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What exactly is a curve?

  • 27-04-2008 6:01pm
    #1
    AARRRGH
    Closed Accounts Posts: 12,382 ✭✭✭✭


    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!


Comments

  • #2
    Fremen
    Registered Users, Registered Users 2 Posts: 2,481 ✭✭✭Fremen


    In general, you can think of a "curve" as a set of points which satisfy some property. In the case of elliptic curves, it's the set of points (x,y) which satisfy

    y^2 = x^3 + bx + c

    when x and y are real numbers, this matches our intuition of what a curve should look like, i.e. a smooth-ish line which you can draw with a pencil. If you're working with integers modulo some prime, it becomes a whole lot tougher to visualise. Personally, I'd stop looking to geometry for help and just trust the maths once you stop working with real numbers.

    I'm not too keen on the cannonball explanation of elliptic curves. Yes, it gives a nice demonstration of a practical use for using an elliptic curve to solve a problem, but it doesn't give a deeper insight into what they are.


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