Find Distance to Curve for Arbitrary Point - Nille

[LostInSpace]

Hi!

I have a bezier curve defined by:
$$\vec{b}(t) = (x(t), y(t))$$
where
$$\begin{array}{lcl} x(t) &=& a_xt^3 + b_xt^2 + c_xt + x_0 \\ y(t) &=& a_yt^3 + b_yt^2 + c_yt + y_0 \end{array}$$
for $$t \in \lbrack 0, 1 \rbrack$$. All constants are computed from vertices on the curve and control points associated with those vertices.

For an arbitrary point $$\vec{r}\in\mathbb{R}^2$$ I want to find all points $$\vec{b}_{t_0}\in\vec{b}(t)$$ (if any) that satisfies
$$\vec{b}_{t_0} + \nabla\vec{b}(t_0)s = \vec{r}$$
for some $$s$$.

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

I don't know how to explain this in a better way...
For any point $$\vec{r}$$ I want to find all points $$\vec{b}(t_0)$$ such that the normal to $$\vec{b}(t_0)$$ intersects $$\vec{r}$$.

How can I do this?

Thanks in advance,
Nille

 

[Timbuqtu]

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

No, $$\nabla \vec{b}(t)$$ (or rather $$\frac{d \vec{b}}{dt}(t)$$) is the tangent vector to the curve. I suppose you're looking for all $$t_0$$ for which the vector $$\vec{b}(t_0)-\vec{r}$$ is perpendicular to the tangent vector. This just means that there dotproduct is zero:

$$(\vec{b}(t_0)-\vec{r})\cdot\frac{d \vec{b}}{dt}(t_0)=0$$

 

[tacman]

Hi Nille,

Thank you for sharing your question with us. It seems like you are trying to find the distance from an arbitrary point \vec{r} to a bezier curve defined by \vec{b}(t). To find this distance, you can use the formula for the distance between a point and a curve, which is given by:

d = \frac{\lvert \vec{r} - \vec{b}(t_0) \rvert}{\lvert \nabla\vec{b}(t_0) \rvert}

where \lvert \cdot \rvert denotes the magnitude or length. In other words, you need to find the value of t_0 that minimizes the distance between \vec{r} and \vec{b}(t_0). This can be done by setting the derivative of d with respect to t_0 equal to 0 and solving for t_0.

I hope this helps. Let me know if you have any further questions. Good luck!