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ユークリッド幾何の導入

位相幾何学

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使用する用語

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線型代数の基礎

（m次元）単位行列((m×m-)Identity Matrix)：\E: $\overset{[m\times m]}{\mathbf{E}} = [\mathbf{e}_1, \cdots, \mathbf{e}_m]$
m次元ユークリッド空間(m-Euclid Space)：U^m: $U^m \equiv U_{\mathbb{0}}^m [\mathbf{E}]$
（m次元(n+1)）位置行列((m×(n+1)-)Position Matrix)：\P: $\overset{[m\times(n+1)]}{\mathbf{P}} = [\mathbf{p}_0, \cdots, \mathbf{p}_n]$
（m次元n）方向行列((m×n-)Direction Matrix)：\L: $\overset{[m\times n]}{\mathbf{L}} = [\mathbf{l}_1, \cdots, \mathbf{l}_n] = [\mathbf{p}_1, \cdots, \mathbf{p}_n] - \mathbf{p}_0 \mathbb{1}^T$
n次元部分空間(n-Subspace)：~U^n: $U_{\mathbf{p}_0}^n[\mathbf{L}] \equiv \widetilde{U}^n[\mathbf{P}] \equiv \widetilde{U}^n$
n次元単体(n-Simplex)：~A^n: $A_{\mathbf{p}_0}^n[\mathbf{L}] \equiv \widetilde{A}^n[\mathbf{P}] \equiv \widetilde{A}^n$
辺乗行列(Half-squared Edge Matrix)：\~B: $\overset{[(n+1)\times(n+1)]}{\tilde{\mathbf{B}}} = \underset{\tiny j=0\cdots n\\ i=0\cdots n}{\mathbf{M}}\left[{\small [j,i] \:} \frac{(\mathbf{p}_j - \mathbf{p}_i)^T (\mathbf{p}_j - \mathbf{p}_i)}{2}\right]$
（(m-1)次元n元）二次超曲面((n-Radii's (m-1)-)Quadric Hypersurface)：f_Q[\p]=0: $f_Q [\mathbf{p}] = \mathbf{p}^T \mathbf{Q} \mathbf{p} + 2 \mathbf{p}^T \mathbf{q}_y + q_{yy} = 0$
（n元）(n-1)次元超球（面）・n次元超球体((n-Radii's) (n-1)-Hypersphere・n-Hyperball)：S^(n-1)・~S^n: $S^{(n-1)} \subseteq \tilde{S}^n$
（n元）(n-1)次元超楕円（面）・n次元超楕円体((n-Radii's) (n-1)-Hyperellipse・n-Hyperellipsoid)：S'^(n-1)・~S'^n: $S'^{(n-1)} \subseteq \tilde{S'}^n$

