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このブログは、数学とアニメとlinuxに傾いています。

2013年11月

鹿目まどかのグラフを作ってみた

madoka_graph

グラフの式

先日、鹿目まどかのグラフを見ました。探してみたら、他のキャラクターのグラフもあるようです。初音ミクのグラフ

これらのグラフに共通して言えるのは、グラフが一筆書きということです。リサージュ図形でフーリエ変換??残念ながら私は理解することが出来ませんでした。

そこで、私はこれとは別のアプローチで鹿目まどかのグラフを表そうと考えました。以前から構想はあったので、どの姿をグラフにしようかは考えていました。こちらのページの右上の鹿目まどかです。グラフをつくるにあたって以下の3つの方法が思い浮かんだので紹介します。


①ブロックごとで考えてグラフにする

$a≦x≦b,c≦y≦d$を塗りつぶすようなグラフは、
\[(((x-a)(x-b)+|(x-a)(x-b)|)^2+((y-c)(y-d)+|(y-c)(y-d)|)^2)=0\]で表すことが出来ます。
この式を利用して最初は鹿目まどかのグラフを描こうとしました。しかしながら、必要な式の数が圧倒的に大きく、また心が折れてしまったので断念しました。以下にあるのはこの方式で作った参考画像です。

\ナッモリーン/まどか途中まで



左の画像の詳細についてはなもり先生のグラフを書いてみたに載っています。

②3次関数を組み合わせることでグラフにする

これは、座標とその点においてのベクトルを決めるだけで決まります。しかしながら、点Aと点Bの間でグラフの傾きが0になることがあるので意図していた形と違うものが出力されるのでこの案は却下しました。

具体的な方法:点A$(x_{1},y_{1})$,傾きが$z_{1}$,点B$(x_{2},y_{2})$,傾きが$z_{2}$,これを満たす3次関数$y=a_{1}x^3+a_{2}x^2+a_{3}x+a_{4}$について、$a_{1},a_{2},a_{3},a_{4}$を、$x_{1},x_{2},y_{1},y_{2},z_{1},z_{2}$を用いて表す事を考える。ただし、$x_{1}{\neq}x_{2}$一般化には成功したものの前述の理由で却下。

③2次関数にパラメーターを加えることでグラフにする

2次関数の場合、2点の傾きの合計が一定になるので、一方の点の傾きを定めれば、ある程度意図した形のグラフをつくることができます。

具体的な方法:
点A$(x_{1},y_{1})$,点B$(x_{2},y_{2})$これを満たす2次関数$y=a_{1}x^2+a_{2}x+a_{3}$について考える。この時、2点の傾きの合計$z_{1}+z_{2}$は、$2\times\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$に等しくなる。$a_{1},a_{2},a_{3}$を定めるため、点Aの傾き$z_{1}$を、$z_{1}=(1-k)\times\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$とおく。こうすると、点Bの傾き$z_{2}$は、$z_{2}=(1+k)\times\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$となり、都合が良い。ちなみに、$k=0$とおけば、直線になる。後は、$x_{1},x_{2},y_{1},y_{2},k$を用いて$a_{1},a_{2},a_{3}$を表す。ただし、$x_{1}{\neq}x_{2}$。これも一般化に成功した。

手作業でグラフの式を出すことも可能ですが、時間がかかってしまうのでプログラムをつくって、座標と$k$を入力するだけで、グラフの式を出力するようにしました。

以下が鹿目まどかの式です。コピーアンドペーストすることでグラスソフトで表示を行うことも出来ますが式の読み込みを行うと5分以上グラフソフトがフリーズすると考えてください。取り扱いには十分ご注意ください。全部で64119文字です。テキストで表示する

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劇場版 魔法少女まどかマギカ[新編]叛逆の物語と文化祭の感想

科学の甲子園のあとで見るはずだった、劇場版 魔法少女まどかマギカ[新編]叛逆の物語を見に行きました。

劇場版[前編][後編]では数学の部分が削られていましたが(確かTV版1話で登場していた問題がなかった)、今回は新しい場面が登場しました。しかも、見滝原中学校での授業風景として映ったものは数学が唯一だと思います。

描かれたものですが、今までに見られていたような問題形式なのではなく、授業をやっているようでした。確認できたところでは、分数関数 積分 三角形
madmagi
←こんな感じ

一瞬でしたので記憶は曖昧です。

話は変わりますが、映画館に行った所予想以上に女性の割合が多いことに気づきました。なんか小学生男女のカップル?も見かけた気がしましたが…。

サンプル数がかなり少ないので、本当はやってはいけないのですが、突然ですがここで比率の検定を行いたいと思います。

帰無仮説H0を、「劇場版まどか☆マギカの観客動員数において、9割が男性である。」とします。すなわち男性の比率pが0.9ということです。(p0=0.9とおきます。)
対立仮説H1を、「劇場版まどか☆マギカの観客動員数において、9割未満が男性である。」すなわち比率p<0.9とします。これは、左側検定になります。
サンプル数nは6で、男性の人数は、x=3すなわち比率p'=0.5です。

本来は、n>30,np>5,n(1-p)>5であることが条件です。(村上正康・安田正實『統計学演習』(培風館、1989年)、117ページ)

p'=x/n=3/6=0.5
z=(p'-p0)/√(p0(1-p0)/n)=(0.5-0.9)/√(0.9*0.1/6)≒-3.266
片側検定なので、-z(0.025)=-1.96。
即ち、z<-z(0.025)が成り立つので、zは棄却域に落ちるから、H0は棄却です。


ネタバレになるかもしれませんが、なんか印象に残ったワード及びつぶやきを、
見滝原循環/見滝原3丁目/見滝原2丁目/風見野/マザーグースのような歌/ラーメン/今までのまどマギの-1を掛けた感じ/キュウ/閉ざされた扉開けよう→扉は閉じられた/第三形態ほむら/マミ→杏子→さやか→ほむら→まどかの変身の順番/影絵の変身シーンは良かったと思う/あざみ野と風見野は1字違いだ/なす角が60度の2直線が動いているシーン問題に出来そう

映画を見た後、つっこまの文化祭に行ってきて、TMO2013を解きました。
2時間ほどしかいることが出来ませんでした。6問解いて計280点でした。ファイルを5枚もとることが出来ました。
62と2013がゲシュタルト崩壊しそうでした。

記事の形になっていないと思いますがこれで終わりにしたいと思います。
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