cos(nτ) (.4.3) n=1.4.3 Bessel (.4.9) τ = ϕ Bessel 1 + J n (ne) = 1 1 e (.4.5) τ = 1 Jn 1 (ne) J n+1 (ne) n[ ] = 1 + e n=1 Bessel n=1 (.4.31) (.4.3) e J n 1 (z) J n+1 (z) = d dz J n(z) (.4.33) n=1 n J n(ne) = e + e 4 (.4.34) (.4.31) (.4.3) Kapteyn 5),6) (.4.34) 3 n 3 e 1 e < 1 e > 1 e < 1 1 < e

31.5.5.1 e 1 13 11 1 11 9. e 1 F 1, F P. ae(e > 1) F P F 1 P a F 1 P r F P a + r F 1 P θ θ.1 P D F DP (ae r cos θ) + (r sin θ) = (a + r) (.5.1) l r = 1 + e cos θ, l = a(e 1) (.5.) π/ π θ θ θ θ < θ < θ cos θ = 1/e (.5.3) ϕ (..9) cosh ϕ = cos θ + e 1 + e cos θ (.5.4) θ ( θ, θ ) ϕ (, ) (.5.) cos θ = cosh ϕ e 1 e cosh ϕ (.5.5) r = a(e cosh ϕ 1) (.5.6) (..11) (.5.5) sin θdθ = (e 1) sinh ϕ dϕ (.5.7) (1 e cosh ϕ)

3 (.5.5) sin θ sin θ = 1 cos θ = e 1 sinh ϕ e cosh ϕ 1 (.5.8) (.5.7) dθ = e 1 dϕ (.5.9) e cosh ϕ 1 (.3.4) (.5.6) r a e 1(e cosh ϕ 1)dϕ = hdt (.5.1) e sinh ϕ ϕ = τ (.5.11) τ ht τ = a e 1 (.5.1) (.5.11) Kepler e < 1 1 < e τ ϕ ϕ ϕ τ τ e ϕ Fourier-cosine 8 K(k) e ϕ = K(k) = π K(k) = πk K(k) cos(kτ)dk (.5.13) Kepler τ ϕ K(k) = πk (.5.13) e ϕ = π 1 [ k e ϕ cos(kτ)dτ (.5.14) e ϕ sin(kτ) dϕ dτ (.5.15) dτ e ϕ sin [ k(e sinh ϕ ϕ) ] dϕ (.5.16) e ϕ sin [ k(e sinh ϕ ϕ ) ] dϕ ] cos(kτ)dk (.5.17) ϕ Bessel ϕ k k 8 e e, e

33.5. e 1.3 Bessel e 1 l F F P FP r F FP θ.3 r = l 1 + cos θ, π < θ < π (.5.18) (.3.14) e = 1, δ = (.3.4) r t ht = θ l (1 + cos θ ) dθ (.5.19) t F (.5.18) r z = tan(θ/) (.5.) dθ = dz (.5.1) 1 + z r = l (1 + z ) (.5.) τ l (1 + z ) dz dt = h (.5.3) l ( z + 1 3 z3) = ht (.5.4) z 3 + 3z = τ (.5.5) τ = 3h l t (.5.6)

34 (.5.5) Kepler 3 Cardano z = ( τ + 1 + τ ) 1/3 ( τ + 1 τ ) 1/3 (.5.7) (.5.) r r = l [ ( τ + 1 + τ ) /3 + ( τ + 1 τ ) /3 1 ] (.5.8) r.6 Kepler θ ϕ Kepler Bessel ϕ Bessel Kepler Bessel = 1) G.N.Watson, A Treatise of the Bessel Functions, nd ed. Cambridge Univ. Press. (1966). ) (1954). 3) (196). 4) 3 1 (13.3) -8. 5) A. Erdéli et.al., Higher Transcendental Functions, Vol., Bateman Manuscript Project, McGraw-Hill Book Co. Inc. (1953) p.13. 6) 3 (196), pp. 13-14.

35 3 3 (14.4.16)