3 ms·

But I want to know more about the equations! What does y^2 = x^3 + ... have to do with an ellipse or the arc length of it?

by jaekwon 12y ago

deleted 12y ago
[deleted]
ChrisLomont 12y ago
Using elementary calculus you can derive the arc length of a circle. Trying the same method leads to an integral you cannot solve in elementary functions (for many definitions of "elementary"). These integrals are called elliptic integrals [1]. They satisfy many relationships, one of which is what is algebraically called an elliptic curve. Trying to gain insight into these integrals, people generalized them to a larger class of similar integrals, which can be thought of as functions mapping complex numbers to complex numbers, that satisfy certain periodicity relationships. They all are defined on a parallelogram, and then the values are repeated over the entire plane. This parallelogram can be cut out, and edges glued together suitably to form a torus (doughnut), which has one hole, called "genus 1" in many areas of mathematics. A sphere, which has no holes, is genus 0. A general way to study these is the Weierstrass elliptic functions [2], which are inverses of this mapping. These functions satisfy a relationship which (under a suitable class of isomorphism) is the equation y=x(x-1)(x-a), which got the name elliptic curve [3]. Another (connected reason) is there is a way to define the genus of a curve in algebraic geometry, which over the complex numbers relates to the surfaces and holes idea above, but this definition can be defined over any field (reals, rational, finite fields like those used in crypto, infinite fields of finite characteristic, and probably many more structures besides fields). Using this definition of genus, lines and quadratics turn out to be genus 0. Curves of the form y=(x-a)(x-b)(x-c) have genus 1, which is already called elliptic in the complex number case so this is another way to think of them. In grad school my math PhD thesis used elliptic curves, so I know a little of the math, but less of the history. I can try to explain more if people want. Next, do you want to hear why fire engines are red? It has a similarly convoluted story :) [1] http://en.wikipedia.org/wiki/Elliptic_integral http://en.wikipedia.org/wiki/Elliptic_integral [2] http://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functions http://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functi... [3] http://en.wikipedia.org/wiki/Elliptic_curve http://en.wikipedia.org/wiki/Elliptic_curve
- zmanian 12y ago
  :clapping: Interested in more...
- JadeNB 12y ago
  I'm a mathematician, and technically know many of these connexions, but had never seen them laid out all at once, and so clearly. Thank you, and bravo! > Next, do you want to hear why fire engines are red? It has a similarly convoluted story :) If that wasn't a joke, yes!
  - ChrisLomont 12y ago
    It's an old joke kids tell each other [1] [1] https://answers.yahoo.com/question/index?qid=20090403214314AAdZjAp https://answers.yahoo.com/question/index?qid=20090403214314A...

But I want to know more about the equations! What does y^2 = x^3 + ... have to do with an ellipse or the arc length of it?

by jaekwon 12y ago

But I want to know more about the equations! What does y^2 = x^3 + ... have to do with an ellipse or the arc length of it?

deleted 12y ago

[deleted]

ChrisLomont 12y ago

Using elementary calculus you can derive the arc length of a circle. Trying the same method leads to an integral you cannot solve in elementary functions (for many definitions of "elementary"). These integrals are called elliptic integrals [1]. They satisfy many relationships, one of which is what is algebraically called an elliptic curve. Trying to gain insight into these integrals, people generalized them to a larger class of similar integrals, which can be thought of as functions mapping complex numbers to complex numbers, that satisfy certain periodicity relationships. They all are defined on a parallelogram, and then the values are repeated over the entire plane. This parallelogram can be cut out, and edges glued together suitably to form a torus (doughnut), which has one hole, called "genus 1" in many areas of mathematics. A sphere, which has no holes, is genus 0. A general way to study these is the Weierstrass elliptic functions [2], which are inverses of this mapping. These functions satisfy a relationship which (under a suitable class of isomorphism) is the equation y=x(x-1)(x-a), which got the name elliptic curve [3]. Another (connected reason) is there is a way to define the genus of a curve in algebraic geometry, which over the complex numbers relates to the surfaces and holes idea above, but this definition can be defined over any field (reals, rational, finite fields like those used in crypto, infinite fields of finite characteristic, and probably many more structures besides fields). Using this definition of genus, lines and quadratics turn out to be genus 0. Curves of the form y=(x-a)(x-b)(x-c) have genus 1, which is already called elliptic in the complex number case so this is another way to think of them. In grad school my math PhD thesis used elliptic curves, so I know a little of the math, but less of the history. I can try to explain more if people want. Next, do you want to hear why fire engines are red? It has a similarly convoluted story :) [1] http://en.wikipedia.org/wiki/Elliptic_integral http://en.wikipedia.org/wiki/Elliptic_integral [2] http://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functions http://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functi... [3] http://en.wikipedia.org/wiki/Elliptic_curve http://en.wikipedia.org/wiki/Elliptic_curve

zmanian 12y ago
:clapping: Interested in more...
JadeNB 12y ago
I'm a mathematician, and technically know many of these connexions, but had never seen them laid out all at once, and so clearly. Thank you, and bravo! > Next, do you want to hear why fire engines are red? It has a similarly convoluted story :) If that wasn't a joke, yes!
- ChrisLomont 12y ago
  It's an old joke kids tell each other [1] [1] https://answers.yahoo.com/question/index?qid=20090403214314AAdZjAp https://answers.yahoo.com/question/index?qid=20090403214314A...