解析幾何の基礎

（k次元）部分単体・部分対面(Sub-simplex・complementary Sub-simplex)：~A^k_ψ・~A^(n-k-1)_{≠ψ}: 部分単体 $\tilde{A}^k_\psi \equiv \tilde{A}^k [\mathbf{P}_\psi]$ に対する部分対面 $\tilde{A}^{(n-k-1)}_{\neq\psi} \equiv \tilde{A}^{(n-k-1)} [\mathbf{P}_{\neq\psi}]$ 、特に、i-頂点 $\tilde{A}^0_i$ に対する部分対面（i-対面）は $\tilde{A}^{(n-1)}_{\neq i}$ 。
（k次元）部分単体頂点集合・部分単体全体集合(Vertex Subset)：ψ・{ψ}: 例えば、0頂点からk頂点までの頂点集合 $\psi=\{0,\cdots,k\}$ 、k次元部分単体全体集合 $\{\psi\}$
正規直交基底(Orthonormal Basis)：\S: $\mathbf{S}=\mathbf{S}[\mathbf{L}] = \tilde{\mathbf{S}}[\mathbf{P}]$
特異値分解(Singular Value Decomposition)：\X = \S \Σ \^A^T: $\mathbf{X}=\mathbf{S} \mathbf{\Sigma} \hat{\mathbf{A}}^T$
擬似逆行列(Pseudo Inverse Matrix)：\X^†: $\mathbf{X}^\dagger=\hat{\mathbf{A}} \mathbf{\Sigma}^{-1} \mathbf{S}^T$ , $\mathbf{X}^\ddagger=\mathbf{S} \mathbf{\Sigma}^{-1} \hat{\mathbf{A}}^T = (\mathbf{X}^\dagger)^T$
（位置内積）逆射行列(Extended Inverse Matrix)：\Φ: $|[\mathbf{X}]| \neq 0$ のとき、 $\mathbf{\Phi}[\mathbf{X}]=\frac{\mathbf{C} [\mathbf{X}]}{|[\mathbf{X}]|}=\mathbf{X}^{-1}$ 。特に、 $\mathbf{\Phi}=\mathbf{\Phi}[\mathbf{P}^T \mathbf{P}]$
（位置内積）逆影行列(Projective Inverse Matrix)：\~Φ: $\tilde{\mathbb{1}}^T \mathbf{C} [\mathbf{X}] \tilde{\mathbb{1}} \neq 0$ のとき、 $\tilde{\mathbf{\Phi}}[\mathbf{X}]=\frac{\tilde{\mathbf{C}} [\mathbf{X}]}{\tilde{\mathbb{1}}^T \mathbf{C} [\mathbf{X}] \tilde{\mathbb{1}}}$ 。特に、 $\tilde{\mathbf{\Phi}}=\tilde{\mathbf{\Phi}}[\mathbf{P}^T \mathbf{P}]$
位底垂線(Position Basis Perpendicular)：\p_y: $\tilde{\mathbf{a}}_y=\frac{\mathbf{C} [\mathbf{P}^T \mathbf{P}] \tilde{\mathbb{1}}}{\tilde{\mathbb{1}}^T \mathbf{C} [\mathbf{P}^T \mathbf{P}] \tilde{\mathbb{1}}}=\frac{\mathbf{\Phi} \tilde{\mathbb{1}}}{\tilde{\mathbb{1}}^T \mathbf{\Phi} \tilde{\mathbb{1}}}$ , $\mathbf{p}_y = \mathbf{P} \tilde{\mathbf{a}}_y$
位底擬似逆行列(Pseudo Inverse Matrix of Position Basis)：\P^†: $\mathbf{P}^\dagger = \frac{\tilde{\mathbf{a}}_y \mathbf{p}_y^T}{\mathbf{p}_y^T \mathbf{p}_y} + \tilde{\mathbf{\Phi}} \mathbf{P}^T = \left(\frac{\mathbf{\Phi} \tilde{\mathbb{1}} \tilde{\mathbb{1}}^T \mathbf{\Phi}}{\tilde{\mathbb{1}}^T \mathbf{\Phi} \tilde{\mathbb{1}}} + \tilde{\mathbf{\Phi}}\right) \mathbf{P}^T$

