Finding intersection of two algebraic curves

In summary, the conversation discusses the difficulty of finding intersections between two algebraic curves, particularly for higher degree curves. The Abel-Ruffini theorem states that there is no general, numeric approach for finding these intersections, even for quintic equations. One potential solution could be to use the resultant of the two polynomials, as suggested by another participant in the conversation. However, this may not be suitable for finding intersections between the real and imaginary sheets of the curves, as demonstrated in a specific example provided by the original poster.
  • #1
aheight
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TL;DR Summary
Is there a standard numeric approach to finding the intersection of two algebraic curves?
Given two algebraic curves:

##f_1(z,w)=a_0(z)+a_1(z)w+\cdots+a_n(z)w^n=0##
##f_2(z,w)=b_0(z)+b_1(z)w+\cdots+b_k(z)w^k=0##

Is there a general, numeric approach to finding where the first curve ##w_1(z)## intersects the second curve ##w_2(z)##? I know for low degree like quadratic or cubics can find the intersection by brute force but was wondering if there is a more general approach for higher degrees say 10 or 12 each?
 
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  • #2
Not expressed with radicals, at least.
If there would be then quintic equations would have such a solution. They do not in general: Abel-Ruffini theorem
 
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  • #3
mfb said:
Not expressed with radicals, at least.
If there would be then quintic equations would have such a solution. They do not in general: Abel-Ruffini theorem

It appears to be a difficult problem even numerically. Was just wondering how others might approach it (numerically).
 
  • #4
@aheight -- You should look at the resultant of the two polynomials. See the Resultant - Wikipedia article for more information, especially the section on Algebraic Geometry. HTH
 
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  • #5
Petek said:
@aheight -- You should look at the resultant of the two polynomials. See the Resultant - Wikipedia article for more information, especially the section on Algebraic Geometry. HTH
Thanks for that. However, perhaps I should have stated above I'm interested in computing the intersections of the real and imaginary sheets of both algebraic curves. Consider a simple case I'm working on:

$$
\begin{align*}
f1(z,w)&=(-6z/5-2z^2+z^4)+(2/5-2 z^2/5)w+1/25 w^2=0\\
f2(z,w)&=(-1+z^2)-6/5 w=0
\end{align*}
$$
If I plot the real sheets of ##w_1## from f1 as the yellow and red surfaces and the real sheet of ##w_2## from f2 as the orange surface in the plot below, I obtain their intersections as the white curves. The white curves I computed by solving simple simultaneous equations for this simple case which I would not be able to do with higher degree curves and was wondering if there is a systematic way to find the intersections for the higher degree cases.

Guess I mean I don't see how the resultant can be used to find the white curves.
intersectionPlot.jpg
 
Last edited:

Related to Finding intersection of two algebraic curves

1. How do you find the intersection of two algebraic curves?

The intersection of two algebraic curves can be found by solving the system of equations formed by setting the two curves equal to each other. This can be done by using algebraic methods such as substitution or elimination.

2. What is the significance of finding the intersection of two algebraic curves?

Finding the intersection of two algebraic curves can provide valuable information about the relationship between the two curves. It can also help in solving real-world problems, such as finding the point of intersection between two moving objects.

3. Can two algebraic curves intersect at more than one point?

Yes, two algebraic curves can intersect at more than one point. In fact, it is possible for two curves to intersect at an infinite number of points, depending on the nature of the curves.

4. Is it always possible to find the intersection of two algebraic curves?

No, it is not always possible to find the intersection of two algebraic curves. Some curves may not intersect at all, while others may have complex or imaginary solutions that cannot be easily determined.

5. Are there any alternative methods for finding the intersection of two algebraic curves?

Yes, there are alternative methods for finding the intersection of two algebraic curves, such as using graphical methods or computer software. These methods can be helpful in cases where the equations are too complex to solve algebraically.

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