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2021-02-28 12:28:41 -08:00
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18
docs/chapters/pointvectors/content.en-GB.md
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@@ -4,9 +4,9 @@ If you want to move objects along a curve, or "away from" a curve, the two vecto
\[
\begin{matrix}
tangent_x(t) = B'_x(t) \\
\textit{tangent}_x(t) = B'_x(t) \\
\\
tangent_y(t) = B'_y(t)
\textit{tangent}_y(t) = B'_y(t)
\end{matrix}
\]
@@ -14,14 +14,14 @@ This gives us the directional vector we want. We can normalize it to give us uni
\[
\begin{matrix}
d = \left \| tangent(t) \right \| = \sqrt{B'_x(t)^2 + B'_y(t)^2} \\
d = \left \| \textit{tangent}(t) \right \| = \sqrt{B'_x(t)^2 + B'_y(t)^2} \\
\\
\hat{x}(t) = \left \| tangent_x(t) \right \|
=\frac{tangent_x(t)}{ \left \| tangent(t) \right \| }
\hat{x}(t) = \left \| \textit{tangent}_x(t) \right \|
=\frac{\textit{tangent}_x(t)}{ \left \| \textit{tangent}(t) \right \| }
= \frac{B'_x(t)}{d} \\
\\
\hat{y}(t) = \left \| tangent_y(t) \right \|
= \frac{tangent_y(t)}{ \left \| tangent(t) \right \| }
\hat{y}(t) = \left \| \textit{tangent}_y(t) \right \|
= \frac{\textit{tangent}_y(t)}{ \left \| \textit{tangent}(t) \right \| }
= \frac{B'_y(t)}{d}
\end{matrix}
\]
@@ -30,9 +30,9 @@ The tangent is very useful for moving along a line, but what if we want to move
\[
\begin{array}{l}
normal_x(t) = \hat{x}(t) \cdot \cos{\frac{\pi}{2}} - \hat{y}(t) \cdot \sin{\frac{\pi}{2}} = - \hat{y}(t) \\
\textit{normal}_x(t) = \hat{x}(t) \cdot \cos{\frac{\pi}{2}} - \hat{y}(t) \cdot \sin{\frac{\pi}{2}} = - \hat{y}(t) \\
\\
normal_y(t) = \underset{quarter~circle~rotation} {\underbrace{ \hat{x}(t) \cdot \sin{\frac{\pi}{2}} + \hat{y}(t) \cdot \cos{\frac{\pi}{2}} }} = \hat{x}(t)
\textit{normal}_y(t) = \underset{\textit{quarter circle rotation}} {\underbrace{ \hat{x}(t) \cdot \sin{\frac{\pi}{2}} + \hat{y}(t) \cdot \cos{\frac{\pi}{2}} }} = \hat{x}(t)
\end{array}
\]