単体座標関係

（単体）位置座標((Areal )Simplex Position Coordinates)：\~α: $\mathbf{p}_X = \mathbf{P} \tilde{\mathbf{\alpha}}_X$ , $\tilde{\mathbf{\alpha}}_X = \tilde{\mathbf{a}}_y \left(\frac{\mathbf{p}_y^T \mathbf{p}_X}{\mathbf{p}_y^T \mathbf{p}_y}\right) + \tilde{\mathbf{\Phi}} \mathbf{P}^T \mathbf{p}_X$
単体超体積公式(Simplex Hypervolume Theorem)：v^n=…: $v^n = \frac{\sqrt{|[\mathbf{L}^T \mathbf{L}]|}}{n!} = \frac{\sqrt{\tilde{\mathbb{1}}^T \mathbf{C} [\mathbf{P}^T \mathbf{P}] \tilde{\mathbb{1}}}}{n!} = \frac{\sqrt{\tilde{\mathbb{1}}^T \mathbf{C} [-\tilde{\mathbf{B}}] \tilde{\mathbb{1}}}}{n!}$; $\det[\mathbf{L}^T \mathbf{L}] = \mathbf{1}^T \mathbf{C}[\mathbf{P}^T \mathbf{P}] \mathbf{1} = \mathbf{1}^T \mathbf{C}[- \tilde{\mathbf{B}}] \mathbf{1}$
分積座標((Homogeneous )Barycentric Coordinates)：\~a: 単体内部点 $\mathbf{p} = \frac{\mathbf{P} \tilde{\mathbf{a}}}{\tilde{\mathbb{1}}^T \tilde{\mathbf{a}}}$ とi-対面で作られるn次元単体の超体積（i-分積）は $\tilde{a}_i = \tilde{\mathbf{e}}_i^T (\tilde{\mathbf{a}}_y + \tilde{\mathbf{\Phi}} \mathbf{P}^T \mathbf{p}) v^n$ 。
i-対面超体積公式(i-Facet Hypervolume Theorem)：v_i^(n-1)=…: $v_0^{(n-1)} = \frac{\sqrt{\mathbb{1}^T \mathbf{C}[\mathbf{L}^T \mathbf{L}] \mathbb{1}}}{(n-1)!}$ 、i=1…nについて $v_i^{(n-1)} = \frac{\sqrt{\mathbf{e}_i^T \mathbf{C}[\mathbf{L}^T \mathbf{L}] \mathbf{e}_i}}{(n-1)!}$ 、i=0…nについて $v_i^{(n-1)} = \frac{\sqrt{\tilde{\mathbf{e}}_i^T \tilde{\mathbf{C}} [\mathbf{P}^T \mathbf{P}] \tilde{\mathbf{e}}_i}}{(n-1)!} = \frac{\sqrt{\tilde{\mathbf{e}}_i^T \tilde{\mathbf{C}} [-\tilde{\mathbf{B}}] \tilde{\mathbf{e}}_i}}{(n-1)!}$
分面座標((Exact-trilinear )Divide-facets Coordinates)：\~j: 単体内部点 $\mathbf{p} = \frac{\mathbf{P} \mathbf{\Sigma}^{\frac{1}{2}} [ \tilde{\mathbf{C}}[\mathbf{P}^T \mathbf{P}] ] \tilde{\mathbf{j}}}{\mathbb{1}^T \mathbf{\Sigma}^{\frac{1}{2}} [ \tilde{\mathbf{C}}[\mathbf{P}^T \mathbf{P}] ] \tilde{\mathbf{j}}}$ からi-対面への垂線長は $\tilde{j}_i = \tilde{\mathbf{e}}_i^T (\tilde{\mathbf{a}}_y + \tilde{\mathbf{\Phi}} \mathbf{P}^T \mathbf{p}) \left(\frac{n v^n}{v^{(n-1)}_i}\right)$ 。
辺乗座標(Half-squared Edge Coordinates)：\~b: 単体内部点 $\mathbf{p} = \mathbf{p}_y + \frac{\mathbf{P} \tilde{\mathbf{\Phi}} \tilde{\mathbf{b}}}{\tilde{\mathbb{1}}^T \mathbf{\Phi} \tilde{\mathbf{b}}}$ について、 $\frac{\tilde{\mathbf{b}}}{\tilde{\mathbb{1}}^T \mathbf{\Phi} \tilde{\mathbf{b}}} = \mathbf{P}^T \mathbf{p} = \frac{\mathbf{P}^T \mathbf{P} \tilde{\mathbf{a}}}{\tilde{\mathbb{1}}^T \tilde{\mathbf{a}}}$
分点座標((Abstract-tripolar )Divide-vertices Coordinates)：\~t: $\mathbf{p} - \mathbf{p}_O = \mathbf{P} (\tilde{\mathbf{a}} - \tilde{\mathbf{a}}_O) = \mathbf{P} \tilde{\mathbf{\Phi}} (\tilde{\mathbf{b}} - \tilde{\mathbf{b}}_O) = - \mathbf{P} \tilde{\mathbf{\Phi}} \tilde{\mathbf{B}} \tilde{\mathbf{a}} \propto \mathbf{P} \tilde{\mathbf{\Phi}} \left(\frac{\tilde{\mathbf{t}} \odot \tilde{\mathbf{t}}}{2}\right)$

