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XeLaTeX interpration of LaTeX for localization
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@@ -27,32 +27,32 @@ However as we've seen in the section on aligning, aligning lets us simplify thin
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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:
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\[
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\begin{align*}
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\begin{aligned}
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& Bézier(t) = x_1(1-t)^3 + 3x_2(1-t)^2t + 3x_3(1-t)t^2 + x_4t^3 \\
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& Bézier^\prime(t) = a(1-t)^2 + 2b(1-t)^t + ct^2\ \left\{ a=3(x_2-x_1),b=3(x_3-x_2),c=3(x_4-x_3) \right\} \\
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& Bézier^{\prime\prime}(t) = u(1-t) + vt\ \left\{ u=2(b-a),v=2(c-b) \right\}\
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\end{align*}
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\end{aligned}
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\]
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And of course the same functions for *y*:
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\[
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\begin{align*}
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\begin{aligned}
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& Bézier(t) = y_1(1-t)^3 + 3y_2(1-t)^2t + 3y_3(1-t)t^2 + y_4t^3 \\
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& Bézier^\prime(t) = d(1-t)^2 + 2e(1-t)^t + ft^2\\
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& Bézier^{\prime\prime}(t) = w(1-t) + zt
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\end{align*}
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\end{aligned}
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\]
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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:
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\[
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\begin{array}
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-18 t^2 x_2 y_1+36 t^2 x_3 y_1-18 t^2 x_4 y_1+18 t^2 x_1 y_2-54 t^2 x_3 y_2 \\
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+36 t^2 x_4 y_2-36 t^2 x_1 y_3+54 t^2 x_2 y_3-18 t^2 x_4 y_3+18 t^2 x_1 y_4 \\
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-36 t^2 x_2 y_4+18 t^2 x_3 y_4+36 t x_2 y_1-54 t x_3 y_1+18 t x_4 y_1-36 t x_1 y_2 \\
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+54 t x_3 y_2-18 t x_4 y_2+54 t x_1 y_3-54 t x_2 y_3-18 t x_1 y_4+18 t x_2 y_4 \\
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-18 x_2 y_1+18 x_3 y_1+18 x_1 y_2-18 x_3 y_2-18 x_1 y_3+18 x_2 y_3
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\begin{array}{lclclclclcl}
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-18 t^2 x_2 y_1 &+& 36 t^2 x_3 y_1 &-& 18 t^2 x_4 y_1 &+& 18 t^2 x_1 y_2 &-& 54 t^2 x_3 y_2 &&\\
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+36 t^2 x_4 y_2 &-& 36 t^2 x_1 y_3 &+& 54 t^2 x_2 y_3 &-& 18 t^2 x_4 y_3 &+& 18 t^2 x_1 y_4 &&\\
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-36 t^2 x_2 y_4 &+& 18 t^2 x_3 y_4 &+& 36 t x_2 y_1 &-& 54 t x_3 y_1 &+& 18 t x_4 y_1 &-& 36 t x_1 y_2 \\
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+54 t x_3 y_2 &-& 18 t x_4 y_2 &+& 54 t x_1 y_3 &-& 54 t x_2 y_3 &-& 18 t x_1 y_4 &+& 18 t x_2 y_4 \\
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-18 x_2 y_1 &+& 18 x_3 y_1 &+& 18 x_1 y_2 &-& 18 x_3 y_2 &-& 18 x_1 y_3 &+& 18 x_2 y_3
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\end{array}
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\]
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