2 T ax 2 + 2bxy + cy 2 + dx + ey + f = 0 a + b + c > 0 a, b, c A xy ( ) ( ) ( ) ( ) u = u 0 + a cos θ, v = v 0 + b sin θ 0 θ 2π u = u 0 ± a
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1 2 T ax 2 + 2bxy + cy 2 + dx + ey + f = 0 a + b + c > 0 a, b, c A xy u = u 0 + a cos θ, v = v 0 + b sin θ 0 θ 2π u = u 0 ± a cos θ, v = v 0 + b tan θ π 2 < θ < π 2 u = u 0 + 2pt, v = v 0 + pt 2 < t < ac b 2 > 0 λ 1λ 2 > 0 b ac b 2 = 0 ac b 2 = 0 λ 1 λ 2 = 0 d e f f b ac b 2 < 0 λ 1λ 2 < 0 f 2
2 2018 T140073
3
4 1 1 2 CG 1 2 OpenGL 1
5 1.1 a, b, c, d, e, f x y ax 2 + 2bxy + cy 2 + dx + ey + f = 0 x, y a + b + c > 0 [1] ax 2 + 2bxy + cy 2 + dx + ey + f = 0 a + b + c > 0 2
6 A = a b b c, B = d e, f ax 2 + 2bxy + cy 2 + dx + ey + f = 0 x y a b b c x y + d e x y + f = 0 x y A x y + B x y + f = 0 [2] 3
7 2.2 A = a b b c det λ a b = 0 b λ c λ a b b λ c = 0 2 λ aλ c b 2 = 0 λ 2 aλ cλ + ac b 2 = 0 λ 2 a + cλ + ac b 2 = 0 2 λ λ λ = a+c± a+c 2 4ac b 2 2 a + c 2 4ac b 2 = 0 4
8 2.3 A A λ 1 = a + c + a + c 2 4ac b 2 2 λ 2 = a + c a + c 2 4ac b 2 2 λ 1, A 1 A 1 = x 1 y 1 λ 1 a b b λ 1 c x 1 y 1 = 0 λ 1 ax 1 by 1 = 0 bx 1 + λ 1 cy 1 = b 0 y 1 = λ 1 ax 1 b 2.2 5
9 2.2 A 1 = 1 λ 1 a b x 1 b = 0 A = a 0 0 c λ 2 λ 1 λ 2 A 2 = x 2 y 2 λ 2 a b b λ 2 c x 2 y 2 = 0 6
10 2.3 b 0 λ 2 ax 2 by 2 = 0 bx 2 + λ 2 cy 2 = 0 x 2 = λ 2 cy 2 b 2.4 A 2 = λ2 c b 1 y b = 0 a 0 A = 0 c 2.4 A A λ 1, λ 2 A A 1 = x 1 y 1, A 2 = x 2 y 2 P = j k k j 7
11 2.1 x y A x y + BP x y + f = 0 x y = P u v t P AP = u v u v + BP u v + f = 0 [3] λ 1 u 2 + λ 2 v 2 + dj + eku + dk + ejv + f = 0 uv 8
12 2.5 2 A = a b b c a + b + c > 0 λ 1, λ 2 ax 2 + 2bxy + cy 2 + dx + ey + f = 0 λ 1 λ 2 > 0 λ 1 λ 2 < 0 λ 1 λ 2 = 0 [2] A 2 a + c + ac b 2 = 0 A a, b, c 2 a, b, c ac b 2 > 0 λ 1 λ 2 > 0 ac b 2 < 0 λ 1 λ 2 < 0 ac b 2 = 0 λ 1 λ 2 = 0 9
13 ac b 2 > 0 λ 1 λ 2 > 0 λ 1 u 2 + λ 2 v 2 + dj + eku + dk + ejv + f = x a + y2 = 1 2 a > 0, b > b2 λ 1 u+ dj + ek 2 +λ 2 u+ dk + ej 2 +f λ 1 2λ 2 u 0 = dj+ek, v 0 = dk+ej 2λ 2 dj + ek 2 dk + ej λ 2 2 = 0 2λ λ 1 u + u λ 2 u + v f λ 1 u 0 2 λ 2 v 0 2 = f λ 1 u 0 2 λ 2 v 0 2 λ 1 u + u λ 2 u + v 0 2 λ 1 u λ 2 v 0 2 f = 1 10
14 λ 1 u + u 0 2 λ 1 u λ 2 v 0 2 f + λ 2 u + v 0 2 λ 1 u λ 2 v 0 2 f = x = a cos θ y = b sin θ 3.5 a 1 = λ 1 dj+ek 2 +λ 2 dk+ej 2λ 2 f 2 λ 1, b 1 = λ 1 dj+ek 2 +λ 2 dk+ej 2λ 2 f 2 λ 2 u = u 0 + a 1 cos θ 0 θ 2π v = v 0 + b 1 sin θ 0 θ 2π θ u 0, v