単体重心関係

重心(Centroid)：\p_G: $\mathbf{p}_G = \frac{\mathbf{P} \tilde{\mathbb{1}}}{\tilde{\mathbb{1}}^T \tilde{\mathbb{1}}}$
重線行列(Medianline Matrix)：\G: $\mathbf{G} = \mathbf{P} \tilde{\mathbf{G}} = - \frac{n+1}{n} \mathbf{P} \left(\tilde{\mathbf{E}} - \frac{\tilde{\mathbb{1}} \tilde{\mathbb{1}}^T}{\tilde{\mathbb{1}}^T \tilde{\mathbb{1}}} \right)$
垂足座標行列(Median Coordinate Matrix)：\~A_G: $\tilde{\mathbf{A}}_G = \frac{\tilde{\mathbb{1}} \tilde{\mathbb{1}}^T - \tilde{\mathbf{E}}}{n}$
重足単体(Median Simplex)：\P_G: $\mathbf{P}_G = \mathbf{P} + \mathbf{G} = \mathbf{P} \tilde{\mathbf{A}}_G$
重均半径・重均偏差(Centroid 0-Facetargeted Circum-radius・Circum-deviation)：r_G・ε_G: $r_G = \frac{\sqrt{\tilde{\mathbb{1}}^T \tilde{\mathbf{B}} \tilde{\mathbb{1}}}}{\tilde{\mathbb{1}}^T \tilde{\mathbb{1}}}$ , $r_G+\epsilon_G \ge r_O \ge r_G$
重均超球（面）(Centroid 0-Facetargeted Hypersphere)：S_G: $S_G \equiv S_{\mathbf{p}_G}^{(n-1)} [r_G[\pm\epsilon_G] \mathbf{S}]$
重中k次元面接半径行列(Centroid k-Facescribed Radii-matrix)：\R_{G_k}: $\mathbf{R}_{G_k} \mathbf{R}_{G_k}^T = \frac{n-k}{(n+1)(k+1)} \mathbf{P} \left(\mathbf{E} - \frac{\mathbb{1} \mathbb{1}^T}{\mathbb{1}^T \mathbb{1}} \right) \mathbf{P}^T$
重中k次元面接超楕円（面）(Centroid k-Facescribed Hyperellipse)：S'_{G_k}: $S'_{G_k} \equiv S_{\mathbf{p}_G}^{(n-1)} [\mathbf{R}_{G_k}]$

