{"id":7021,"date":"2023-11-17T15:41:18","date_gmt":"2023-11-17T06:41:18","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?p=7021"},"modified":"2023-11-17T15:47:26","modified_gmt":"2023-11-17T06:47:26","slug":"%e6%a5%95%e5%86%86%e8%bb%8c%e9%81%93%e4%b8%8a%e3%81%ae%e6%99%82%e5%88%bb%e3%81%94%e3%81%a8%e3%81%ae%e4%bd%8d%e7%bd%ae%e3%82%92-python-%e3%81%a7%e6%95%b0%e5%80%a4%e7%9a%84%e3%81%ab%e6%b1%82%e3%82%81","status":"publish","type":"post","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/7021\/","title":{"rendered":"\u6955\u5186\u8ecc\u9053\u4e0a\u306e\u6642\u523b\u3054\u3068\u306e\u4f4d\u7f6e\u3092 Python \u3067\u6570\u5024\u7684\u306b\u6c42\u3081\u308b"},"content":{"rendered":"

\u300c\u6955\u5186\u8ecc\u9053\u4e0a\u306e\u6642\u523b\u3054\u3068\u306e\u4f4d\u7f6e\u3092\u6c42\u3081\u308b\u305f\u3081\u306e\u4e0b\u3054\u3057\u3089\u3048<\/a>\u300d\u3067\u4e0b\u3054\u3057\u3089\u3048\u3057\u305f\u5f0f\u3092 Python \u3067\u6570\u5024\u7684\u306b\u89e3\u304f\u3002
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\u30e9\u30a4\u30d6\u30e9\u30ea\u306e import<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[1]:<\/div>\n
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import<\/span> numpy<\/span> as<\/span> np<\/span>\r\nimport<\/span> matplotlib.pyplot<\/span> as<\/span> plt<\/span>\r\n\r\n# \u4ee5\u4e0b\u306f\u30b0\u30e9\u30d5\u3092 SVG \u3067 Notebook \u306b\u30a4\u30f3\u30e9\u30a4\u30f3\u8868\u793a\u3055\u305b\u308b\u8a2d\u5b9a<\/span>\r\n%<\/span>config<\/span> InlineBackend.figure_formats = ['svg']\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u7121\u6b21\u5143\u5316\u3055\u308c\u305f\u5f0f<\/h3>\n
\u300c\u6955\u5186\u8ecc\u9053\u4e0a\u306e\u6642\u523b\u3054\u3068\u306e\u4f4d\u7f6e\u3092\u6c42\u3081\u308b\u305f\u3081\u306e\u4e0b\u3054\u3057\u3089\u3048<\/a>\u300d\u3067\u4e0b\u3054\u3057\u3089\u3048\u3057\u305f\u5f0f\uff08\u7121\u6b21\u5143\u5316\u6e08\u307f\uff09\u3002<\/p>\n
scipy.integrate.solve_ivp()<\/code> \u306f $\\displaystyle \\frac{dy}{dt} = f(t, y)$ \u306e\u5f62\u306e\u5f0f\u3092\u89e3\u304f\u306e\u3067\uff0c\u5909\u6570\u3092\u63c3\u3048\u308b\u3002<\/p>\n
$ \\phi \\rightarrow y, \\ T \\rightarrow t$<\/p>\n
\\begin{eqnarray}
\n\\frac{dy}{dt} &=& f(t, y) = 2 \\pi \\frac{(1+e \\cos y)^2}{(1-e^2)^{3\/2}} \\tag{1}\\\\
\nR(y) &\\equiv& \\frac{r}{a} = \\frac{1-e^2}{1+e\\cos y} \\tag{2}\\\\
\nX(y) &\\equiv& \\frac{x}{a} = R(y) \\cos y \\tag{3}\\\\
\nY(y) &\\equiv& \\frac{y}{a} = R(y) \\sin y \\tag{4}
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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# \u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u53f3\u8fba \u72ec\u7acb\u5909\u6570\u306f t, y \u6c7a\u3081\u6253\u3061<\/span>\r\ndef<\/span> f<\/span>(<\/span>t<\/span>,<\/span> y<\/span>):<\/span>\r\n    global<\/span> e<\/span>\r\n    return<\/span> 2<\/span>*<\/span>np<\/span>.<\/span>pi<\/span>*<\/span>(<\/span>1<\/span>+<\/span>e<\/span>*<\/span>np<\/span>.