Cubic Polynomial/ Circle intersection(s)

[markraz]

Hi, How does one find the intersection of a Cubic Polynomial and a Circle?? Is this something that can be done will simple algebra?
I could only find parabola/circle intersection.

Thanks in advance

 

[Dr.Peterson]

You'd write the equations and try to solve them as a system. It may or may not be solvable; you may end up with a 6th degree equation that can't be solved exactly, but if it reduces to a cubic polynomial, the work may be difficult but theoretically possible. Or you might get a special case that is easy.

Can you give a specific example of what you want to do (and perhaps why)?

 

[JeffM]

There is no guarantee that there even is an intersection between a cubic and any conic section. You need to put more specifications on the problem. At this point, you have not even said whether the cubic has local extrema or not. If the cubic has no local extrema, there are no, 1 or at most 2 points of intersection with a circle. If the cubic has local extrema, there may be as many as 6 points of intersection and as few as none.

 

[markraz]

You'd write the equations and try to solve them as a system. It may or may not be solvable; you may end up with a 6th degree equation that can't be solved exactly, but if it reduces to a cubic polynomial, the work may be difficult but theoretically possible. Or you might get a special case that is easy.

Can you give a specific example of what you want to do (and perhaps why)?

Thanks appreciate it. So I have a bunch of points that I am fitting to a cubic polynomial ax^3+bx^2+cx+d. I was able to find some good examples of curve fitting using Vandermonde method. After I fit the polynomial there will be circles that will intersect this polynomial. So i need to find these intersection points as seen below since the polynomial will clip these circles. Thanks in advance

 

[JeffM]

I am still unsure what you are asking.

These cubics look as though they have slopes close to zero, the circles have very small radii relative to the scale at which you are looking at the cubic, and the number of intersection points exceeds zero but does not exceed 2. Is that what you are looking for: a mathematical description of the properties of the infinite number of circles arising in such circumstances?

I doubt how you estimated the cubic has anything to do with defining the set of circles with a radius less than or equal to some arbitrary positive number and two intersections with the cubic although the method of estimation may be important with respect to the reliability of the solution set.

 

[markraz]

I am still unsure what you are asking.

These cubics look as though they have slopes close to zero, the circles have very small radii relative to the scale at which you are looking at the cubic, and the number of intersection points exceeds zero but does not exceed 2. Is that what you are looking for: a mathematical description of the properties of the infinite number of circles arising in such circumstances?

Hi thanks, I think I am trying to do something like this quadratic curve/circle intersection.

Crossing Curves: Finding the Intersections of Parabolas and Circles - dummies

When a parabola and circle intersect, the possibilities for their meeting are many and varied. The two curves can intersect in as many as four different points, or maybe three, or just two or even just one point. Keep your options open and be alert for as many common solutions as possible (right...

www.dummies.com

In my case I want to use a cubic curve

Thanks for you help. appreciate it

 

[JeffM]

Thanks for the citation. It helps understand where you are coming from.

It points out that there are five possible situations with the intersections of a quadratic and a circle. There are seven with respect to the intersections of a cubic and a circle. Moreover, as Dr. Peterson has pointed out, it is not soluble in general because it involves finding the roots of a polynomial of degree 6.

Your picture, however, implied that you were looking for something less than a totally general solution. I suspect that the problem becomes at least a bit more tractable if we put some restraints on it. For an example of what I am thinking about, if you are looking for exactly one or two intersections in a domain where the slope of the cubic is nowhere negative and the circle is fully contained within that domain, I have an idea that MAY work with some help from calculus. I haven't tried it out because I have no idea whether that fits your needs or not.

 

[markraz]

Thanks for the citation. It helps understand where you are coming from.

It points out that there are five possible situations with the intersections of a quadratic and a circle. There are seven with respect to the intersections of a cubic and a circle. Moreover, as Dr. Peterson has pointed out, it is not soluble in general because it involves finding the roots of a polynomial of degree 6.

Your picture, however, implied that you were looking for something less than a totally general solution. I suspect that the problem becomes at least a bit more tractable if we put some restraints on it. For an example of what I am thinking about, if you are looking for exactly one or two intersections in a domain where the slope of the cubic is nowhere negative and the circle is fully contained within that domain, I have an idea that MAY work with some help from calculus. I haven't tried it out because I have no idea whether that fits your needs or not.

Thanks appreciate it. This math problem is just something I am using within a computer program, that I am writing for fun. So using computer algorithms I was able to solve the problem without issues. But as a learning experience I was just curious how to do this by hand using algebra (or whatever else). So on that note I appreciate your help and time. At this point I think this would be way beyond my math skills to solve manually. even if you outlined it for me. Again I appreciate all your help and Dr. Peterson's help