15 [5] ac b 2 < 0 λ 1 λ 2 < 0 λ 1 u 2 + λ 2 v 2 + dj + eku + dk + ejv + f = x 2 a y2 = 1 2 a > 0, b > b2 3.1 λ 1 u+ dj + ek 2 +λ 2 u+ dk + ej 2 +f λ 1 2λ 2 u 0 = dj+ek, v 0 = dk+ej 2λ 2 12 dj + ek 2 dk + ej λ 2 2 = 0 2λ
16 λ 1 u + u λ 2 v + v f λ 1 u 0 2 λ 2 v 0 2 = f λ 1 u 0 2 λ 2 v 0 2 f λ 1 u 0 2 λ 2 v 0 2 λ 1 u + u λ 2 v + v 0 2 λ 1 u λ 2 v 0 2 f = 1 λ 1 u + u 0 2 λ 1 u λ 2 v 0 2 f + λ 2 v + v 0 2 λ 1 u λ 2 v 0 2 f = λ 2 < x = a cos θ y = b tan θ 3.10 a 1 = λ 1 dj+ek 2 +λ 2 dk+ej 2λ 2 f 2 λ 1, b 1 = λ 1 dj+ek 2 +λ 2 dk+ej 2λ 2 f 2 λ 2 u = u 0 + a 1 cos θ π 2 < θ < π 2 v = v 0 + b 1 tan θ π 2 < θ < π 2 θ u 0, v
17 [5] ac b 2 = 0 λ 1 λ 2 = 0 λ 1 > λ 2 λ 1 > 0, λ 2 = 0 λ 1 u 2 + dj + eku + dk + ejv + f = y 2 = 4px
18 λ 1 u + dj + ek 2 dj + ek + dk + ejv + f λ 1 2 = u 0 = dj+ek 3.13 λ 1 u + u dk + ejv + f λ 1 u 0 2 = λ 1 u + u 0 2 = dk + ejv f + λ 1 u 0 2 λ 1 u + u 0 2 = dk + ejv f + λ 1u 0 2 λ x = pt 2 y = 2pt 3.15 p 1 = dk+ej 4λ 1, v 0 = λ 1 f λ 1 +u 0 2 dk+ej u = u 0 + 2p 1 t 0 t 100 v = v 0 + p 1 t 2 0 t 100 t u 0, v 0 λ 1 > λ 2 λ 1 = 0, λ 2 < 0 15
19 λ 2 v 2 + dj + eku + dk + ejv + f = y 2 = 4px 3.17 λ 2 v + dk + ej 2 dk + ej + dj + eku + f λ 2 2 = λ 2 2λ 2 v 0 = dk+ej 2λ λ 2 v + v dj + eku + f λ 2 v 0 2 = λ 2 v + v 0 2 = dj + eku f + λ 2 v 0 2 λ 2 v + v 0 2 = dj + eku f + λ 2v 0 2 λ x = pt 2 y = 2pt 3.20 p 1 = dj+ek 4λ 2, u 0 = λ 2 f λ 2 +v 0 2 dj+ek u = u 0 + p 1 t 2 0 t 100 v = v 0 + 2p 1 t 0 t
20 t v 0, u
21 3.3.2 λ 1 > λ 2 λ 1 > 0, λ 2 = 0 dk + ej = 0 λ 1 > λ 2 λ 1 = 0, λ 2 < 0 dj + ek = 0 λ 1 > λ 2 λ 1 > 0, λ 2 = 0 dk + ej = λ 1 u 2 + dj + eku + f = λ 1 u + dj + ek 2 dj + ek λ f = 0 λ 1 λ 1 u + dj + ek 2 dj + ek = λ 1 2 f = 0 u 0 = dj+ek u + u 0 2 = λ 1u 0 2 f u = u 0 ± λ 1 λ 1 u 0 2 f λ
22 3.22 u = u 0 + u = u 0 λ 1 u 0 2 f λ 1 1 u 0 2 f 1 λ 1 > λ 2 λ 1 = 0, λ 2 < 0 dj + ek = 0 λ 2 v 2 + dk + ejv + f = λ 2 v + dk + ej 2 dk + ej λ f = 0 2λ 2 2λ 2 λ 2 λ 2 v + dk + ej 2 dk + ej = λ 2 2 f = 0 2λ 2 2λ 2 v 0 = dk+ej 2λ 2 v + v 0 2 = λ 2v 0 2 f λ 2 λ 2 v 0 v = v 0 ± 2 f 3.23 λ 2 19
23 3.22 v = v 0 + v = v 0 λ 2 v 0 2 f λ 2 λ 2 v 0 2 f λ
24 4 2 ax 2 + 2bxy + cy 2 + dx + ey + f = 0 a + b + c > > 0 ac b 2 >
25 b ac b 2 = = 0 6 e ac b 2 = 0 dj + ek = 0 λ 1 > λ 2 λ 1 > 0, λ 2 =
26 f 1 8 f 9 23
27 b ac b 2 < < 0 10 f
28 f facilitator ax 2 + 2bxy + cy 2 + dx + ey + f = 0 display display A 25
