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chapter0/bezier-curves.md

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# Bézier curves
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*Author: Ahmed Ratnani*
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We recall the definition of a Bézier curve:
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chapter0/bezier.md

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# Bernstein polynomials
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Without loss of generality, we restrict to the case of the unit interval, namely $a=0$ and $b=1$.
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In figure (Fig. \ref{fig:bernstein-polynomials}), we plot the first sixth Bernstein polynomials.

chapter0/bsplines-curves.md

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# B-Splines curves
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Let $(\mathbf{P}_i)_{ 0 \leqslant i \leqslant n}\in \mathbb{R}^d$ be a sequence of control points. Following the same approach as for Bézier curves, we define B-Splines curves as
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chapter0/bsplines-operations.md

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# Fundamental geometric operations for B-Splines
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Having more control on a curve, adding new control points, can be done in two different ways:
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chapter0/bsplines-surfaces.md

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# B-Splines surfaces
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The B-spline surface in $\mathbb{R}^d$ associated to knots $(T_u, T_v)$ where $T_u=(u_i)_{0\leqslant i \leqslant n_u + p_u + 1}$ and $T_v=(v_i)_{0\leqslant i \leqslant n_v + p_v + 1}$, and control points $(\mathbf{P}_{ij})_{ 0 \leqslant i \leqslant n_u, 0 \leqslant j \leqslant n_v}$ is defined by :
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chapter0/bsplines.md

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# B-Splines
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Given a subdivision $\{x_0 < x_1 < \cdots < x_r\}$ of the interval $I = [x_0, x_r]$, the \textbf{Schoenberg space} is the space of piecewise polynomials of degree $p$, on the interval $I$ and given regularities $\{k_1, k_2, \cdots, k_{r-1}\}$ at the internal points $\{x_1, x_2, \cdots, x_{r-1}\}$.
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chapter0/cad.md

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# Computer Aided Design
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TODO

chapter0/data-structure.md

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# Data Structure
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In the sequel, we shall use **StencilMatrix** and **StencilVector** from the **psydac** library.
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chapter0/fem.md

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# Introduction to B-Splines FEM
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Let $\Omega \subset \mathbb{R}^d$ be a computational domain that is the image of a logical domain $\mathcal{P}$, *i.e.* a unit line (in *1d*), square (in *2d*) or a cube (in *3d*) with a **mapping** function
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chapter0/howto.md

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# What to expect from IGA-Python
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