2008-06-10

等分する(2)

幾何立体角球

この記事の続き。式を整理する。

- $\bf{EqDiv(n)}=\{eqdiv(n)_1,...,eqdiv(n)_{n+1}\}$
- $eqdiv(n)_{k}=\sqrt{\frac{n+1}{n}}(\sum_{i=k}^{n}(\frac{1}{i}\sqrt{\frac{i}{i+1}}e_i)-\sqrt{\frac{k-1}{k}}e_{k-1})$ , $k=1,2,...,n+1$
- ,
  - ただし、 $e_0=0$ とする
たしかめエクセル

ちょっと書き換えて

- $Eqdiv(n)_{k}=\sqrt{\frac{n+1}{n}}(\sqrt{\frac{n-k}{n-k+1}}e_{k}-\sum_{i=0}^{k-1}(\sqrt{\frac{1}{(n-i)(n+1-i)}}e_i))$ , $k=0,1,2,...,n$
- さらにjavaで確かめてみよう

public static double[][] eqdiv(int n){
		double[][] ret = new double[n][n-1];
		for(int i=0;i<ret.length;i++){
			if(i<n-1){
				ret[i][i]=Math.sqrt((double)(n)/(double)(n-1))*
				Math.sqrt((double)(n-i-1)/(double)(n-i));
			}
			
			for(int j=0;j<i;j++){
				ret[i][j]=-Math.sqrt((double)(n)/(double)(n-1))*
				Math.sqrt(1/(double)((n-1-j)*(n-j)));
			}
		}
		return ret;
	}

- 19次元、20分割の場合。行が、20分割のベクトル、列は、19次元を張る格子単位ベクトルの成分。各行について、格子単位ベクトル成分の自乗を足し合わせると１になる(分割ベクトルが単位ベクトルであることを示している)。また、各列を足し合わせると０になる(全分割ベクトルは「放射状」に「均等」に広がっていて、全部足すと、ゼロベクトルとなることを示している)。

0.9999999999999998	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	
-0.05263157894736842	0.9986139979479091	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	
-0.05263157894736842	-0.0554785554415505	0.9970717360268645	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	
-0.05263157894736842	-0.0554785554415505	-0.058651278589815566	0.9953452035868788	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	
-0.05263157894736842	-0.0554785554415505	-0.058651278589815566	-0.06220907522417993	0.9933992677987827	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	
-0.05263157894736842	-0.0554785554415505	-0.058651278589815566	-0.06220907522417993	-0.06622661785325218	0.991189255566704	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	
-0.05263157894736842	-0.0554785554415505	-0.058651278589815566	-0.06220907522417993	-0.06622661785325218	-0.07079923254047886	0.9886574781098639	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	
-0.05263157894736842	-0.0554785554415505	-0.058651278589815566	-0.06220907522417993	-0.06622661785325218	-0.07079923254047886	-0.0760505752392203	0.9857281161802883	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	
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-0.05263157894736842	-0.0554785554415505	-0.058651278589815566	-0.06220907522417993	-0.06622661785325218	-0.07079923254047886	-0.0760505752392203	-0.08214400968169068	-0.08929995329613663	-0.09782319760890368	-0.108147614087175	-0.12091270835166859	-0.13710212427677043	-0.15831189671532586	-0.18731716231633877	0.917662935482247	0.0	0.0	0.0	
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