

A Curve by Any Other Name - zmanian
https://whispersystems.org/blog/a-curve-by-any-other-name/

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jaekwon
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?

~~~
ChrisLomont
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_functi...](http://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functions)

[3]
[http://en.wikipedia.org/wiki/Elliptic_curve](http://en.wikipedia.org/wiki/Elliptic_curve)

~~~
JadeNB
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
It's an old joke kids tell each other [1]

[1]
[https://answers.yahoo.com/question/index?qid=20090403214314A...](https://answers.yahoo.com/question/index?qid=20090403214314AAdZjAp)

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rab_oof
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.)

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dnautics
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.

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AceJohnny2
That was an unusually roundabout and over-flourished way of explaining the
name origin of "Elliptic Curves".

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senorprogrammer
Also known as "telling a story".

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AceJohnny2
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.

~~~
rosser
"Anecdata, party of one: your table is ready."

