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zE3Z_ooNPR7>3Dbr>HZJCQ;)Ua_M#BIoH2YM5MDD8pcJo(QZbEM8$XQo4UPA;@;4f` zZ)iu{tGQxo)vuYOTxyDSamyh>yO!SYB>#SyfY!`^MOK<#@HgAb=@MUQ0&j;fn zivxu^^W9tn6mSR1V%ho+2z8@=aF!-s$55ol)Qi)fP(Gt+;dX+oQg^~pDR2b&`FVsv L%*^r{3ZVZ5#;>z( diff --git a/exercise3/Theorie.tex b/exercise3/Theorie.tex index 5e3339b..b2e08f0 100644 --- a/exercise3/Theorie.tex +++ b/exercise3/Theorie.tex @@ -25,10 +25,24 @@ The more obtuse the triangle, the more acute the angle of the corner of the voro \subsection*{(b)} The formula for calculating the surface of a triangle is: $\frac{1}{2}*h*b$ As for the red triangle: $\frac{1}{2}*h*b$ -with: $h=\frac{||p_i - p_j||}{2}$ and $b=b_1 + b_2$ where $b_1$ is left part of the baseline facing $\alpha_{ij}$ and $b_2$ is the right one facing $\beta_{ij}$.\\ - -\subsection*{(c)} - +We define: $h=\frac{||\underline{p}_i - \underline{p}_j||}{2}$ and $b=b_1 + b_2$ where $b_1$ is left part of the baseline facing $\alpha_{ij}$ and $b_2$ is the right one facing $\beta_{ij}$. In addition we define the vertex of the traingle facing $\alpha_{ij}$ to be $A$ and the vertex of the triangle facing $\beta_{ij}$ to be $B$. The distance between $p_i$ and $A$ (R) is equal to $\frac{||\underline{p}_i-\underline{p}_j||}{2 sin(\alpha_{ij}}$. It can be assumed that the angle $\underline{p}_S \underline{A} \underline{p}_i=\alpha_{ij}$ with $p_S$ as the vertex where the red triangle crosses the edge from $\underline{p}_i$ to $\underline{p}_j$. As a result it can be assumed that $cos(\alpha_{ij})= \frac{b_1}{R}$ and also $b_1 = cos(\alpha_{ij}) \times R$\\ +As a result the surface of the left part of the red triangle can be calculated: +\begin{align*} +A_{\alpha}(i,j)&=\frac{1}{2}\times b_1 \times \frac{||\underline{p}_i- \underline{p}_j||}{2}\\ +A_{\alpha}(i,j)&=\frac{1}{4}\times cos(\alpha_{ij}) \times R\times ||\underline{p}_i- \underline{p}_j||\\ +A_{\alpha}(i,j)&=\frac{1}{4}\times \frac{cos(\alpha_{ij})}{2\times sin(\alpha_{ij})} \times R \times {||\underline{p}_i- \underline{p}_j||}^2\\ +A_{\alpha}(i,j)&=\frac{1}{8}\times cot(\alpha_{ij}) \times R \times {||\underline{p}_i- \underline{p}_j||}^2\\ +\end{align*} +The same goes for the right side and as a result if we combine both: +\begin{align*} +A(i,j)&=\frac{1}{8}\times cot(\alpha_{ij}) \times R \times {||\underline{p}_i- \underline{p}_j||}^2 + \frac{1}{8}\times cot(\beta_{ij}) \times R \times {||\underline{p}_i- \underline{p}_j||}^2\\ +A(i,j)&=\frac{1}{8}\times (cot(\alpha_{ij})+cot(\beta_{ij})) \times R \times {||\underline{p}_i- \underline{p}_j||}^2\\ +\end{align*} +If we sum up all the triangles T with the index $j$ of the voronoi-area this is the result: +\begin{align*} +A_{voronoi-area}(i)&=\sum_{j\in T(i)} \frac{1}{8}\times (cot(\alpha_{ij})+cot(\beta_{ij})) \times R \times {||\underline{p}_i- \underline{p}_j||}^2\\ +A_{voronoi-area}(i)&=\frac{1}{8}\times \sum_{j\in T(i)} (cot(\alpha_{ij})+cot(\beta_{ij})) \times R \times {||\underline{p}_i- \underline{p}_j||}^2\\ +\end{align*} \subsection*{3.1.2} \subsection*{(a)} We know every vertex($v$) has three edges($e$) and 2 faces($f$). In addition every edge($e$) can separated into two half edges ($e_h$), this means $e_h = 2*e = 6*v$.\\