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So I was doing some problems looking for curves of an intersection between two surfaces. I was wondering how do I know when i "got" it? What is the algorithm of finding these curves?

For example,

Take the surfaces $z=4x^2+y^2$ and $y=x^2$. So the curve would be $z=4x^2+x^4$ if we just plug in one of the equations But this is not where the story ends. Now to fully represent the curve lets parametrize it. $x=t, y=t^2, z=4t^2+t^4$.

So here i had to plug in of the equations into the other one. Is this always how you solve such problems? Sometimes we want to parametrize first and then do some plugging but in the end, we have to somehow plug one equation into the other one. Am i correct? I am sorry if the question is vague but i am looking for a more or less universal way of finding a solution to such a problem.

Sorfosh
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  • This is related to the commutative algebra of polynomial rings over some given field. The equations determine an "ideal" in the corresponding polynomial ring, in our case it is - say - $\Bbb Q[x,y,z]$, and there are algorithms to work in such structures. (E.g. Groebner bases, elimination ideal, etc.) – dan_fulea Jun 12 '18 at 17:33
  • The concept of a surface is very broad, and I get a strong feeling that you do not intend such breadth, but its very hard to tell exactly what you intend. Are you able to specify more precisely what you mean by a surface? For example, do you mean "the solution set of an equation $F(x,y,z)=c$ in three real variables $x,y,z$ such that $c$ is a regular value of $F$"? – Lee Mosher Jun 12 '18 at 17:35
  • @LeeMosher I mean a "thing" That is somehow described by a single equation of at most three variables. So x=z is a surface, x^2+y^2=1 is a surface. Now i looking for ways to find the intersections of these. Not sure if that's good enough of an exmaplantion. – Sorfosh Jun 12 '18 at 17:37

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Curves and surfaces can be described in different ways, among others with implicit, explicit and parametric equations.

If the two surfaces are given by explicit or implicit equations, like in your example, just forming a system with the two equations

$$\begin{cases}z=4x^2+y^2,\\y=x^2\end{cases}$$ is a representation of the curve and you can stop here. This representation allows you to check if a given point $(x,y,z)$ belongs to the curve.

If you are requested to generate points on the curve, then a parametric representation is more appropriate. You can obtain one by seeing one of the variables as independent, and solving the system of equations for the other two variables. For instance

$$(x,x^2,4x^2+x^4)$$ describes the given curve.

In the case of general equations, there is no systematic method and sometimes no analytical solution at all.

  • Thanks for the answer. So I can see how the system of equations is a solution. Then from there, we just solve it by treating a variable as an independent. As you did with $x$ set it to $t$ and we have paratremized the curve. Now this got me thinking, since in my book they find the line of interesction of planes by finding it's vector using cross product. But it seems that all i have to do is take the two equations, and simply do the same thing as we are doing here, right? – Sorfosh Jun 12 '18 at 17:54
  • @Sorfosh: in fact, the cross product is an alternative technique to solve the system of (linear) equations. Specifically, the components of the cross-product are just the minors of the $2\times3$ matrix. You might as well use Gaussian elimination. –  Jun 12 '18 at 18:06