L-systemを用いた幾何学模様と木の生成.pptx

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学校の実験で作成した成果物に関する説明資料です。 L-system と呼ばれる再起的アルゴリズムを用いて、幾何学模様や木を自動生成するアルゴリズムを開発しました。元となったアルゴリズムやどう実装に落とし込んだか、また自分で追加したギミックなどについて説明をしています。

Engineering

やったこと
openglを使い、「L-system」を用いることで、
幾何学模様と木を生成！！

やったこと
openglを使い、「L-system」を用いることで、
幾何学模様と木を生成！！
こんな感じ→

L-sysyemとは
・再帰的な構造を文字列で表現
・文字列をあるルールに従って再帰的に進化

L-sysyemの例
例1: 木の構造
= 枝が3本に分かれる
=枝が3本に分かれる
→ 再帰的！

L-sysyemの例
これを例えば、
初期文字列 = “FA”
ルール : ’A’ = “[&FA]/[&FA]/[&FA]”
F: 進む
[]: 一つのかたまり
/ : 回転 &: 倒す

L-sysyemの例
これを例えば、
初期文字列 = “FA”
ルール : ’A’ = “[&FA]/[&FA]/[&FA]”
F: 進む
[]: 一つのかたまり
/ : 回転 &: 倒す
2世代目の文字列: F [&FA]/[&FA]/[&FA] ←Aから進化

やったこと
この文字列を前から読んでいく
→ 線を一本ずつ描画
F [&FA]/[&FA]/[&FA]

やったこと
この文字列を前から読んでいく
→ 線を一本ずつ描画
・始点, 長さ, 角度がわかれば線が描ける！
→これらの情報を保持、変更していく
F [&FA]/[&FA]/[&FA]

やったこと
1世代目 2世代目 3世代目
描画するとこんな感じ
枝を3本生やして、その先でそれぞれ3本生やして…

L-sysyemの例
例2:ペンローズタイル
右図… ペンローズタイル
非周期的な平面充填構造
※ ロジャー・ペンローズ: 2020にノーベル物理学賞受賞

L-sysyemの例
例2: ペンローズタイル
二つのタイルのみを使って、
再帰的に大きくしていく
…

L-sysyemの例
例2: ペンローズタイル
二つのタイルのみを使って、
再帰的に大きくしていく
→ L-system で実装
…

苦労した点①
・3次元の枝の回転の処理
※glRotatef()では、回転した先の座標が
分からない → 次の世代に繋げられない
𝜃
←座標は？

工夫した点①
3次元のベクトル回転
…3種類存在（右図）
＋任意軸回転
ロール
ピッチ

工夫した点①
・任意軸回転
→「ロドリゲスの回転行列」
を右のベクトル 𝑟 に掛けて、回転
解決策

工夫した点②
葉っぱのモデリング
右図: 四角の平面を葉っぱとして描画
→光の当たり方、形が不自然

工夫した点②
・葉っぱを4つの平面に分割
・y軸方向の膨らみをつける
→本物の葉っぱに近づけた！
解決策

推しポイント①
「再帰的」という性質に着目
「L-system」という一貫したアルゴリズムで描写！

推しポイント②
ギミックを多数用意
・幾何学模様が回転 → 木が生成
・幾何学模様、木ともに複数種類切り替え可能
・幾何学模様の回転のスピードアップ/ダウン
・光源の位置変更
etc…

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