[Asteroid AI] Asteroid Modeling : Basic Structure

이거 꾸준히 하고 있었는데 뭔가 기록을 해 놓아야 할 스케일까지 와서 다시 기록을 하려고 합니다.

 

대충 변경사항을 요약해보자면

- RL말고 AI 자체로 접근하기로 함 : RL을 쓸 수도 있지만 그것만 쓰지는 않을 듯

- 모델링부터 다시

- 선행연구 리뷰도 해볼 예정

- 목표 : 25년 봄 천문학회 포스터 발표

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Asteroid Modeling : Mathmetical Expression

Spherical coordinate is represented as

so the relation between spherical coord. and cartesian coord. is

$$\begin{cases} x=r \sin\theta \cos\phi \\ y=r \sin\theta \sin\phi \\ z = r \cos\theta \end{cases} \quad \text{and} \quad \begin{cases} r= (x^2+y^2+z^2)^{1/2} \\ \phi = \text{arctan2}(x, y) \in [0, 2\pi] \\ \theta = \frac{\pi}{2}-\text{arctan}(\sqrt{x^2+y^2}, z) \in [0, \phi] \end{cases}$$

 

We will consider an asteroid as a set of facets that forms a closed surface.

Vertices Coordinate Rule
- variable : $r_{ij}$
- $\theta$ and $\phi$ are fixed : with constant interval
- interval is determined by the number of division $N_{\phi}, N_{\theta}$
- neiboring rows are pulled as half of $\phi$ interval

$$\begin{cases} r_{ij} \text{ : variable} \\ \phi_{ij} = (j\mod 2)\frac{\Delta\phi}{2}+i\Delta\phi \\ \theta_{ij}=j\Delta\theta \end{cases}$$

where $\Delta\phi:=\frac{2\pi}{N_{\phi}}, \Delta\theta:=\frac{\pi}{N_{\theta}}, i\in\mathbb{Z_{N_{\phi}}}, j\in\mathbb{Z_{N_{\theta}}}$

indexing

 

Facet Indexing : to calculate reflecting flux, we need to calculate the normal vector of each facets

in array of surface vectors, each vectors are recorded in [i, j] index, and sub-index follows the below picture

- lower triangle : 1 / upper triangle : 2

- leftmost vertex of lower triangle has corr. $i, j$ index

 

each normal vectors are calculated by cross product.

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Basic Structure : Ellipsoid

for cartesian : $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+1$

 

$r$ value for fixed $\theta, \phi$ : 

$$r(\phi, \theta)=\frac{abc}{\sqrt{ (bc\cos\phi\sin\theta)^2 + (ca\sin\phi\sin\theta)^2 + (ab\cos\theta)^2 }}$$