Sphere

113 days ago by Cornu

var('sa,n11,sb,n21,d,n01,q1,n12,n22,n02,q2,n13,n23,n03,q3') 
        
(sa, n11, sb, n21, d, n01, q1, n12, n22, n02, q2, n13, n23, n03, q3)
(sa, n11, sb, n21, d, n01, q1, n12, n22, n02, q2, n13, n23, n03, q3)
eqn = [sa*n11-sb*n21-d*n01==q1, sa*n12-sb*n22-d*n02==q2, sa*n13-sb*n23-d*n03==q3] 
       
s = solve(eqn, sa, sb, d) 
       
print latex(s) 
        
\left[\left[\mbox{sa} = -\frac{{\left(n_{22} q_{3} - n_{23}
q_{2}\right)} n_{01} - {\left(n_{02} q_{3} - n_{03} q_{2}\right)} n_{21}
+ {\left(n_{02} n_{23} - n_{03} n_{22}\right)} q_{1}}{{\left(n_{12}
n_{23} - n_{13} n_{22}\right)} n_{01} - {\left(n_{02} n_{23} - n_{03}
n_{22}\right)} n_{11} + {\left(n_{02} n_{13} - n_{03} n_{12}\right)}
n_{21}}, \mbox{sb} = -\frac{{\left(n_{12} q_{3} - n_{13} q_{2}\right)}
n_{01} - {\left(n_{02} q_{3} - n_{03} q_{2}\right)} n_{11} +
{\left(n_{02} n_{13} - n_{03} n_{12}\right)} q_{1}}{{\left(n_{12} n_{23}
- n_{13} n_{22}\right)} n_{01} - {\left(n_{02} n_{23} - n_{03}
n_{22}\right)} n_{11} + {\left(n_{02} n_{13} - n_{03} n_{12}\right)}
n_{21}}, d = -\frac{{\left(n_{22} q_{3} - n_{23} q_{2}\right)} n_{11} -
{\left(n_{12} q_{3} - n_{13} q_{2}\right)} n_{21} + {\left(n_{12} n_{23}
- n_{13} n_{22}\right)} q_{1}}{{\left(n_{12} n_{23} - n_{13}
n_{22}\right)} n_{01} - {\left(n_{02} n_{23} - n_{03} n_{22}\right)}
n_{11} + {\left(n_{02} n_{13} - n_{03} n_{12}\right)}
n_{21}}\right]\right]
\left[\left[\mbox{sa} = -\frac{{\left(n_{22} q_{3} - n_{23} q_{2}\right)} n_{01} - {\left(n_{02} q_{3} - n_{03} q_{2}\right)} n_{21} + {\left(n_{02} n_{23} - n_{03} n_{22}\right)} q_{1}}{{\left(n_{12} n_{23} - n_{13} n_{22}\right)} n_{01} - {\left(n_{02} n_{23} - n_{03} n_{22}\right)} n_{11} + {\left(n_{02} n_{13} - n_{03} n_{12}\right)} n_{21}}, \mbox{sb} = -\frac{{\left(n_{12} q_{3} - n_{13} q_{2}\right)} n_{01} - {\left(n_{02} q_{3} - n_{03} q_{2}\right)} n_{11} + {\left(n_{02} n_{13} - n_{03} n_{12}\right)} q_{1}}{{\left(n_{12} n_{23} - n_{13} n_{22}\right)} n_{01} - {\left(n_{02} n_{23} - n_{03} n_{22}\right)} n_{11} + {\left(n_{02} n_{13} - n_{03} n_{12}\right)} n_{21}}, d = -\frac{{\left(n_{22} q_{3} - n_{23} q_{2}\right)} n_{11} - {\left(n_{12} q_{3} - n_{13} q_{2}\right)} n_{21} + {\left(n_{12} n_{23} - n_{13} n_{22}\right)} q_{1}}{{\left(n_{12} n_{23} - n_{13} n_{22}\right)} n_{01} - {\left(n_{02} n_{23} - n_{03} n_{22}\right)} n_{11} + {\left(n_{02} n_{13} - n_{03} n_{12}\right)} n_{21}}\right]\right]
view(s) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[\mbox{sa} = -\frac{{\left(n_{22} q_{3} - n_{23} q_{2}\right)} n_{01} - {\left(n_{02} q_{3} - n_{03} q_{2}\right)} n_{21} + {\left(n_{02} n_{23} - n_{03} n_{22}\right)} q_{1}}{{\left(n_{12} n_{23} - n_{13} n_{22}\right)} n_{01} - {\left(n_{02} n_{23} - n_{03} n_{22}\right)} n_{11} + {\left(n_{02} n_{13} - n_{03} n_{12}\right)} n_{21}}, \mbox{sb} = -\frac{{\left(n_{12} q_{3} - n_{13} q_{2}\right)} n_{01} - {\left(n_{02} q_{3} - n_{03} q_{2}\right)} n_{11} + {\left(n_{02} n_{13} - n_{03} n_{12}\right)} q_{1}}{{\left(n_{12} n_{23} - n_{13} n_{22}\right)} n_{01} - {\left(n_{02} n_{23} - n_{03} n_{22}\right)} n_{11} + {\left(n_{02} n_{13} - n_{03} n_{12}\right)} n_{21}}, d = -\frac{{\left(n_{22} q_{3} - n_{23} q_{2}\right)} n_{11} - {\left(n_{12} q_{3} - n_{13} q_{2}\right)} n_{21} + {\left(n_{12} n_{23} - n_{13} n_{22}\right)} q_{1}}{{\left(n_{12} n_{23} - n_{13} n_{22}\right)} n_{01} - {\left(n_{02} n_{23} - n_{03} n_{22}\right)} n_{11} + {\left(n_{02} n_{13} - n_{03} n_{12}\right)} n_{21}}\right]\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[\mbox{sa} = -\frac{{\left(n_{22} q_{3} - n_{23} q_{2}\right)} n_{01} - {\left(n_{02} q_{3} - n_{03} q_{2}\right)} n_{21} + {\left(n_{02} n_{23} - n_{03} n_{22}\right)} q_{1}}{{\left(n_{12} n_{23} - n_{13} n_{22}\right)} n_{01} - {\left(n_{02} n_{23} - n_{03} n_{22}\right)} n_{11} + {\left(n_{02} n_{13} - n_{03} n_{12}\right)} n_{21}}, \mbox{sb} = -\frac{{\left(n_{12} q_{3} - n_{13} q_{2}\right)} n_{01} - {\left(n_{02} q_{3} - n_{03} q_{2}\right)} n_{11} + {\left(n_{02} n_{13} - n_{03} n_{12}\right)} q_{1}}{{\left(n_{12} n_{23} - n_{13} n_{22}\right)} n_{01} - {\left(n_{02} n_{23} - n_{03} n_{22}\right)} n_{11} + {\left(n_{02} n_{13} - n_{03} n_{12}\right)} n_{21}}, d = -\frac{{\left(n_{22} q_{3} - n_{23} q_{2}\right)} n_{11} - {\left(n_{12} q_{3} - n_{13} q_{2}\right)} n_{21} + {\left(n_{12} n_{23} - n_{13} n_{22}\right)} q_{1}}{{\left(n_{12} n_{23} - n_{13} n_{22}\right)} n_{01} - {\left(n_{02} n_{23} - n_{03} n_{22}\right)} n_{11} + {\left(n_{02} n_{13} - n_{03} n_{12}\right)} n_{21}}\right]\right]
s 
        
