40 lines
1.7 KiB
TeX
40 lines
1.7 KiB
TeX
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The splines are a necessary element for everything that will follow. This is
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why we tried to make it as generic as possible. For this, we used a C++11
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feature called \emph{variadic templates}. A spline is composed of control
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points (represented as C++ tuples), a nodes vector and degree. The class we
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made looks like this
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\begin{lstlisting}[language=c++]
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template<typename... Types>
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class Spline
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{
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public:
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std::tuple<Types...> operator()(float t);
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std::tuple<Types...> prime(float t);
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private:
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std::vector<std::tuple<Types...>> controlPoints;
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std::vector<float> nodes;
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int degree;
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};
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\end{lstlisting}
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We redefined the \texttt{operator()} to be able to \emph{evaluate} a spline at
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a certain time, and the result will be an object which is a barycenter of those
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control points. We also made a member function \texttt{prime} to be able to
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compute the derivative of a spline.
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The \texttt{operator()} computes the barycent of the control points like this
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$$C(t) = \sum_{i=0}^m P_i N_i^n(t)$$
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where $(P_i)_{i\in [[0,m]]}$ are the $m+1$ control points, $n$ is the degree of
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the spline and
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$$N_i^0(t) = \left\{\begin{matrix}1&\text{if } t \in [t_i, t_{i+1}] \\ 0 & \text{otherwise} \end{matrix} \right.$$
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$$ N_i^k(t) = \frac{t-t_i}{t_{i+k}-t_{i}} N_i^{k-1}(t) + \frac{t_{i+k+1} - t}{t_{i+k+1}-t_{i+1}} N_{i+1}^{k-1}(t) \quad\text{assuming} \quad \frac{0}{0} = 0$$
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The \texttt{prime} member function computes the derivative like this
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$$C'(t) = \sum_{i=0}^m N_i^{\prime n} (t) P_i $$
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with
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$$ N_i^{\prime n}(t) = \frac{n}{t_{i+n}-t_{i}}N_i^{n-1}(t) - \frac{n}{t_{i+n+1}-t_{i+1}}N_{i+1}^{n-1}(t)$$
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These formulas have been taken from the reference \cite{spline}.
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