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Math Help - What exactly is a curve?

  1. #1
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    What exactly is a curve?

    I am reading about curves (elliptic curves) at the moment.

    Some of the curves cannot be visualised, i.e. they do not look like a curve, they are just a bunch of random dots.

    Does this mean curves have some mathemaical properties in common, rather than actually having to look like a curve?

    Also, I've read the following explanation of elliptic curves:

    You have a pile of cannonballs grouped together to form a pyramid.

    On the first layer are nine cannonballs.
    On the second layer are four cannonballs.
    On the top later is one cannonball.

    To calculate how many cannonballs are required for a pyramid of height x, the following equation can be used:

    1^2 + 2^2 + 3^2 + ... x^2 = x(x+1)(2x+1) / 6

    If we want the base to be a perfect square, we need to find a solution to:

    y^2 = x(x+1)(2x+1) / 6, where x and y are positive integers. This type of equation represents an elliptic curve.

    ...

    I do not understand this. How does a pyramid of cannonballs having a square base represent an elliptic curve equation?

    Maybe these are stupid questions, but I'm finding it hard to find "basic" information on what an elliptic curve really is.

    So to summarise, my questions are as follows:

    1. If an elliptic curve does not look like a curve, how is it still a curve? Does it just have to satisfy some equation?
    2. Why does a calculation to see if the base of a pyramid is square represent an elliptic curve equation?

    Any help greatly appreciated.

    Thanks!
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  2. #2
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    Ignore the stuff about cannonballs. That is simply intended as motivation for introducing the equation y^2 = x(x+1)(2x+1)/6. If x and y represent numbers of cannonballs then of course they must be integers. But in the equation y^2 = x(x+1)(2x+1)/6, x and y are to be thought of as real numbers (maybe the book doesn't emphasise this transition from whole numbers to real numbers). So you should think of the equation as representing the graph of a variable y=\pm\sqrt{x(x+1)(2x+1)/6}; in other words, a curve.

    If you want to know why it's an elliptic curve, you'll find the definition here.
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  3. #3
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    According to one of my professors (he does number theory) the reason why it is called "elliptic" though not actually looking like an ellipse is because functions of the form y^2 = x^3+ax+b arise in elliptic integrals. I remember asking a question like that once too, I was also confused about the reason we call them elliptic.

    I realized you are reading Lawrence C. Washington.
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  4. #4
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    Thank you both for your replies.

    I have another question if you don't mind.

    I am studying elliptic curves with regard to elliptic curve cryptography.

    Instead of using real numbers for the points, I am using polynomials (within a finite field.)

    I can easily visualise using real number as points on a curve, but using polynomials?!

    Do the polynomials just represent points, i.e. if I solved the polynomial I would have a number for x and y which I could use as points? Or are the polynomials just polynomials, and should I stop trying to bring them back to something "normal" I can visualise?

    Thank you.
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  5. #5
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    If you're working over a finite field then the "curve" will only have finitely many points on it. But the algebraic formalism works in the same way as it does when the scalars are the reals. The idea is to use the geometric picture from the real case to motivate the techniques that you use over a finite field.

    For example, if the curve is y^2 = x(x+1)(2x+1)/6 then (using calculus) you can find that the tangent at the point (1,1) is 13x-12y=1 (in the real case). Suppose you're working over the field \mathbb{Z}_7. Then this equation becomes 6x-5y=1, which you can still think of as the "tangent" to the "curve" at this point, even though the "curve" contains only finitely many points. This analogy is a very powerful one, leading to all sorts of algebraic results that you would be unlikely to discover without the geometric motivation.
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