[[sa == -((n22*q3 - n23*q2)*n01 - (n02*q3 - n03*q2)*n21 + (n02*n23 -
n03*n22)*q1)/((n12*n23 - n13*n22)*n01 - (n02*n23 - n03*n22)*n11 +
(n02*n13 - n03*n12)*n21), sb == -((n12*q3 - n13*q2)*n01 - (n02*q3 -
n03*q2)*n11 + (n02*n13 - n03*n12)*q1)/((n12*n23 - n13*n22)*n01 -
(n02*n23 - n03*n22)*n11 + (n02*n13 - n03*n12)*n21), d == -((n22*q3 -
n23*q2)*n11 - (n12*q3 - n13*q2)*n21 + (n12*n23 - n13*n22)*q1)/((n12*n23
- n13*n22)*n01 - (n02*n23 - n03*n22)*n11 + (n02*n13 - n03*n12)*n21)]]
[[sa == -((n22*q3 - n23*q2)*n01 - (n02*q3 - n03*q2)*n21 + (n02*n23 - n03*n22)*q1)/((n12*n23 - n13*n22)*n01 - (n02*n23 - n03*n22)*n11 + (n02*n13 - n03*n12)*n21), sb == -((n12*q3 - n13*q2)*n01 - (n02*q3 - n03*q2)*n11 + (n02*n13 - n03*n12)*q1)/((n12*n23 - n13*n22)*n01 - (n02*n23 - n03*n22)*n11 + (n02*n13 - n03*n12)*n21), d == -((n22*q3 - n23*q2)*n11 - (n12*q3 - n13*q2)*n21 + (n12*n23 - n13*n22)*q1)/((n12*n23 - n13*n22)*n01 - (n02*n23 - n03*n22)*n11 + (n02*n13 - n03*n12)*n21)]]
view(eqn) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[-d n_{01} + n_{11} \mbox{sa} - n_{21} \mbox{sb} = q_{1}, -d n_{02} + n_{12} \mbox{sa} - n_{22} \mbox{sb} = q_{2}, -d n_{03} + n_{13} \mbox{sa} - n_{23} \mbox{sb} = q_{3}\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[-d n_{01} + n_{11} \mbox{sa} - n_{21} \mbox{sb} = q_{1}, -d n_{02} + n_{12} \mbox{sa} - n_{22} \mbox{sb} = q_{2}, -d n_{03} + n_{13} \mbox{sa} - n_{23} \mbox{sb} = q_{3}\right]
sa = s[0][0] sb = s[0][1] d = s[0][2] 
       