<\/span>cos<\/span>(<\/span>y<\/span>))<\/span>**<\/span>2<\/span>\/<\/span>((<\/span>1<\/span>-<\/span>e<\/span>**<\/span>2<\/span>)<\/span>**<\/span>(<\/span>3<\/span>\/<\/span>2<\/span>))<\/span>\r\n\r\ndef<\/span> R<\/span>(<\/span>y<\/span>):<\/span>\r\n    global<\/span> e<\/span>\r\n    return<\/span> (<\/span>1<\/span>-<\/span>e<\/span>**<\/span>2<\/span>)<\/span>\/<\/span>(<\/span>1<\/span>+<\/span>e<\/span>*<\/span>np<\/span>.<\/span>cos<\/span>(<\/span>y<\/span>))<\/span>\r\n\r\ndef<\/span> X<\/span>(<\/span>y<\/span>):<\/span>\r\n    global<\/span> e<\/span>\r\n    return<\/span> R<\/span>(<\/span>y<\/span>)<\/span>*<\/span>np<\/span>.<\/span>cos<\/span>(<\/span>y<\/span>)<\/span>\r\n\r\ndef<\/span> Y<\/span>(<\/span>y<\/span>):<\/span>\r\n    global<\/span> e<\/span>\r\n    return<\/span> R<\/span>(<\/span>y<\/span>)<\/span>*<\/span>np<\/span>.<\/span>sin<\/span>(<\/span>y<\/span>)<\/span>\r\n\r\n# \u96e2\u5fc3\u7387<\/span>\r\ne<\/span> =<\/span> 0.6<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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solve_ivp()<\/code> \u3067\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u304f<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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from<\/span> scipy.integrate<\/span> import<\/span> solve_ivp<\/span>\r\n\r\n# T \u306e\u521d\u671f\u5024 <\/span>\r\nt0<\/span> =<\/span> 0<\/span>\r\n# T \u306e\u7d42\u4e86\u5024<\/span>\r\nt1<\/span> =<\/span> 1<\/span>\r\nt_span<\/span> =<\/span> [<\/span>t0<\/span>,<\/span> t1<\/span>]<\/span>\r\n# phi \u306e\u521d\u671f\u5024<\/span>\r\ny_ini<\/span> =<\/span> [<\/span>0<\/span>]<\/span>\r\n\r\n# \u5206\u5272\u6570<\/span>\r\nNdiv<\/span> =<\/span> 36<\/span>\r\n\r\nt_list<\/span> =<\/span> np<\/span>.<\/span>linspace<\/span>(<\/span>t0<\/span>,<\/span> t1<\/span>,<\/span> Ndiv<\/span>+<\/span>1<\/span>)<\/span>\r\nanswer<\/span> =<\/span> solve_ivp<\/span>(<\/span>f<\/span>,<\/span> t_span<\/span>,<\/span> y_ini<\/span>,<\/span> \r\n                   t_eval<\/span> =<\/span> t_list<\/span>,<\/span> \r\n                   # \u8a08\u7b97\u7cbe\u5ea6\u306e\u8a2d\u5b9a<\/span>\r\n                   rtol<\/span> =<\/span> 1.e-13<\/span>,<\/span> atol<\/span> =<\/span> 1.e-15<\/span>)<\/span>\r\n\r\n# \u8a08\u7b97\u7d50\u679c\u306f .y \u3068\u3057\u3066\u53d6\u308a\u51fa\u3057\u307e\u3059\u3002<\/span>\r\nphii<\/span> =<\/span> answer<\/span>.<\/span>y<\/span>[<\/span>0<\/span>]<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u3061\u3087\u3046\u30691\u5468\u671f $T = T_{\\rm end}$ \u305f\u3066\u3070 $\\phi = 2 \\pi$ \u306b\u306a\u3063\u3066\u3044\u308b\u306f\u305a\u3060\u304b\u3089\uff0c\u6570\u5024\u8aa4\u5dee\u3092\u78ba\u8a8d\u3059\u308b\u3002<\/p>\n
$$\\phi(T_{\\rm end}) – 2 \\pi$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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# [-1] \u306f\u6700\u5f8c\u306e\u9805<\/span>\r\n\r\nphii<\/span>[<\/span>-<\/span>1<\/span>]<\/span> -<\/span> 2<\/span>*<\/span>np<\/span>.<\/span>pi<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[4]:<\/div>\n