単体垂心関係

正射影行列(Orthogonal Projection Matrix)：\~W: $\mathbf{W}[\mathbf{L}] = \mathbf{L} \mathbf{\Phi}[\mathbf{L}^T \mathbf{L}] \mathbf{L}^T = \mathbf{P} \tilde{\mathbf{\Phi}} \mathbf{P}^T = \tilde{\mathbf{W}}[\mathbf{P}]$ ,
直交射行列(Orthogonal Complement Matrix)：\~Y: $\mathbf{Y}[\mathbf{L}] = \mathbf{E} - \mathbf{W}[\mathbf{L}] = \mathbf{E} - \tilde{\mathbf{W}}[\mathbf{P}] = \tilde{\mathbf{Y}}[\mathbf{P}]$
i垂線(i-Perpendicular)：\h_i: $\mathbf{h}_i = - \frac{\mathbf{P} \tilde{\mathbf{C}}[\mathbf{P}^T \mathbf{P}] \tilde{\mathbf{e}}_i}{\tilde{\mathbf{e}}_i^T \tilde{\mathbf{C}}[\mathbf{P}^T \mathbf{P}] \tilde{\mathbf{e}}_i}$
逆垂線総和定理(inversion of Simplex Perpendiculars is zero-sum vector)：Σ\h_i^‡=\0: $\sum_{i=0\cdots n} \mathbf{h}_i^\ddagger = \sum \frac{\mathbf{h}_i}{\mathbf{h}_i^T \mathbf{h}_i} = \mathbb{0}$
垂線行列(Perpendiculars Matrix)：\H: $\mathbf{H} = \mathbf{P} \tilde{\mathbf{H}} = - \mathbf{P} \tilde{\mathbf{C}}[\mathbf{P}^T \mathbf{P}] \mathbf{\Sigma}^{-1}\left[ \tilde{\mathbf{C}}[\mathbf{P}^T \mathbf{P}] \right]$
垂足座標行列(Orthic Coordinate Matrix)：\~A_H: $\tilde{\mathbf{A}}_H = \tilde{\mathbf{E}} - \left( \tilde{\mathbf{C}}[\mathbf{P}^T \mathbf{P}] \mathbf{\Sigma}^{-1}\left[ \tilde{\mathbf{C}}[\mathbf{P}^T \mathbf{P}] \right] \right)$
垂足単体(Orthic Simplex)：\P_H: $\mathbf{P}_H = \mathbf{P} + \mathbf{H} = \mathbf{P} \tilde{\mathbf{A}}_H$
位置等内積行列(Position Inner-product Equivalence Matrix)：\~N: $\tilde{\mathbf{N}} = \mathbf{\Sigma}[\tilde{\mathbf{\nu}}] + \tilde{\mathbb{1}} \left(\tilde{\mathbf{b}}_O - \frac{\tilde{\mathbf{\nu}}}{2}\right)^T + \left(\tilde{\mathbf{b}}_O - \frac{\tilde{\mathbf{\nu}}}{2}\right) \tilde{\mathbb{1}}^T$
垂心((Constrained )Orthocenter)：\p_H: $\mathbf{P}^T \mathbf{P} = \tilde{\mathbf{N}}$ の場合に限り、 $\mathbf{p}_H = \frac{\mathbf{P} \begin{pmatrix} \frac{1}{\nu_0} \\ \vdots \\ \frac{1}{\nu_n} \end{pmatrix}}{\tilde{\mathbb{1}}^T \begin{pmatrix} \frac{1}{\nu_0} \\ \vdots \\ \frac{1}{\nu_n} \end{pmatrix}}$
広義垂心(Extended Orthocenter)：\p_H': $\mathbf{p}_{H'}=\mathbf{p}_y + \mathbf{P} \tilde{\mathbf{\Phi}} \begin{pmatrix} \mathbf{p}_0^T \mathbf{p}_{0G} \\ \vdots \\ \mathbf{p}_n^T \mathbf{p}_{nG} \end{pmatrix} \frac{n}{n-1}$
逆垂心(Lemoine-Symmedian Center)：\p_{/H}: $\mathbf{p}_{/H} = \frac{\mathbf{P} \mathbf{\Sigma}[\tilde{\mathbf{C}}[\mathbf{P}^T \mathbf{P}]] \tilde{\mathbb{1}}}{\tilde{\mathbb{1}}^T \mathbf{\Sigma}[\tilde{\mathbf{C}}[\mathbf{P}^T \mathbf{P}]] \tilde{\mathbb{1}}}= \frac{\mathbf{P} \begin{pmatrix} \frac{1}{\mathbf{h}_0^T \mathbf{h}_0} \\ \vdots \\ \frac{1}{\mathbf{h}_n^T \mathbf{h}_n} \end{pmatrix}}{\tilde{\mathbb{1}}^T \begin{pmatrix} \frac{1}{\mathbf{h}_0^T \mathbf{h}_0} \\ \vdots \\ \frac{1}{\mathbf{h}_n^T \mathbf{h}_n} \end{pmatrix}}$
逆垂半径・逆垂偏差(Symmedian (n-1)-Facetargeted In-radius・In-deviation)：r_{/H}・ε_{/H}: $r_{/H}=\sqrt{\frac{\tilde{\mathbb{1}}^T \mathbf{C}[- \tilde{\mathbf{B}}] \tilde{\mathbb{1}}}{\tilde{\mathbb{1}}^T \mathbf{\Sigma}[ \tilde{\mathbf{C}}[- \tilde{\mathbf{B}}] ] \tilde{\mathbb{1}}}}$ , $r_{/H}+\epsilon_{/H} \ge r_I \ge r_{/H}$
逆垂超球(Symmedian (n-1)-Facetargeted Hypersphere)：S_{/H}: $S_{/H} \equiv S_{\mathbf{p}_{/H}}^{(n-1)} [r_{/H}[\pm \epsilon_{/H}] \mathbf{S}]$

