From 082e5348971e4f974538fda07b803f67e273d585 Mon Sep 17 00:00:00 2001
From: Thomas FORGIONE <thomas.forgione@gmail.com>
Date: Thu, 12 Mar 2015 21:43:03 +0100
Subject: [PATCH] Splines.tex

---
 report/subsections/splines.tex | 38 +++++++++++++++++++++++++++++++++-
 1 file changed, 37 insertions(+), 1 deletion(-)

diff --git a/report/subsections/splines.tex b/report/subsections/splines.tex
index e965047..91750d3 100644
--- a/report/subsections/splines.tex
+++ b/report/subsections/splines.tex
@@ -1 +1,37 @@
-Hello
+The splines are a necessary element for everything that will follow. This is
+why we tried to make it as generic as possible. For this, we used a C++11
+feature called \emph{variadic templates}. A spline is composed of control
+points (represented as C++ tuples), a nodes vector and degree. The class we
+made looks like this
+
+\begin{lstlisting}[language=c++]
+template<typename... Types>
+class Spline
+{
+    public:
+        std::tuple<Types...> operator()(float t);
+        std::tuple<Types...> prime(float t);
+
+    private:
+        std::vector<std::tuple<Types...>> controlPoints;
+        std::vector<float> nodes;
+        int degree;
+};
+\end{lstlisting}
+
+We redefined the \texttt{operator()} to be able to \emph{evaluate} a spline at
+a certain time, and the result will be an object which is a barycenter of those
+control points. We also made a member function \texttt{prime} to be able to
+compute the derivative of a spline.
+
+The \texttt{operator()} computes the barycent of the control points like this
+$$C(t) = \sum_{i=0}^m P_i N_i^n(t)$$
+where $(P_i)_{i\in [[0,m]]}$ are the $m+1$ control points, $n$ is the degree of
+the spline and
+$$N_i^0(t) = \left\{\begin{matrix}1&\text{if } t \in [t_i, t_{i+1}] \\ 0 & \text{otherwise} \end{matrix} \right.$$
+$$ N_i^k(t) = \frac{t-t_i}{t_{i+k}-t_{i}} N_i^{k-1}(t) + \frac{t_{i+k+1} - t}{t_{i+k+1}-t_{i+1}} N_{i+1}^{k-1}(t) \quad\text{assuming} \quad \frac{0}{0} = 0$$
+
+The \texttt{prime} member function computes the derivative like this
+$$C'(t) = \sum_{i=0}^m N_i^{\prime n} (t) P_i $$
+with
+$$ N_i^{\prime n}(t) = \frac{n}{t_{i+n}-t_{i}}N_i^{n-1}(t) - \frac{n}{t_{i+n+1}-t_{i+1}}N_{i+1}^{n-1}(t)$$