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-2.5348612098241574e-12<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4e00\u5b9a\u6642\u9593\u9593\u9694\u3054\u3068\u306e\u4f4d\u7f6e\u3092\u30b0\u30e9\u30d5\u306b<\/h3>\n
\u4e00\u5b9a\u6642\u9593\u9593\u9694 $\\displaystyle \\Delta T = \\frac{1}{N_{\\rm div}}$ \u3054\u3068\u306e\u4f4d\u7f6e\uff1a<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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fig<\/span> =<\/span> plt<\/span>.<\/span>figure<\/span>(<\/span>figsize<\/span>=<\/span>[<\/span>6<\/span>,<\/span>6<\/span>])<\/span>\r\n# \u7e26\u6a2a\u306e\u30a2\u30b9\u30da\u30af\u30c8\u6bd4\u3092\u7b49\u3057\u304f<\/span>\r\nplt<\/span>.<\/span>axes<\/span>()<\/span>.<\/span>set_aspect<\/span>(<\/span>'equal'<\/span>)<\/span>\r\n# \u8ef8\u306e\u8a2d\u5b9a<\/span>\r\nplt<\/span>.<\/span>axis<\/span>([<\/span>-<\/span>2<\/span>,<\/span> 1<\/span>,<\/span> -<\/span>1.5<\/span>,<\/span> 1.5<\/span>])<\/span>\r\n\r\nplt<\/span>.<\/span>scatter<\/span>(<\/span>X<\/span>(<\/span>phii<\/span>),<\/span> Y<\/span>(<\/span>phii<\/span>),<\/span> c<\/span>=<\/span>\"blue\"<\/span>,<\/span> s<\/span>=<\/span>20<\/span>);<\/span>\r\n\r\n# x\u8ef8 y\u8ef8<\/span>\r\nplt<\/span>.<\/span>axhline<\/span>(<\/span>0<\/span>,<\/span> color<\/span>=<\/span>'black'<\/span>,<\/span> dashes<\/span>=<\/span>(<\/span>3<\/span>,<\/span> 3<\/span>),<\/span> linewidth<\/span>=<\/span>0.6<\/span>)<\/span>\r\nplt<\/span>.<\/span>axvline<\/span>(<\/span>0<\/span>,<\/span> color<\/span>=<\/span>'black'<\/span>,<\/span> dashes<\/span>=<\/span>(<\/span>3<\/span>,<\/span> 3<\/span>),<\/span> linewidth<\/span>=<\/span>0.6<\/span>);<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u53c2\u8003\uff1a\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\u3068\u306e\u6bd4\u8f03<\/h3>\n
\u300c\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u3044\u3066\u30b1\u30d7\u30e9\u30fc\u306e\u7b2c2\u6cd5\u5247\u3092\u8996\u899a\u7684\u306b\u78ba\u8a8d\u3059\u308b<\/a>\u300d\u306b\u307e\u3068\u3081\u3066\u3044\u308b\u3088\u3046\u306b\uff0c\u6955\u5186\u8ecc\u9053\u306e $x, y$ \u3092\u8ecc\u9053\u9577\u534a\u5f84 $a$ \u3067\u898f\u683c\u5316\u3057\u305f $X_u, Y_u$ \u3092\uff0c\u96e2\u5fc3\u8fd1\u70b9\u96e2\u89d2 $u$ \u3092\u4f7f\u3063\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3042\u3089\u308f\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n
\\begin{eqnarray}
\nX_u &\\equiv& \\frac{x}{a} = \\cos u \u2013 e\\\\
\nY_u &\\equiv& \\frac{y}{a} = \\sqrt{1-e^2} \\sin u
\n\\end{eqnarray}<\/p>\n
\u96e2\u5fc3\u8fd1\u70b9\u96e2\u89d2 $u$ \u3068\u5468\u671f $P$ \u3067\u898f\u683c\u5316\u3055\u308c\u305f\u6642\u9593 $T$ \u3068\u306e\u95a2\u4fc2\u306f\uff0c\u4ee5\u4e0b\u306e\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u3044\u3066\u5f97\u3089\u308c\u308b\u3002<\/p>\n