単体内心・傍心関係

内心(Incenter)：\p_I: $\mathbf{p}_I = \frac{\mathbf{P} \mathbf{\Sigma}^{\frac{1}{2}} [ \tilde{\mathbf{C}}[\mathbf{P}^T \mathbf{P}] ] \tilde{\mathbb{1}}}{\tilde{\mathbb{1}}^T \mathbf{\Sigma}^{\frac{1}{2}} [ \tilde{\mathbf{C}}[\mathbf{P}^T \mathbf{P}] ] \tilde{\mathbb{1}}}$
内接半径(Inradius)：r_I: $r_I = \frac{\sqrt{\tilde{\mathbb{1}}^T \mathbf{C}[- \tilde{\mathbf{B}}] \tilde{\mathbb{1}}}}{\tilde{\mathbb{1}}^T \mathbf{\Sigma}^{\frac{1}{2}} [ \tilde{\mathbf{C}}[- \tilde{\mathbf{B}}] ] \tilde{\mathbb{1}}}$
内接超球（面）(Inscribed Hypersphere)：S_I: $S_I \equiv S_{\mathbf{p}_I}^{(n-1)} [r_I \mathbf{S}]$
（広義）傍心((Extended )Excenter)：\p_{J_j}: $\mathbf{p}_{J_j} = \frac{\mathbf{P} \mathbf{\Sigma}^{\frac{1}{2}} [ \tilde{\mathbf{C}}[\mathbf{P}^T \mathbf{P}] ] \tilde{\mathbf{\delta}'}_j}{\tilde{\mathbb{1}}^T \mathbf{\Sigma}^{\frac{1}{2}} [ \tilde{\mathbf{C}}[\mathbf{P}^T \mathbf{P}] ] \tilde{\mathbf{\delta}'}_j}$
広義傍接超球（面）(Extended Exscribed Hypersphere)：S_{J_j}: $S_{J_j} \equiv S_{\mathbf{p}_{J_j}}^{(n-1)} [r_{J_j} \mathbf{S}]$
分面心(Divide-facets Center)：\p_J: $\mathbf{p}_J = \frac{\mathbf{P} \mathbf{\Sigma}^{\frac{1}{2}} [ \tilde{\mathbf{C}}[\mathbf{P}^T \mathbf{P}] ] \tilde{\mathbf{j}}}{\tilde{\mathbb{1}}^T \mathbf{\Sigma}^{\frac{1}{2}} [ \tilde{\mathbf{C}}[\mathbf{P}^T \mathbf{P}] ] \tilde{\mathbf{j}}$

