Hi!(adsbygoogle = window.adsbygoogle || []).push({});

I have a bezier curve defined by:

[tex]

\vec{b}(t) = (x(t), y(t))

[/tex]

where

[tex]

\begin{array}{lcl}

x(t) &=& a_xt^3 + b_xt^2 + c_xt + x_0 \\

y(t) &=& a_yt^3 + b_yt^2 + c_yt + y_0

\end{array}

[/tex]

for [tex]t \in \lbrack 0, 1 \rbrack[/tex]. All constants are computed from vertices on the curve and control points associated with those vertices.

For an arbitrary point [tex]\vec{r}\in\mathbb{R}^2[/tex] I want to find all points [tex]\vec{b}_{t_0}\in\vec{b}(t)[/tex] (if any) that satisfies

[tex]\vec{b}_{t_0} + \nabla\vec{b}(t_0)s = \vec{r}[/tex]

for some [tex]s[/tex].

Not sure I'm correct here, but as fas as I remember, [tex]\nabla \vec{b}(t)[/tex] is the normal to the curve, right?

I don't know how to explain this in a better way...

For any point [tex]\vec{r}[/tex] I want to find all points [tex]\vec{b}(t_0)[/tex] such that the normal to [tex]\vec{b}(t_0)[/tex] intersects [tex]\vec{r}[/tex].

How can I do this?

Thanks in advance,

Nille

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# Distance to curve

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