\\begin{eqnarray}
\n2\\pi T &=& u \u2013 e \\sin u, \\quad T \\equiv \\frac{t}{P}
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3<\/h4>\n
$$g(u_i) \\equiv u_i \u2013 e \\sin u_i – 2\\pi T_i =0, \\quad T_i = \\frac{i}{N_{\\rm div}}$$<\/p>\n
\u3092 $u_i$ \u306b\u3064\u3044\u3066 root_scalar()<\/code> \u95a2\u6570\u3067\u6570\u5024\u7684\u306b\u89e3\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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from<\/span> scipy.optimize<\/span> import<\/span> root_scalar<\/span>\r\n\r\n# root_scalar \u3092\u4f7f\u3046\u3068\u304d\u306f\uff0c\u5909\u6570\u306f x \u6c7a\u3081\u6253\u3061<\/span>\r\ndef<\/span> g<\/span>(<\/span>x<\/span>):<\/span>\r\n    global<\/span> e<\/span>,<\/span> T<\/span>\r\n    return<\/span> x<\/span> -<\/span> e<\/span>*<\/span>np<\/span>.<\/span>sin<\/span>(<\/span>x<\/span>)<\/span> -<\/span> 2<\/span>*<\/span>np<\/span>.<\/span>pi<\/span>*<\/span>T<\/span>\r\n\r\nTlist<\/span> =<\/span> np<\/span>.<\/span>linspace<\/span>(<\/span>0<\/span>,<\/span> 1<\/span>,<\/span> Ndiv<\/span>+<\/span>1<\/span>)<\/span>\r\n\r\nui<\/span> =<\/span> []<\/span>\r\n\r\nfor<\/span> T<\/span> in<\/span> Tlist<\/span>:<\/span>\r\n    ans<\/span> =<\/span> root_scalar<\/span>(<\/span>g<\/span>,<\/span> bracket<\/span> =<\/span> [<\/span>0<\/span>,<\/span> 2<\/span>*<\/span>np<\/span>.<\/span>pi<\/span>],<\/span> xtol<\/span> =<\/span> 1.e-14<\/span>)<\/span>\r\n    ui<\/span>.<\/span>append<\/span>(<\/span>ans<\/span>.<\/span>root<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4e00\u5b9a\u6642\u9593\u9593\u9694\u3054\u3068\u306e\u4f4d\u7f6e<\/h4>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[7]:<\/div>\n
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def<\/span> Xu<\/span>(<\/span>u<\/span>):<\/span>\r\n    global<\/span> e<\/span>\r\n    return<\/span> np<\/span>.<\/span>cos<\/span>(<\/span>u<\/span>)<\/span> -<\/span> e<\/span>\r\n\r\ndef<\/span> Yu<\/span>(<\/span>u<\/span>):<\/span>\r\n    global<\/span> e<\/span>\r\n    return<\/span> np<\/span>.<\/span>sqrt<\/span>(<\/span>1<\/span>-<\/span>e<\/span>**<\/span>2<\/span>)<\/span> *<\/span> np<\/span>.<\/span>sin<\/span>(<\/span>u<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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2\u3064\u306e\u89e3\u306e\u6bd4\u8f03<\/h4>\n
$\\phi$ \u306b\u95a2\u3059\u308b1\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u3044\u305f\u89e3\u3067\u3042\u308b X(phii)<\/code>, Y(phii)<\/code> \u3068\uff0c\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u3044\u305f\u89e3\u3067\u3042\u308b Xu(ui)<\/code>, Yu(ui)<\/code> \u306f\uff0c\u5f53\u7136\u306a\u304c\u3089\u6570\u5024\u8aa4\u5dee\u306e\u7bc4\u56f2\u3067\u4e00\u81f4\u3057\u3066\u308b\u306f\u305a\u3067\u3042\u308b\u3002<\/p>\n