29 idle main int main int argc, char *argv[] { } glutinit&argc, argv; glutinitdisplaymodeglut_rgba GLUT_DOUBLE GLUT_DEPTH; glutinitwindowposition0, 0; xsize = 600; ysize = 600; glutinitwindowsizexsize, ysize; glutcreatewindow"quadratic curve"; glclearcolor1.0, 1.0, 1.0, 1.0; facilitator; glutdisplayfuncdisplay; glutreshapefuncreshape; glutidlefuncidle; glutmainloop; return 0; 26
30 5 xy P 13 27
31 ax 2 + 2bxy + cy 2 + dx + ey + f = 0 a + b + c > 0 a, b, c A xy u = u 0 + a cos θ, v = v 0 + b sin θ 0 θ 2π u = u 0 ± a cos θ, v = v 0 + b tan θ π 2 < θ < π 2 28
32 u = u 0 + 2pt, v = v 0 + pt 2 < t < ac b 2 > 0 λ 1 λ 2 > 0 b ac b 2 = 0 ac b 2 = 0 λ 1 λ 2 = 0 d e f f b ac b 2 < 0 λ 1 λ 2 < 0 f 2 f [1] 2003 [2] tsutomu/la1/16/lecture16 6.pdf [3] tsutomu/la1/16/lecture16 5.pdf [4] START DASH!! 2015 [5]
6 2 2 x y x y t P P = P t P = I P P P ( ) ( ) ,, ( ) ( ) cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ y x θ x θ P
6 x x 6.1 t P P = P t P = I P P P 1 0 1 0,, 0 1 0 1 cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ x θ x θ P x P x, P ) = t P x)p ) = t x t P P ) = t x = x, ) 6.1) x = Figure 6.1 Px = x, P=, θ = θ P
() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
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yoshi@image.med.osaka-u.ac.jp http://www.image.med.osaka-u.ac.jp/member/yoshi/ II Excel, Mathematica Mathematica Osaka Electro-Communication University (2007 Apr) 09849-31503-64015-30704-18799-390 http://www.image.med.osaka-u.ac.jp/member/yoshi/
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高等学校学習指導要領解説 数学編
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[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s
![[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s](/thumbs/94/118579915.jpg "[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s")
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ac b 0 r = r a 0 b 0 y 0 cy 0 ac b 0 f(, y) = a + by + cy ac b = 0 1 ac b = 0 z = f(, y) f(, y) 1 a, b, c 0 a 0 f(, y) = a ( ( + b ) ) a y ac b + a y
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0.6 A = ( 0 ),. () A. () x n+ = x n+ + x n (n ) {x n }, x, x., (x, x ) = (0, ) e, (x, x ) = (, 0) e, {x n }, T, e, e T A. (3) A n {x n }, (x, x ) = (,
[ ], IC 0. A, B, C (, 0, 0), (0,, 0), (,, ) () CA CB ACBD D () ACB θ cos θ (3) ABC (4) ABC ( 9) ( s090304) 0. 3, O(0, 0, 0), A(,, 3), B( 3,, ),. () AOB () AOB ( 8) ( s8066) 0.3 O xyz, P x Q, OP = P Q =
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Chap10.dvi
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