Bayangan Titik akibat Pencerminan oleh Cermin Berbentuk Permukaan Bola

[Roni]

Kula Nuwun.

\section{Bayangan Titik akibat Pencerminan oleh Cermin Berbentuk Permukaan Bola}

Andaikan di ruang $\mathbb{R}^3$, ada sebuah cermin berbentuk permukaan bola, yaitu
$$S^2(\vec{r}_0, R) := \{ \vec{r} \in \mathbb{R}^3 ~|~ |\vec{r} - \vec{r}_0| = R \}$$
di mana $\vec{r}_0 \in \mathbb{R}^3$ adalah pusat dari $S^2(\vec{r}_0, R)$, serta $R \in \mathbb{R}^+$ adalah jari-jari dari $S^2(\vec{r}_0, R)$.

Selanjutnya, titik $\vec{r} \in \mathbb{R}^3$ akan dicerminkan oleh $S^2(\vec{r}_0, R)$, sehingga menghasilkan dua buah bayangan, yaitu $\vec{r}'_1, \vec{r}'_2 \in \mathbb{R}^3$.
$$\vec{r}'_1 = \vec{r}_0 + (R + s'_1)\frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|}.$$
$$s'_1 := \frac{s_1f_1}{s_1 - f_1}, ~~~~~ s_1 := |\vec{r} - \vec{r}_0| - R, ~~~~~ f_1 := -\frac{1}{2}R.$$
$$\vec{r}'_1 = \vec{r}_0 + \left(R + \frac{s_1f_1}{s_1 - f_1}\right)\frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|}.$$
$$\vec{r}'_1 = \vec{r}_0 + \left(R + \frac{(|\vec{r} - \vec{r}_0| - R)(-R/2)}{|\vec{r} - \vec{r}_0| - R/2}\right)\frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|}.$$
$$\vec{r}'_2 = \vec{r}_0 + (R - s'_2)\frac{\vec{r}_0 - \vec{r}}{|\vec{r}_0 - \vec{r}|}.$$
$$\vec{r}'_2 = \vec{r}_0 + (s'_2 - R)\frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|}.$$
$$s'_2 := \frac{s_2f_2}{s_2 - f_2}, ~~~~~ s_2 := |\vec{r} - \vec{r}_0| + R, ~~~~~ f_2 := \frac{1}{2}R.$$
$$\vec{r}'_2 = \vec{r}_0 + \left(\frac{s_2f_2}{s_2 - f_2} - R\right)\frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|}.$$
$$\vec{r}'_2 = \vec{r}_0 + \left(\frac{(|\vec{r} - \vec{r}_0| + R)R/2}{|\vec{r} - \vec{r}_0| + R/2} - R\right)\frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|}.$$
$$\vec{r}'_1 = \vec{r}_0 + \frac{R(|\vec{r} - \vec{r}_0| - R/2) + (|\vec{r} - \vec{r}_0| - R)(-R/2)}{|\vec{r} - \vec{r}_0| - R/2}\frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|}.$$
$$\vec{r}'_1 = \vec{r}_0 + \frac{R|\vec{r} - \vec{r}_0|/2}{|\vec{r} - \vec{r}_0| - R/2}\frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|}.$$
$$\vec{r}'_1 = \vec{r}_0 + R\frac{\vec{r} - \vec{r}_0}{2|\vec{r} - \vec{r}_0| - R}.$$
$$\vec{r}'_2 = \vec{r}_0 + \frac{(|\vec{r} - \vec{r}_0| + R)R/2 - R(|\vec{r} - \vec{r}_0| + R/2)}{|\vec{r} - \vec{r}_0| + R/2}\frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|}.$$
$$\vec{r}'_2 = \vec{r}_0 + \frac{(-R/2)|\vec{r} - \vec{r}_0|}{|\vec{r} - \vec{r}_0| + R/2}\frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|}.$$
$$\vec{r}'_2 = \vec{r}_0 - R\frac{\vec{r} - \vec{r}_0}{2|\vec{r} - \vec{r}_0| + R}.$$

Jadi, bayangan dari titik $\vec{r}$ akibat pencerminan oleh cermin berbentuk permukaan bola $S^2(\vec{r}_0, R)$ adalah
$$\vec{r}' = \vec{r}'_\pm = \vec{r}_0 \pm R\frac{\vec{r} - \vec{r}_0}{2|\vec{r} - \vec{r}_0| \mp R}.$$
Bayangannya ada dua, yaitu $\vec{r}'_+$ dan $\vec{r}'_-$.

Terpujilah Kristus.