x= d/sa view(x) view(solve(x, q1)) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{d}{\mbox{sa}} = \frac{{\left(n_{22} q_{3} - n_{23} q_{2}\right)} n_{11} - {\left(n_{12} q_{3} - n_{13} q_{2}\right)} n_{21} + {\left(n_{12} n_{23} - n_{13} n_{22}\right)} q_{1}}{{\left(n_{22} q_{3} - n_{23} q_{2}\right)} n_{01} - {\left(n_{02} q_{3} - n_{03} q_{2}\right)} n_{21} + {\left(n_{02} n_{23} - n_{03} n_{22}\right)} q_{1}}
\newcommand{\Bold}[1]{\mathbf{#1}}\left[q_{1} = -\frac{{\left({\left(n_{11} n_{23} - n_{13} n_{21}\right)} q_{2} - {\left(n_{11} n_{22} - n_{12} n_{21}\right)} q_{3}\right)} \mbox{sa} - {\left(d n_{01} n_{23} - d n_{03} n_{21}\right)} q_{2} + {\left(d n_{01} n_{22} - d n_{02} n_{21}\right)} q_{3}}{d n_{02} n_{23} - d n_{03} n_{22} - {\left(n_{12} n_{23} - n_{13} n_{22}\right)} \mbox{sa}}\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{d}{\mbox{sa}} = \frac{{\left(n_{22} q_{3} - n_{23} q_{2}\right)} n_{11} - {\left(n_{12} q_{3} - n_{13} q_{2}\right)} n_{21} + {\left(n_{12} n_{23} - n_{13} n_{22}\right)} q_{1}}{{\left(n_{22} q_{3} - n_{23} q_{2}\right)} n_{01} - {\left(n_{02} q_{3} - n_{03} q_{2}\right)} n_{21} + {\left(n_{02} n_{23} - n_{03} n_{22}\right)} q_{1}}
\newcommand{\Bold}[1]{\mathbf{#1}}\left[q_{1} = -\frac{{\left({\left(n_{11} n_{23} - n_{13} n_{21}\right)} q_{2} - {\left(n_{11} n_{22} - n_{12} n_{21}\right)} q_{3}\right)} \mbox{sa} - {\left(d n_{01} n_{23} - d n_{03} n_{21}\right)} q_{2} + {\left(d n_{01} n_{22} - d n_{02} n_{21}\right)} q_{3}}{d n_{02} n_{23} - d n_{03} n_{22} - {\left(n_{12} n_{23} - n_{13} n_{22}\right)} \mbox{sa}}\right]
A = matrix([[n01, n11, n21], [n02, n12, n22], [n03, n13, n23]]) 
       
