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

2 T140073 1 2 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

2018 T140073

1 1 1.1.................................................. 2 2 2 2.1....................................... 2 2.2................................................ 4 2.3............................................ 5 2.4........................................ 7 2.5 2............................................... 9 3 2 10 3.1................................................... 10 3.2.................................................. 12 3.3.................................................. 14 4 2 21 4.1........................................ 25 5 27 6 28 6.1................................................... 28 6.2..................................................... 29

1 1 2 CG 1 2 OpenGL 1

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] 2 2.1 ax 2 + 2bxy + cy 2 + dx + ey + f = 0 a + b + c > 0 2

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

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

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 = 0 2.1 2.1 b 0 y 1 = λ 1 ax 1 b 2.2 5

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

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 2 2.3 2.4 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

2.1 x y A x y + BP x y + f = 0 x y = P u v t P AP = 1 0 0 2 u v 1 0 0 2 u v + BP u v + f = 0 [3] λ 1 u 2 + λ 2 v 2 + dj + eku + dk + ejv + f = 0 uv 8

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

3 2 3.1 3.1.1 1 ac b 2 > 0 λ 1 λ 2 > 0 λ 1 u 2 + λ 2 v 2 + dj + eku + dk + ejv + f = 0 3.1 x 2 3.1 a + y2 = 1 2 a > 0, b > 0 3.2 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 2 3.3 3.3 λ 1 u + u 0 2 + λ 2 u + v 0 2 + f λ 1 u 0 2 λ 2 v 0 2 = 0 3.4 f λ 1 u 0 2 λ 2 v 0 2 λ 1 u + u 0 2 + λ 2 u + v 0 2 λ 1 u 0 2 + λ 2 v 0 2 f = 1 10

λ 1 u + u 0 2 λ 1 u 0 2 + λ 2 v 0 2 f + λ 2 u + v 0 2 λ 1 u 0 2 + λ 2 v 0 2 f = 1 3.5 3.5 3.2 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 0 1 11

3.1.2 3.2 3.2.1 [5] ac b 2 < 0 λ 1 λ 2 < 0 λ 1 u 2 + λ 2 v 2 + dj + eku + dk + ejv + f = 0 3.6 x 2 a y2 = 1 2 a > 0, b > 0 3.7 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 2 3.8 3.3

λ 1 u + u 0 2 + λ 2 v + v 0 2 + f λ 1 u 0 2 λ 2 v 0 2 = 0 3.9 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 0 2 + λ 2 v + v 0 2 λ 1 u 0 2 + λ 2 v 0 2 f = 1 λ 1 u + u 0 2 λ 1 u 0 2 + λ 2 v 0 2 f + λ 2 v + v 0 2 λ 1 u 0 2 + λ 2 v 0 2 f = 1 3.10 λ 2 < 0 3.10 3.7 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 0 3.2.2 3.2.1 13

2 3.3 3.3.1 1 [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 = 0 3.11 3.11 y 2 = 4px 3.12 14

λ 1 u + dj + ek 2 dj + ek + dk + ejv + f λ 1 2 = 0 3.13 u 0 = dj+ek 3.13 λ 1 u + u 0 2 + dk + ejv + f λ 1 u 0 2 = 0 3.14 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 1 3.15 3.15 3.12 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

λ 2 v 2 + dj + eku + dk + ejv + f = 0 3.16 3.16 y 2 = 4px 3.17 λ 2 v + dk + ej 2 dk + ej + dj + eku + f λ 2 2 = 0 3.18 2λ 2 2λ 2 v 0 = dk+ej 2λ 2 3.18 λ 2 v + v 0 2 + dj + eku + f λ 2 v 0 2 = 0 3.19 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 2 3.20 3.20 3.17 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 100 16

t v 0, u 0 3 17

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 = 0 3.21 λ 1 u 2 + dj + eku + f = 0 3.21 λ 1 u + dj + ek 2 dj + ek λ 1 2 + 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 1 3.22 18

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 = 0 3.23 λ 2 v + dk + ej 2 dk + ej λ 2 2 + 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

3.22 v = v 0 + v = v 0 λ 2 v 0 2 f λ 2 λ 2 v 0 2 f λ 2 4 3.3.3 20

4 2 ax 2 + 2bxy + cy 2 + dx + ey + f = 0 a + b + c > 0 1 2 > 0 ac b 2 > 0 5 21

b ac b 2 = 0 1 2 = 0 6 e ac b 2 = 0 dj + ek = 0 λ 1 > λ 2 λ 1 > 0, λ 2 = 0 7 22

f 1 8 f 9 23

b ac b 2 < 0 1 2 < 0 10 f 2 11 24

f 12 4.1 facilitator ax 2 + 2bxy + cy 2 + dx + ey + f = 0 display display A 25

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

5 xy P 13 27

14 15 6 6.1 1 2 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

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 2 6.2 [1] 2003 [2] http://www.math.ryukoku.ac.jp/ tsutomu/la1/16/lecture16 6.pdf [3] http://www.math.ryukoku.ac.jp/ tsutomu/la1/16/lecture16 5.pdf [4] START DASH!! 2015 [5] 1998 29