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\input{psfig}
\title{\vspace{2.0in} 3 DIMENSIONAL DYNAMICAL SURFACE CONSTRUCTION WITH
BEZIER PATCHES \vspace{1.3in} }
\author{ Hasan Timucin OZDEMIR }
\date{\vspace{2.0in} \today}


\makeatother
\newenvironment{tex2html_wrap}{}{}
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\def\lthtmltypeout#1{{\let\protect\string\immediate\write\lthtmlwrite{#1}}}%
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\begin{document}
\pagestyle{empty}
{\newpage
\clearpage
\samepage \(O(n\log{n})\)
}

{\newpage
\clearpage
\samepage \(O(n)\)
}

{\newpage
\clearpage
\samepage \(P_{i} : (x_{i},y_{i})\)
}

{\newpage
\clearpage
\samepage \(P_{j} : (x_{j}, y_{j}) \)
}

{\newpage
\clearpage
\samepage \( P_{k} : (x_{k}, y_{k}) \)
}

{\newpage
\clearpage
\samepage \begin{displaymath}L(i,j) : \frac{x-x_{i}}{y-y_{i}} = \frac{x_{i}-x_{j}}{y_{i}-y_{j}} \end{displaymath}
}

{\newpage
\clearpage
\samepage \(P_{k}\)
}

{\newpage
\clearpage
\samepage \(P_{k}^{\prime}\)
}

{\newpage
\clearpage
\samepage \(P_{k}^{\prime}\)
}

{\newpage
\clearpage
\samepage \(x_{k}\)
}

{\newpage
\clearpage
\samepage \(P_{k}^{\prime}\)
}

{\newpage
\clearpage
\samepage \begin{displaymath}\frac{x-x_{i}}{y-y_{i}} = \frac{x_{i}-x_{j}}{y_{i}-y_{j}} \end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\frac{y_{i}-y_{j}}{x_{i}-x_{j}} \cdot (x_{k}-x_{i}) = y-y_{i} \end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\frac{y_{i}-y_{j}}{x_{i}-x_{j}} \cdot (x_{k}-x_{i}) + y_{i} = y\end{displaymath}
}

{\newpage
\clearpage
\samepage \(d_{v}\)
}

{\newpage
\clearpage
\samepage \begin{displaymath}d_{v} = \mid y-y_{k} \mid = \mid \frac{y_{i}-y_{j}}{x_{i}-x_{j}}
 \cdot (x_{k}-x_{i}) + y_{i} - y_{k} \mid \end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}L(i,j) : \frac{y_{i}-y_{j}}{x_{i}-x_{j}} \cdot x -
\frac{y_{i}-y_{j}}{x_{i}-x_{j}} \cdot x_{i}) + y_{i} = y\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}L(i,j,k) : y = \frac{y_{i}-y_{j}}{x_{i}-x_{j}} \cdot x + b\end{displaymath}
}

{\newpage
\clearpage
\samepage \(P_{k}\)
}

{\newpage
\clearpage
\samepage \(P_{k}\)
}

{\newpage
\clearpage
\samepage \begin{displaymath}y_{k} = \frac{y_{i}-y_{j}}{x_{i}-x_{j}} \cdot x_{k} + b
\Rightarrow b = y_{k} - \frac{y_{i}-y_{j}}{x_{i}-x_{j}} \cdot x_{k}\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}D_{L(i,j)} : \left( \frac{y_{i}-y_{j}}{x_{i}-x_{j}} ,
y_{i}-\frac{y_{i}-y_{j}}{x_{i}-x_{j}} \cdot x_{k} \right) \end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}D_{L(i,j,k)} : \left( \frac{y_{i}-y_{j}}{x_{i}-x_{j}} ,
y_{k}-\frac{y_{i}-y_{j}}{x_{i}-x_{j}} \cdot x_{k} \right) \end{displaymath}
}

{\newpage
\clearpage
\samepage \(d_{v}^{\prime}\)
}

{\newpage
\clearpage
\samepage \begin{displaymath}d_{v}^{\prime} = \mid y_{i}-\frac{y_{i}-y_{j}}{x_{i}-x_{j}} \cdot x_{i} -
y_{k} + \frac{y_{i}-y_{j}}{x_{i}-x_{j}} \cdot x_{k} \mid \end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}d_{v}^{\prime} = \mid y_{i} + \frac{y_{i}-y_{j}}{x_{i}-x_{j}} \cdot ( x_{k}-x_{i} ) - y_{k} \mid \end{displaymath}
}

{\newpage
\clearpage
\samepage \(d_{v}\)
}

{\newpage
\clearpage
\samepage \(d_{v}^{\prime}\)
}

{\newpage
\clearpage
\samepage \(O(\log{n})\)
}

{\newpage
\clearpage
\samepage \(O(n)\)
}

{\newpage
\clearpage
\samepage \( O(n)\)
}

{\newpage
\clearpage
\samepage \(O(\log{n})\)
}

\stepcounter{section}

\end{document}