単体k次元面心関係

面因子行列(Face-factor Matrix)：\Ψ_k: $\overset{[(n-k)\times(n-k)]}{\mathbf{\Psi}_{(k+1)}} = \underset{\tiny j\not\in\psi \\ i\not\in\psi}{\mathbf{M}}\left[{\small [j,i] \:} (-1)^{(j+i)} \frac{\tilde{\mathbb{1}}^T \mathbf{C}[\mathbf{P}_{j\psi}^T \mathbf{P}_{i\psi}] \tilde{\mathbb{1}}}{\tilde{\mathbb{1}}^T \mathbf{C}[\mathbf{P}_{j\psi}^T \mathbf{P}_{j\psi}] \tilde{\mathbb{1}}}\right]$
面因子平均行列(Face-factor Average Matrix)：\~Ψ_k: $\overset{[(n+1)\times(n+1)]}{\tilde{\mathbf{\Psi}}_{(k+1)}} = \frac{\sum_{\psi \in \{\psi\}} (\mathbf{E}_{\neq\psi} \mathbf{\Psi}_{(k+1)} \mathbf{E}_{\neq\psi}^T)}{_n C_{(k+1)}}$
k次元面心(Constrained k-Facescribed Midcenter)：\p_{K_k}: $\mathbf{\Psi}_{(k+1)} \underset{\tiny [\not\in \psi]}{\tilde{\mathbf{a}}_{K_k}} = \underset{\tiny [\not\in \psi]}{\tilde{\mathbf{\epsilon}}_{\psi k}}$ が全てのφで成り立つ場合に限り、 $\mathbf{p}_{K_k} = \frac{\mathbf{P} \mathbf{C}^T[\tilde{\mathbf{\Psi}}_(k+1)] \tilde{\mathbb{1}}}{\tilde{\mathbb{1}}^T \mathbf{C}[\tilde{\mathbf{\Psi}}_(k+1)] \tilde{\mathbb{1}}} + \mathbf{P} \tilde{\mathbf{C}}^T [\tilde{\mathbf{\Psi}}_{(k+1)}] \tilde{\mathbf{\epsilon}}_k$
k次元面接半径(Constrained k-Facescribed Midradius)：r_{K_k}: $r_{K_k} =$
k次元面接超球（面）(Constrained k-Facescribed Hypersphere)：S_{K_k}: $S_{K_k} \equiv S_{\mathbf{p}_{K_k}}^{(n-1)} [r_{K_k} \mathbf{S}]$
k次元面均心(Least-Square k-Facetargeted Mid-center)：\p_{Ψ_k}: $\mathbf{p}_{\Psi_k} = \frac{\mathbf{P} \mathbf{C}^T[\tilde{\mathbf{\Psi}}_k] \tilde{\mathbb{1}}}{\tilde{\mathbb{1}}^T \mathbf{C}[\tilde{\mathbf{\Psi}}_k] \tilde{\mathbb{1}}}$
k次元面均半径・偏差(Least-Square k-Facetargeted Mid-radius・Mid-deviation)：r_{Ψ_k}・ε_{Ψ_k}: $r_{\Psi_k}+\epsilon_{\Psi_k} \ge r_{K_k} \ge r_{\Psi_k}$
k次元面均超球（面）(Least-Square k-Facetargeted Hypersphere)：S_{Ψ_k}: $S_{\Psi_k} \equiv S_{\mathbf{p}_{\Psi_k}}^{(n-1)} [r_{\Psi_k}[\pm \epsilon_{\Psi_k}] \mathbf{S}]$

単体外心関係

外心(Circumcenter)：\p_O: $\mathbf{p}_O = \mathbf{p}_y + \mathbf{P} \tilde{\mathbf{\Phi}} \begin{pmatrix} \mathbf{p}_0^T \mathbf{p}_0 \\ \vdots \\ \mathbf{p}_n^T \mathbf{p}_n \end{pmatrix} \frac{1}{2}$
外接半径(Circumradius)：r_O: $r_O = \sqrt{\frac{-|[- \tilde{\mathbf{B}}]|}{\tilde{\mathbb{1}}^T \mathbf{C}[- \tilde{\mathbf{B}}] \tilde{\mathbb{1}}}}$
外接超球（面）(Circumscribed Hypersphere)：S_O: $S_O \equiv S_{\mathbf{p}_O}^{(n-1)}[r_O \mathbf{S}]$
（内・外）分点心((Interior・Exterior )Divide-vertices Center)：\p_{T-}・\p_{T+}: $\mathbf{p}_{T-} = \mathbf{p}_{TO} - \hat{\mathbf{x}}_T \epsilon_T$ , $\mathbf{p}_{T+} = \mathbf{p}_{TO} + \hat{\mathbf{x}}_T \epsilon_T$
分点中心(Divide-vertices Midpoint)：\p_{TO}: $\mathbf{p}_{TO} = \mathbf{p}_O + \hat{\mathbf{x}}_T r_T$
分点心向外線(Divide-vertices Direction)：\x_T: $\mathbf{x}_T = \frac{\mathbf{P} \widetilde{\mathbf{C}}[\mathbf{P}^T \mathbf{P}] \widetilde{\mathbf{b}}_t}{\widetilde{\mathbb{1}}^T \mathbf{C}[\mathbf{P}^T \mathbf{P}] \widetilde{\mathbb{1}}}$
分点偏差(Extended Apollonius Radius)：ε_T: $\epsilon_T =$
分点心補超球（面）(Extended Apollonius Hypersphere)：S_T: $S_T \equiv S_{\mathbf{p}_{TO}}^{(m-n)}[\epsilon_T \mathbf{S}]$

