Analízis

]> Analízis Gömb: a következő mátrixok írják le:

(\begin{matrix} 1 & 0 \\ - \frac{n_{2} - n_{1}}{R_{1}} & 1 \end{matrix}) \cdot (\begin{matrix} 1 & \frac{2 R_{1}}{n_{2}} \\ 0 & 1 \end{matrix}) \cdot (\begin{matrix} 1 & 0 \\ \frac{n_{1} - n_{2}}{R_{1}} & 1 \end{matrix})

a szorzás eredménye:

(\begin{matrix} \frac{2 n_{1}}{n_{2}} - 1 & \frac{2 R_{1}}{n_{2}} \\ \frac{2 n_{1} (n_{1} - n_{2})}{n_{2} R_{1}} & \frac{2 n_{1}}{n_{2}} - 1 \end{matrix})

Egy félgömb alakú vastag lencse mátrixa:

(\begin{matrix} 1 & 0 \\ - \frac{n_{2} - n_{1}}{R_{1}} & 1 \end{matrix}) \cdot (\begin{matrix} 1 & \frac{R_{1}}{n_{2}} \\ 0 & 1 \end{matrix}) = (\begin{matrix} 1 & \frac{R_{1}}{n_{2}} \\ \frac{n_{1} - n_{2}}{R_{1}} & \frac{n_{1}}{n_{2}} \end{matrix})

.
Egy másik irányba fordítotté:

(\begin{matrix} 1 & \frac{R_{1}}{n_{2}} \\ 0 & 1 \end{matrix}) \cdot (\begin{matrix} 1 & 0 \\ \frac{n_{1} - n_{2}}{R_{1}} & 1 \end{matrix}) = (\begin{matrix} \frac{n_{1}}{n_{2}} & \frac{R_{1}}{n_{2}} \\ \frac{n_{1} - n_{2}}{R_{1}} & 1 \end{matrix})

Két egymás felé fordított lencséé:

(\begin{matrix} 1 & \frac{R_{2}}{n_{3}} \\ 0 & 1 \end{matrix}) \cdot (\begin{matrix} 1 & 0 \\ \frac{n_{1} - n_{3}}{R_{2}} & 1 \end{matrix}) \cdot (\begin{matrix} 1 & 0 \\ - \frac{n_{2} - n_{1}}{R_{1}} & 1 \end{matrix}) \cdot (\begin{matrix} 1 & \frac{R_{1}}{n_{2}} \\ 0 & 1 \end{matrix}) = (\begin{matrix} \frac{n_{1} (R_{1} + R_{2}) - n_{2} R_{2}}{n_{3} R_{1}} & \frac{n_{1} (R_{1} + R_{2})}{n_{2} n_{3}} \\ \frac{n_{1} - n_{2}}{R_{1}} + \frac{n_{1} - n_{3}}{R_{2}} & \frac{n_{1} (R_{1} + R_{2}) - n_{3} R_{1}}{n_{2} R_{2}} \end{matrix})

n_{1} = 1, n_{2} = n_{3} = n, R_{1} = R_{2} = R

(\begin{matrix} \frac{2}{n} - 1 & \frac{2 R}{n^{2}} \\ \frac{2 - 2 n}{R} & \frac{2}{n} - 1 \end{matrix})

.
Egy irányba fordítotté:

(\begin{matrix} 1 & 0 \\ - \frac{n_{3} - n_{1}}{R_{2}} & 1 \end{matrix}) \cdot (\begin{matrix} 1 & \frac{R_{2}}{n_{3}} \\ 0 & 1 \end{matrix}) \cdot (\begin{matrix} 1 & 0 \\ - \frac{n_{2} - n_{1}}{R_{1}} & 1 \end{matrix}) \cdot (\begin{matrix} 1 & \frac{R_{1}}{n_{2}} \\ 0 & 1 \end{matrix}) = (\begin{matrix} \frac{(n_{1} - n_{2}) R_{2}}{n_{3} R_{1}} + 1 & \frac{n_{3} R_{1} + n_{1} R_{2}}{n_{2} n_{3}} \\ \frac{n_{1} (n_{1} - n_{2})}{n_{3} R_{1}} + \frac{n_{1} - n_{3}}{R_{2}} & \frac{R_{2} n_{1}^{2} + (n_{1} - n_{3}) n_{3} R_{1}}{n_{2} n_{3} R_{2}} \end{matrix})

n_{1} = 1, n_{2} = n_{3} = n, R_{1} = R_{2} = R

(\begin{matrix} \frac{1}{n} & \frac{(n + 1) R}{n^{2}} \\ \frac{1 - n^{2}}{n R} & - 1 + \frac{1}{n} + \frac{1}{n^{2}} \end{matrix})

. Másik irányba fordítottaké:

(\begin{matrix} 1 & \frac{R_{2}}{n_{3}} \\ 0 & 1 \end{matrix}) \cdot (\begin{matrix} 1 & 0 \\ \frac{n_{1} - n_{3}}{R_{2}} & 1 \end{matrix}) \cdot (\begin{matrix} 1 & \frac{R_{1}}{n_{2}} \\ 0 & 1 \end{matrix}) \cdot (\begin{matrix} 1 & 0 \\ \frac{n_{1} - n_{2}}{R_{1}} & 1 \end{matrix}) = (\begin{matrix} \frac{R_{1} n_{1}^{2} + (n_{1} - n_{2}) n_{2} R_{2}}{n_{2} n_{3} R_{1}} & \frac{n_{1} R_{1} + n_{2} R_{2}}{n_{2} n_{3}} \\ \frac{n_{1} - n_{2}}{R_{1}} + \frac{n_{1} (n_{1} - n_{3})}{n_{2} R_{2}} & \frac{(n_{1} - n_{3}) R_{1}}{n_{2} R_{2}} + 1 \end{matrix})

, ha

n_{1} = 1, n_{2} = n_{3} = n, R_{1} = R_{2} = R

(\begin{matrix} - 1 + \frac{1}{n} + \frac{1}{n^{2}} & \frac{(n + 1) R}{n^{2}} \\ \frac{1 - n^{2}}{n R} & \frac{1}{n} \end{matrix})

Gömb és félgömb:

(\begin{matrix} \frac{(2 n_{1} - n_{2}) n_{3} R_{1} + 2 n_{1} (n_{1} - n_{2}) R_{2}}{n_{2} n_{3} R_{1}} & \frac{2 n_{3} R_{1} + (2 n_{1} - n_{2}) R_{2}}{n_{2} n_{3}} \\ \frac{\frac{2 (n_{1} - n_{2}) n_{1}^{2}}{n_{3} R_{1}} + \frac{(2 n_{1} - n_{2}) (n_{1} - n_{3})}{R_{2}}}{n_{2}} & \frac{2 (n_{1} - n_{3}) n_{3} R_{1} + n_{1} (2 n_{1} - n_{2}) R_{2}}{n_{2} n_{3} R_{2}} \end{matrix})

n_{1} = 1, n_{2} = n_{3} = n, R_{1} = R_{2} = R

(\begin{matrix} \frac{2}{n^{2}} - 1 & \frac{(n + 2) R}{n^{2}} \\ \frac{(n - 3) n^{2} + 2}{n^{2} R} & \frac{n + 2}{n^{2}} - 2 \end{matrix})

(\begin{matrix} \frac{n_{1} (2 n_{1} (R_{1} + R_{2}) - n_{2} (R_{1} + 2 R_{2}))}{n_{2} n_{3} R_{1}} & \frac{2 n_{1} (R_{1} + R_{2}) - n_{2} R_{2}}{n_{2} n_{3}} \\ \frac{(2 n_{1} - n_{2}) (n_{1} - n_{3}) R_{1} + 2 n_{1} (n_{1} - n_{2}) R_{2}}{n_{2} R_{1} R_{2}} & \frac{- 2 n_{3} R_{1} - n_{2} R_{2} + 2 n_{1} (R_{1} + R_{2})}{n_{2} R_{2}} \end{matrix})

n_{1} = 1, n_{2} = n_{3} = n, R_{1} = R_{2} = R

(\begin{matrix} \frac{4}{n} - 3 & - \frac{(n - 4) R}{n^{2}} \\ \frac{(n - 5) n + 4}{n R} & \frac{4 - 3 n}{n^{2}} \end{matrix})

(\begin{matrix} \frac{- 2 n_{3} R_{1} - n_{2} R_{2} + 2 n_{1} (R_{1} + R_{2})}{n_{2} R_{2}} & \frac{2 n_{1} (R_{1} + R_{2}) - n_{2} R_{2}}{n_{2} n_{3}} \\ \frac{(2 n_{1} - n_{2}) (n_{1} - n_{3}) R_{1} + 2 n_{1} (n_{1} - n_{2}) R_{2}}{n_{2} R_{1} R_{2}} & \frac{n_{1} (2 n_{1} (R_{1} + R_{2}) - n_{2} (R_{1} + 2 R_{2}))}{n_{2} n_{3} R_{1}} \end{matrix})

n_{1} = 1, n_{2} = n_{3} = n, R_{1} = R_{2} = R

(\begin{matrix} \frac{4 - 3 n}{n^{2}} & - \frac{(n - 4) R}{n^{2}} \\ \frac{(n - 5) n + 4}{n R} & \frac{4}{n} - 3 \end{matrix})

(\begin{matrix} \frac{\frac{n_{1} (2 n_{1} - n_{2})}{n_{3}} + \frac{2 (n_{1} - n_{3}) R_{1}}{R_{2}}}{n_{2}} & \frac{2 n_{3} R_{1} + (2 n_{1} - n_{2}) R_{2}}{n_{2} n_{3}} \\ \frac{\frac{2 (n_{1} - n_{2}) n_{1}^{2}}{n_{3} R_{1}} + \frac{(2 n_{1} - n_{2}) (n_{1} - n_{3})}{R_{2}}}{n_{2}} & \frac{(2 n_{1} - n_{2}) n_{3} R_{1} + 2 n_{1} (n_{1} - n_{2}) R_{2}}{n_{2} n_{3} R_{1}} \end{matrix})

, ha

n_{1} = 1, n_{2} = n_{3} = n, R_{1} = R_{2} = R

(\begin{matrix} \frac{n + 2}{n^{2}} - 2 & \frac{(n + 2) R}{n^{2}} \\ \frac{(n - 3) n^{2} + 2}{n^{2} R} & \frac{2}{n^{2}} - 1 \end{matrix})