極変換

【きょくへんかん (polar transformation)】

2次曲線に関する点と直線の双対変換のこと. $d\,$次元空間では, 2次曲面が構文解析に失敗 (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (d+1)\times (d+1)\,} 対称行列$A\,$と$d\,$次元ベクトル${\boldsymbol {x}}\,$を用いて構文解析に失敗 (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle ({\boldsymbol {x}},1)^{\top }A({\boldsymbol {x}},1)=0\,} と表せ, 点構文解析に失敗 (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\boldsymbol {p}}\,} に対して, 方程式構文解析に失敗 (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle ({\boldsymbol {x}},1)^{\top }A({\boldsymbol {p}},1)=0\,} を満たす超平面構文解析に失敗 (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle D(p)\,} を対応させる. 逆に定数ベクトル構文解析に失敗 (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\boldsymbol {p}}\,} を用いて構文解析に失敗 (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle ({\boldsymbol {x}},1)A({\boldsymbol {p}},1)=0\,} と書ける超平面$h\,$に対して点 構文解析に失敗 (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle D(h)=p\,} を対応させる. 明らかに構文解析に失敗 (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle D(D(p))=p\,} , 構文解析に失敗 (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle D(D(h))=h\,} である. 極変換は, 接続関係を保存する.