AC 鋪築量計算公式說明
===

## 示意圖

> 收方測量曲線方程式示意圖


> 鋪築量斷面關係圖


## 參數定義

* define: section number $i$
* 斷面編號
* define: Space of after remove of section $S_{r,i}$
* 刨除後斷面測量點位集合
* space $S_r$ = {Vertex of Leveling} sorted by x
* Vertex = (distance, elevation)
* defint: Space of after finish of section $S_{f,i}$
* 鋪築後斷面測量點位集合
* space $S_f$ = {Vertex of Leveling} sorted by x
* Vertex = (distance, elevation)
* define: curve function of after remove of section $Y_r(x)_i$
* 刨除後斷面測量曲線方程式
* defint: curve function of after finish of section $Y_f(x)_i$
* 鋪築後斷面測量曲線方程式
* define: total Volumn $V_{s,det,i,i+1}$
* 第i,i+1斷面的體積，利用行列式計算
* define: total Volumn $V_{s,int,i,i+1}$
* 第i,i+1斷面的體積，利用積分計算
* define: weight of section length $w_{i,i+1}$
* 長度權重

* $merge(S_1,S_2)$
* make two sorted space into $S_n$ by Determinant rule
* 行列式算面積請參考高中數學

## 計算式

* $V_{s,det,i,i+1} = 0.5 * w_{i,i+1} * (0.5*det(merge(S_{r,i},S_{f,i})) + 0.5*det(merge(S_{r,i+1}, S_{f, i+1})))$
* $V_{s,int,i,i+1} = 0.5 * w_{i,i+1} * (\int (Y_f(x)_{i+1}-Y_r(x)_{i+1}) dx) + \int (Y_f(x)_i-Y_r(x)_i) dx)$
* $V_s = \sum (V_{s,det,i,i+1} + V_{s,int,i,i+1}) / 2$
* i, from section 0 to section n-1
* for all i, $0 < i < n$, step of distance by split