A Curve by Any Other Name(whispersystems.org)
whispersystems.org
A Curve by Any Other Name
https://whispersystems.org/blog/a-curve-by-any-other-name/
12 comments
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
[2] http://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functi...
[3] http://en.wikipedia.org/wiki/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
[2] http://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functi...
[3] http://en.wikipedia.org/wiki/Elliptic_curve
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!
> Next, do you want to hear why fire engines are red? It has a similarly convoluted story :)
If that wasn't a joke, yes!
It's an old joke kids tell each other [1]
[1] https://answers.yahoo.com/question/index?qid=20090403214314A...
[1] https://answers.yahoo.com/question/index?qid=20090403214314A...
:clapping: Interested in more...
[deleted]
There are some concepts in hard sciences that are difficult to relate to using everyday analogies (monads, general relativity, particle physics, EC). In such situations, it may make more sense to look past the compulsion to find concrete (over?)simplifications and start looking at properties, behavior and utility. (Humans want to find patterns and construct conceptual models from these patterns... Even where there is no pattern and also where the pattern is far more complex than even the best minds could hope to imagine.)
It's missing the important parts of the interesting algebra that's created by drawing lines across the curves, how these properties are preserved in the discrete analog, and a brief description of ejy that's useful for crypto.
That was an unusually roundabout and over-flourished way of explaining the name origin of "Elliptic Curves".
Also known as "telling a story".
I guess I don't appreciate being told a story with lots of extraneous details (you don't need to list all pie types. Or describe how you imagine Euler at work) to get a historical explanation. I like my points salient.
"Anecdata, party of one: your table is ready."
What does y^2 = x^3 + ... have to do with an ellipse or the arc length of it?