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2021-02-28 12:28:41 -08:00
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docs/chapters/bsplined/content.en-GB.md
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@@ -16,7 +16,7 @@ Thankfully, much like as was the case for Bézier curves, the derivative of a B-
where
\[
P_i \prime = \frac{d}{knot_{i+d+1} - knot_{i+1}} (P_{i+1} - P_i)
P_i \prime = \frac{d}{\textit{knot}_{i+d+1} - \textit{knot}_{i+1}} (P_{i+1} - P_i)
\]
@@ -28,7 +28,7 @@ As a concrete example, let's look at cubic (=degree 3) B-Spline with five coordi
\begin{array}{l}
d = 3, \\
P = {(50,240), (185,30), (320,135), (455,25), (560,255)}, \\
knots = {0,1,2,3,4,5,6,7,8}
\textit{knots} = {0,1,2,3,4,5,6,7,8}
\end{array}
\]
@@ -38,20 +38,20 @@ Applying the above knowledge, we end up with a new B-Spline of degree <em>d-1</e
\[
\begin{array}{l}
P_0 \prime = \frac{d}{knot_{i+d+1} - knot_{i+1}} (P_{i+1} - P_i)
= \frac{3}{knot_{4} - knot_{1}} (P_1 - P_0)
P_0 \prime = \frac{d}{\textit{knot}_{i+d+1} - \textit{knot}_{i+1}} (P_{i+1} - P_i)
= \frac{3}{\textit{knot}_{4} - \textit{knot}_{1}} (P_1 - P_0)
= \frac{3}{3} (P_1 - P_0)
= (135, -210) \\
P_1 \prime = \frac{d}{knot_{i+d+1} - knot_{i+1}} (P_{i+1} - P_i)
= \frac{3}{knot_{5} - knot_{2}} (P_2 - P_1)
P_1 \prime = \frac{d}{\textit{knot}_{i+d+1} - \textit{knot}_{i+1}} (P_{i+1} - P_i)
= \frac{3}{\textit{knot}_{5} - \textit{knot}_{2}} (P_2 - P_1)
= \frac{3}{3} (P_2 - P_1)
= (135, 105) \\
P_2 \prime = \frac{d}{knot_{i+d+1} - knot_{i+1}} (P_{i+1} - P_i)
= \frac{3}{knot_{6} - knot_{3}} (P_3 - P_2)
P_2 \prime = \frac{d}{\textit{knot}_{i+d+1} - \textit{knot}_{i+1}} (P_{i+1} - P_i)
= \frac{3}{\textit{knot}_{6} - \textit{knot}_{3}} (P_3 - P_2)
= \frac{3}{3} (P_3 - P_2)
= (135, -110) \\
P_3 \prime = \frac{d}{knot_{i+d+1} - knot_{i+1}} (P_{i+1} - P_i)
= \frac{3}{knot_{7} - knot_{4}} (P_4 - P_3)
P_3 \prime = \frac{d}{\textit{knot}_{i+d+1} - \textit{knot}_{i+1}} (P_{i+1} - P_i)
= \frac{3}{\textit{knot}_{7} - \textit{knot}_{4}} (P_4 - P_3)
= \frac{3}{3} (P_4 - P_3)
= (105, 230) \\
\end{array}
@@ -63,7 +63,7 @@ So, we end up with a derivative that has as parameters:
\begin{array}{l}
d = 3, \\
P = {(50,240), (185,30), (320,135), (455,25), (560,255)}, \\
knots = {0,1,2,3,4,5,6,7,8}
\textit{knots} = {0,1,2,3,4,5,6,7,8}
\end{array}
\]