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Pomax
2020-10-18 14:27:57 -07:00
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docs/chapters/bsplines/content.en-GB.md
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@@ -76,7 +76,7 @@ That looks complicated, but it's not. Computing alpha is just a fraction involvi
Of course, the recursion does need a stop condition:
\[
d^k_0(t) = 0, \ d^0_i(t) = N_{i,1}(t) =
d^k_0(t) = 0, ~d^0_i(t) = N_{i,1}(t) =
\left\{\begin{matrix}
1& \text{if } t \in [knot_i,knot_{i+1}) \\
0& \text{otherwise}
@@ -90,39 +90,39 @@ Thanks to Cox and de Boor, we can compute points on a B-Spline pretty easily usi
\[
d^3_3 = \left \{
\begin{aligned}
\alpha^3_3 \times d^2_3, & \ \textit{ with } d^2_3 = \left \{
\alpha^3_3 \times d^2_3, & ~\textit{ with } d^2_3 = \left \{
\begin{aligned}
\alpha^2_3 \times d^1_3, & \ \textit{ with } d^1_3 =
\alpha^2_3 \times d^1_3, & ~\textit{ with } d^1_3 =
\left \{
\begin{aligned}
\alpha^1_3 \times d^0_3, & \ \textit{ with } d^0_3 \textit{ either 0 or 1} \\
\alpha^1_3 \times d^0_3, & ~\textit{ with } d^0_3 \textit{ either 0 or 1} \\
+ & \\
\left ( 1 - \alpha^1_3 \right ) \times d^0_2, & \ \textit{ with } d^0_2 \textit{ either 0 or 1} \\
\left ( 1 - \alpha^1_3 \right ) \times d^0_2, & ~\textit{ with } d^0_2 \textit{ either 0 or 1} \\
\end{aligned}
\right . \\
+ & \\
\left ( 1 - \alpha^2_3 \right ) \times d^1_2, & \ \textit{ with } d^1_2 =
\left ( 1 - \alpha^2_3 \right ) \times d^1_2, & ~\textit{ with } d^1_2 =
\left \{
\begin{aligned}
\alpha^1_2 \times d^0_2 & \\
+ & \\
\left ( 1 - \alpha^1_2 \right ) \times d^0_1, & \ \textit{ with } d^0_1 \textit{ either 0 or 1} \\
\left ( 1 - \alpha^1_2 \right ) \times d^0_1, & ~\textit{ with } d^0_1 \textit{ either 0 or 1} \\
\end{aligned}
\right . \\
\end{aligned}
\right . \\
+ & \\
\left ( 1 - \alpha^3_3 \right ) \times d^2_2, & \ \textit{ with } d^2_2 = \left \{
\left ( 1 - \alpha^3_3 \right ) \times d^2_2, & ~\textit{ with } d^2_2 = \left \{
\begin{aligned}
\alpha^2_2 \times d^1_2 & \\
& \\
+ & \\
\left ( 1 - \alpha^2_2 \right ) \times d^1_1, & \ \textit{ with } d^1_1 =
\left ( 1 - \alpha^2_2 \right ) \times d^1_1, & ~\textit{ with } d^1_1 =
\left \{
\begin{aligned}
\alpha^1_1 \times d^0_1 \\
+ & \\
\left ( 1 - \alpha^1_1 \right ) \times d^0_0, & \ \textit{ with } d^0_0 \textit{ either 0 or 1} \\
\left ( 1 - \alpha^1_1 \right ) \times d^0_0, & ~\textit{ with } d^0_0 \textit{ either 0 or 1} \\
\end{aligned}
\right . \\
\end{aligned}