2\u3064\u306e\u914d\u5217\u306e\u5f15\u304d\u7b97\u3092\u3057\u3066\u5dee\u3092\u8abf\u3079\u308b\u300237\u7d44\u5168\u3066\u8868\u793a\u3059\u308b\u3068\u5197\u9577\u306a\u306e\u3067\uff0c\u4ee3\u8868\u3057\u30663\u7d44\u307b\u3069\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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# 2\u3064\u306e\u6570\u5024\u89e3 X(phii), Xu(ui) \u306e\u5dee<\/span>\r\nprint<\/span>((<\/span>X<\/span>(<\/span>phii<\/span>)<\/span> -<\/span> Xu<\/span>(<\/span>ui<\/span>))[<\/span>10<\/span>])<\/span>\r\nprint<\/span>((<\/span>X<\/span>(<\/span>phii<\/span>)<\/span> -<\/span> Xu<\/span>(<\/span>ui<\/span>))[<\/span>20<\/span>])<\/span>\r\nprint<\/span>((<\/span>X<\/span>(<\/span>phii<\/span>)<\/span> -<\/span> Xu<\/span>(<\/span>ui<\/span>))[<\/span>30<\/span>])<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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3.0744295997919835e-12\r\n-4.904965322793942e-13\r\n-1.5972778655282127e-12\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u5ff5\u306e\u305f\u3081\u306b2\u3064\u306e\u6570\u5024\u89e3\u3092\u30b0\u30e9\u30d5\u306b\u3059\u308b\u3068\uff0c\u5f53\u7136\u306a\u304c\u3089\u91cd\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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fig<\/span> =<\/span> plt<\/span>.<\/span>figure<\/span>(<\/span>figsize<\/span>=<\/span>[<\/span>6<\/span>,<\/span>6<\/span>])<\/span>\r\n# \u7e26\u6a2a\u306e\u30a2\u30b9\u30da\u30af\u30c8\u6bd4\u3092\u7b49\u3057\u304f<\/span>\r\nplt<\/span>.<\/span>axes<\/span>()<\/span>.<\/span>set_aspect<\/span>(<\/span>'equal'<\/span>)<\/span>\r\n# \u8ef8\u306e\u8a2d\u5b9a<\/span>\r\nplt<\/span>.<\/span>axis<\/span>([<\/span>-<\/span>2<\/span>,<\/span> 1<\/span>,<\/span> -<\/span>1.5<\/span>,<\/span> 1.5<\/span>])<\/span>\r\n\r\nplt<\/span>.<\/span>scatter<\/span>(<\/span>X<\/span>(<\/span>phii<\/span>),<\/span> Y<\/span>(<\/span>phii<\/span>),<\/span> c<\/span>=<\/span>\"blue\"<\/span>,<\/span> s<\/span>=<\/span>20<\/span>,<\/span> label<\/span> =<\/span> \"\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\"<\/span>);<\/span>\r\nplt<\/span>.<\/span>scatter<\/span>(<\/span>Xu<\/span>(<\/span>ui<\/span>),<\/span> Yu<\/span>(<\/span>ui<\/span>),<\/span> c<\/span>=<\/span>\"red\"<\/span>,<\/span> s<\/span>=<\/span>2<\/span>,<\/span> label<\/span> =<\/span> \"\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\"<\/span>);<\/span>\r\n\r\n# x\u8ef8 y\u8ef8<\/span>\r\nplt<\/span>.<\/span>axhline<\/span>(<\/span>0<\/span>,<\/span> color<\/span>=<\/span>'black'<\/span>,<\/span> dashes<\/span>=<\/span>(<\/span>3<\/span>,<\/span> 3<\/span>),<\/span> linewidth<\/span>=<\/span>0.6<\/span>)<\/span>\r\nplt<\/span>.<\/span>axvline<\/span>(<\/span>0<\/span>,<\/span> color<\/span>=<\/span>'black'<\/span>,<\/span> dashes<\/span>=<\/span>(<\/span>3<\/span>,<\/span> 3<\/span>),<\/span> linewidth<\/span>=<\/span>0.6<\/span>)<\/span>\r\nplt<\/span>.<\/span>legend<\/span>();<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"
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