Minimum distance between a point and a geometric locus

[aliekor]

Hi guys I have a problem to solve, I'd like to find the minimum distance between a point and a geometric locus described in closed form, for example the intersection of two circles:

p= point coordinate
p1= center coordinate circle 1
p2=center coordinate circle 2
r1=radius of circle 1
r2=radius of circle 2

|p1|^2-r1^2<0 (region inside circle 1)
|p2|^2-r2^2<0 (region inside circle 2)
how can i find an expression that gave me the minimum distance between p and the intersection of the two circles?

thank you in advance

 

[chiro]

Hey aliekor and welcome to the forums.

Depending on the level of knowledge you have you can do it in different ways. If you know calculus you can form what's called a minimization problem and solve for the first derivatives to be 0 and then obtain the solutions and screen out the ones that aren't required.

Have you taken calculus?

 

[aliekor]

no is not possible. I need a closed form for my problem. actually I need to know if p is inside the intersection of the circles, I thought that a minimum distance negative can give me this information.

 

[chiro]

There is no reason why you can't get a closed form answer.

Also do you have an analytic expression (equality or inequality) for p? You have given constraints for everything else except for what p is mathematically.

 

[aliekor]

No I don't. I need just the measure of the distance between a given p and the geometric locus

 

[chiro]

This sounds like a high school problem but I don't know the definition of a locus (and haven't seen it in more than 10 years): can you give the mathematical definition of a locus from a point p?

 

[aliekor]

http://en.wikipedia.org/wiki/Locus_(mathematics )

 

[chiro]

Use the definition of the norm: in two dimensions, the norm is (x - a)^2 + (y - b)^2 from some object defined at a variable position (x,y) to a point (a,b). Use some simple calculus to minimize this distance under your constraints. Minimize the square of the residual since it's easier and will correspond to the minimum of the distance since the quadratic function monotonically increases for all positive values of the distance.

This is the approach used for commonly solving these problems. You need to minimize (x - a)^2 + (y - b)^2 = 0 subject to your constraints that you have given by using Lagrange Multipliers.

 

[aliekor]

I'll find that the minimum distance is 0... Already have the coordinate of the point p. i have to find the minimum distance and not a point p that minimize the distance.

 

[HallsofIvy]

Part of the problem is that you have asked for the distance from p to "intersection of the two circles" without giving any condition that the two circles do intersect! You need that the distance between p1 and p2 is less than r1+ r2. And, then, in general, there will be two points of intersection. Did you intend to require that the two circles be tangent to one another?