view(A) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr}
n_{01} & n_{11} & n_{21} \\
n_{02} & n_{12} & n_{22} \\
n_{03} & n_{13} & n_{23}
\end{array}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr}
n_{01} & n_{11} & n_{21} \\
n_{02} & n_{12} & n_{22} \\
n_{03} & n_{13} & n_{23}
\end{array}\right)
b = [-d, sa, -sb] 
       
view(b) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[-d = \frac{{\left(n_{22} q_{3} - n_{23} q_{2}\right)} n_{11} - {\left(n_{12} q_{3} - n_{13} q_{2}\right)} n_{21} + {\left(n_{12} n_{23} - n_{13} n_{22}\right)} q_{1}}{{\left(n_{12} n_{23} - n_{13} n_{22}\right)} n_{01} - {\left(n_{02} n_{23} - n_{03} n_{22}\right)} n_{11} + {\left(n_{02} n_{13} - n_{03} n_{12}\right)} n_{21}}, \mbox{sa} = -\frac{{\left(n_{22} q_{3} - n_{23} q_{2}\right)} n_{01} - {\left(n_{02} q_{3} - n_{03} q_{2}\right)} n_{21} + {\left(n_{02} n_{23} - n_{03} n_{22}\right)} q_{1}}{{\left(n_{12} n_{23} - n_{13} n_{22}\right)} n_{01} - {\left(n_{02} n_{23} - n_{03} n_{22}\right)} n_{11} + {\left(n_{02} n_{13} - n_{03} n_{12}\right)} n_{21}}, -\mbox{sb} = \frac{{\left(n_{12} q_{3} - n_{13} q_{2}\right)} n_{01} - {\left(n_{02} q_{3} - n_{03} q_{2}\right)} n_{11} + {\left(n_{02} n_{13} - n_{03} n_{12}\right)} q_{1}}{{\left(n_{12} n_{23} - n_{13} n_{22}\right)} n_{01} - {\left(n_{02} n_{23} - n_{03} n_{22}\right)} n_{11} + {\left(n_{02} n_{13} - n_{03} n_{12}\right)} n_{21}}\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[-d = \frac{{\left(n_{22} q_{3} - n_{23} q_{2}\right)} n_{11} - {\left(n_{12} q_{3} - n_{13} q_{2}\right)} n_{21} + {\left(n_{12} n_{23} - n_{13} n_{22}\right)} q_{1}}{{\left(n_{12} n_{23} - n_{13} n_{22}\right)} n_{01} - {\left(n_{02} n_{23} - n_{03} n_{22}\right)} n_{11} + {\left(n_{02} n_{13} - n_{03} n_{12}\right)} n_{21}}, \mbox{sa} = -\frac{{\left(n_{22} q_{3} - n_{23} q_{2}\right)} n_{01} - {\left(n_{02} q_{3} - n_{03} q_{2}\right)} n_{21} + {\left(n_{02} n_{23} - n_{03} n_{22}\right)} q_{1}}{{\left(n_{12} n_{23} - n_{13} n_{22}\right)} n_{01} - {\left(n_{02} n_{23} - n_{03} n_{22}\right)} n_{11} + {\left(n_{02} n_{13} - n_{03} n_{12}\right)} n_{21}}, -\mbox{sb} = \frac{{\left(n_{12} q_{3} - n_{13} q_{2}\right)} n_{01} - {\left(n_{02} q_{3} - n_{03} q_{2}\right)} n_{11} + {\left(n_{02} n_{13} - n_{03} n_{12}\right)} q_{1}}{{\left(n_{12} n_{23} - n_{13} n_{22}\right)} n_{01} - {\left(n_{02} n_{23} - n_{03} n_{22}\right)} n_{11} + {\left(n_{02} n_{13} - n_{03} n_{12}\right)} n_{21}}\right]