点足単体関係

組み合わせ単位行列(Combination Identify Matrix)：\~E'_(_(k+1)^(n+1)): $\overset{[(n+1)\times {}_{(n+1)} C_{(k+1)}]}{\tilde{\mathbf{E}'}}_{{n+1}\choose{k+1}}=\mathbf{M}[{\small [j,i] \:} 0 \text{ or } 1], \tilde{\mathbf{e}'}_{j C_{(k+1)}^{(n+1)}} \neq \tilde{\mathbf{e}'}_{j C_{(k+1)}^{(n+1)}}$
点足座標行列(Cevian Coordinate Matrix)：\~A': $\tilde{\mathbf{A}'} = \left( \tilde{\mathbf{a}} \tilde{\mathbb{1}}^T - \mathbf{\Sigma}[\tilde{\mathbf{a}}] \right) \mathbf{\Sigma}^{-1}[\tilde{\mathbb{1}}-\tilde{\mathbf{a}}]$
点足座標行列(Cevian Coordinate Matrix)：\~A': $\tilde{\mathbf{A}'} = \left( \tilde{\mathbf{a}} \tilde{\mathbb{1}}^T - \mathbf{\Sigma}[\tilde{\mathbf{a}}] \right) \mathbf{\Sigma}^{-1}[\tilde{\mathbb{1}}-\tilde{\mathbf{a}}]$
点足単体(Cevian Simplex)：\P \~A': $0 \le v^n \left[\mathbf{P} \tilde{\mathbf{A}'}\right] = |[ \tilde{\mathbf{A}'} ]| v^n = \left( n \prod_{i=0\cdots n} \frac{\tilde{a}_i}{1-\tilde{a}_i} \right) v^n \le \frac{v^n}{n^n}$ （最大値は $a_0=\cdots =a_n=\frac{1}{n+1}$ のとき）
点反足座標行列(Anticevian Coordinate Matrix)：\~A'^{-1}: $\tilde{\mathbf{A}'}^{-1} = \left( (\tilde{\mathbb{1}} - \tilde{\mathbf{a}}) \frac{\tilde{\mathbb{1}}^T}{n} - \mathbf{\Sigma}[\tilde{\mathbb{1}} - \tilde{\mathbf{a}}] \right) \mathbf{\Sigma}^{-1}[ \tilde{\mathbf{a}} ]$
点反足単体(Anticevian Simplex)：\P \~A'^{-1}: $v^n \left[\mathbf{P} \tilde{\mathbf{A}'}^{-1}\right] = |[ \tilde{\mathbf{A}'}^{-1} ]| v^n = \left( \frac{1}{n} \prod_{i=0\cdots n} \frac{1-\tilde{a}_i}{\tilde{a}_i} \right) v^n \ge n^n v^n$ （最小値は $a_0=\cdots =a_n=\frac{1}{n+1}$ のとき）
点垂足座標行列(Pedal Coordinate Matrix)：\~A'_H: $\tilde{\mathbf{A}'}_H = \tilde{\mathbf{a}} \tilde{\mathbb{1}}^T - \left( \tilde{\mathbf{C}}[\mathbf{P}^T \mathbf{P}] \mathbf{\Sigma}^{-1}\left[ \tilde{\mathbf{C}}[\mathbf{P}^T \mathbf{P}] \right] \mathbf{\Sigma}[ \tilde{\mathbf{a}} ] \right)$
点垂足単体(Pedal Simplex)：\P \~A'_H: $v^n \left[\mathbf{P} \tilde{\mathbf{A}'}_H \right]$