1
0
mirror of https://github.com/Pomax/BezierInfo-2.git synced 2025-08-26 01:44:03 +02:00

XeLaTeX interpration of LaTeX for localization

This commit is contained in:
Pomax
2017-02-26 22:27:36 -08:00
parent a889d16f36
commit ba54cc37f9
216 changed files with 477 additions and 404 deletions

View File

@@ -27,32 +27,32 @@ However as we've seen in the section on aligning, aligning lets us simplify thin
Of course, before we do our aligned check, let's see what happens if we compute the curvature function without axis-aligning. We start with the first and second derivatives, given our basis functions:
\[
\begin{align*}
\begin{aligned}
& Bézier(t) = x_1(1-t)^3 + 3x_2(1-t)^2t + 3x_3(1-t)t^2 + x_4t^3 \\
& Bézier^\prime(t) = a(1-t)^2 + 2b(1-t)^t + ct^2\ \left\{ a=3(x_2-x_1),b=3(x_3-x_2),c=3(x_4-x_3) \right\} \\
& Bézier^{\prime\prime}(t) = u(1-t) + vt\ \left\{ u=2(b-a),v=2(c-b) \right\}\
\end{align*}
\end{aligned}
\]
And of course the same functions for *y*:
\[
\begin{align*}
\begin{aligned}
& Bézier(t) = y_1(1-t)^3 + 3y_2(1-t)^2t + 3y_3(1-t)t^2 + y_4t^3 \\
& Bézier^\prime(t) = d(1-t)^2 + 2e(1-t)^t + ft^2\\
& Bézier^{\prime\prime}(t) = w(1-t) + zt
\end{align*}
\end{aligned}
\]
Asking a computer to now compose the *C(t)* function for us (and to expand it to a readable form of simple terms) gives us this rather overly complicated set of arithmetic expressions:
\[
\begin{array}
-18 t^2 x_2 y_1+36 t^2 x_3 y_1-18 t^2 x_4 y_1+18 t^2 x_1 y_2-54 t^2 x_3 y_2 \\
+36 t^2 x_4 y_2-36 t^2 x_1 y_3+54 t^2 x_2 y_3-18 t^2 x_4 y_3+18 t^2 x_1 y_4 \\
-36 t^2 x_2 y_4+18 t^2 x_3 y_4+36 t x_2 y_1-54 t x_3 y_1+18 t x_4 y_1-36 t x_1 y_2 \\
+54 t x_3 y_2-18 t x_4 y_2+54 t x_1 y_3-54 t x_2 y_3-18 t x_1 y_4+18 t x_2 y_4 \\
-18 x_2 y_1+18 x_3 y_1+18 x_1 y_2-18 x_3 y_2-18 x_1 y_3+18 x_2 y_3
\begin{array}{lclclclclcl}
-18 t^2 x_2 y_1 &+& 36 t^2 x_3 y_1 &-& 18 t^2 x_4 y_1 &+& 18 t^2 x_1 y_2 &-& 54 t^2 x_3 y_2 &&\\
+36 t^2 x_4 y_2 &-& 36 t^2 x_1 y_3 &+& 54 t^2 x_2 y_3 &-& 18 t^2 x_4 y_3 &+& 18 t^2 x_1 y_4 &&\\
-36 t^2 x_2 y_4 &+& 18 t^2 x_3 y_4 &+& 36 t x_2 y_1 &-& 54 t x_3 y_1 &+& 18 t x_4 y_1 &-& 36 t x_1 y_2 \\
+54 t x_3 y_2 &-& 18 t x_4 y_2 &+& 54 t x_1 y_3 &-& 54 t x_2 y_3 &-& 18 t x_1 y_4 &+& 18 t x_2 y_4 \\
-18 x_2 y_1 &+& 18 x_3 y_1 &+& 18 x_1 y_2 &-& 18 x_3 y_2 &-& 18 x_1 y_3 &+& 18 x_2 y_3
\end{array}
\]