Closest line from a point to a curve in R^2

[RBG]

Given a parametrized curve $X(t):I\to\mathbb{R}^2$ I am trying to show given a fixed point $p$, and the closest point on $X$ to $p$, $X(t_0)$, the line between the point and the curve is perpendicular to the curve. My only idea so far is to show that $(p-X(t))\cdot(\frac{X'(t)}{||X'(t)||})=0$. But in general, I don't see why this would be true? It seems clear geometrically, but obviously that's not an argument. Any hints?

 

[Simon Bridge]

Use the definition of "perpendicular".

 

[RBG]

Use the definition of "perpendicular".

Isn't that the dot product is zero? That or that the slopes of the tangent lines are inverse reciprocals of one another. But I don't see how the latter definition can be applied...

 

[Simon Bridge]

Dot products of what?
How can a line be perpendicular to a curve?

 

[S.G. Janssens]

the closest point on $X$ to $p$

I take it that you assume the curve is smooth. Even then, note that the closest point may not be unique. Also, when $I$ is not compact, a closest point is not guaranteed to exist, as $t \mapsto \|p - X(t)\|$ need not assume its infimum over $I$ then.

 

[RBG]

Dot products of what?
How can a line be perpendicular to a curve?

Dot product of the tangent vector, right? So above $(p-X(t_0))$ is the vector between point and curve and $X'(t_0)$ is the tangent vector. But I don't see why should $p\dot X'(t_0)-X(t_0)X'(t_0)=0$

 

[Simon Bridge]

... given a point on the curve, how would you tell that it is the closest point?
Given the curve C and a point P, how would you usually go about finding the closest point on C to P?

 

[RBG]

I take it that you assume the curve is smooth. Even then, note that the closest point may not be unique. Also, when $I$ is not compact, a closest point is not guaranteed to exist.

We are assuming $X(t)$ is a regular parametrized curve and $t_0$ is not an endpoint of $I$.

 

[Simon Bridge]

Dot product of the tangent vector, right? So above $(p-X(t_0))$ is the vector between point and curve and $X'(t_0)$ is the tangent vector. But I don't see why should $p\dot X'(t_0)-X(t_0)X'(t_0)=0$

... why not just work through the expression that arises from the definition you just used?

 

[RBG]

... given a point on the curve, how would you tell that it is the closest point?
Given the curve C and a point P, how would you usually go about finding the closest point on C to P?

You would take the derivative of $||p-X(t)||$ and minimize it. Then check which points are minimal, right?

 

[Ray Vickson]

You would take the derivative of $||p-X(t)||$ and minimize it. Then check which points are minimal, right?

That is how I would do it, but I would make the problem easier by minimizing $|| p - X(t)||^2$ instead of $||p - X(t)||$. These problems are equivalent, in the sense that their $t$-solutions are the same.

 

[RBG]

... why not just work through the expression that arises from the definition you just used?

I really don't understand what you mean by this. I should use the fact that I am minimizing $|p-X(t)||$ somehow to reduce the dot product?

 

[RBG]

I really don't understand what you mean by this. I should use the fact that I am minimizing $|p-X(t)||$ somehow to reduce the dot product?

OOOOOOHHHHH... duh. Nevermind. Right... Thanks! Just do the calculation of taking the derivative