Elliptic curve (7369 bytes)
1: ...hese curves are '''''not''''' [[ellipse]]s: see [[elliptic integral]] for the origin of the term.
3: Elliptic curves are by definition [[non-singular]], meaning they...
5: Typical elliptic curves over the field of [[real number]]s are given by ...
9: ...ptic curve is a non-singular projective algebraic curve over ''K'' with [[genus (mathematics)|genus]] 1.
11: ... of ''K'' is neither 2 nor 3, then every elliptic curve over ''K'' can be written in the form

Elliptic curve cryptography (6835 bytes)
1: ...cs]] of [[elliptic curve]]s. The use of elliptic curves in cryptography was suggested independently by N...
3: ...er, is that encrypting and decrypting in elliptic curve cryptosystems may take longer than other cryptosy...
5: ...ogarithm]] problem for the group of an [[elliptic curve]] over some [[finite field]]. The most popular fi...
7: ...*''k'')*''P'' = ''k''*(''j''*''P''). The elliptic curve discrete logarithm problem (ECDLP) is then to det...
11: ... ''G'' is selected and published for use with the curve ''E''(''q''). A private key ''k'' is selected as ...

Elliptic Curve DSA (746 bytes)
1: ...ignature Algorithm]] which operates on [[elliptic curve]] [[group (mathematics)|groups]].
5: However [[elliptic curve]] groups are not vulnerable to a [[Number Field S...

Semistable elliptic curve (1684 bytes)
1: ...' ''p''. This can be explained by saying that the curve ''E_p'' obtained by reduction of ''E'' ...
11: ...orresponding to prime numbers ''p'', are elliptic curves unless ''p'' is in ''S''. For those exceptional ...

Lenstra elliptic curve factorization (3244 bytes)
1: ...[integer factorization]] which employs [[elliptic curve]]s.
8: ... and is likely to be ''smooth'' for some Elliptic curves.
10: The Lenstra elliptic curve factorization method to find a factor of the give...
12: ...'' on it. Then, we consider the group law on this curve mod ''n'' - this is possible since almost all res...
16: ...pefully, ''eA'' is a zero element of the Elliptic curve group in '''Z''' ''_p'', but not in '''...

Elliptic integral (6030 bytes)
1: In [[integral calculus]], an '''elliptic integral''' is any [[function (mathematics)|funct...
7: ...unctions, and the three canonical forms (i.e. the elliptic integrals of the first, second and third kind).
10: ...nts are completely equivalent (they give the same elliptic integral), but can be confusing due to thier diff...
12: * ''k'' the '''[[elliptic modulus]]'''
16: ...fying one is the same as specifying another. The elliptic integrals will also depend on another argument; t...

Elliptic geometry (1570 bytes)
1: '''Elliptic geometry''' (sometimes known as '''Riemannian geo...
3: ...ic geometry, there are no parallel lines at all. Elliptic geometry has other unusual properties. For examp...
5: The simplest model of elliptic geometry is that of [[spherical geometry]], where...
7: ...term is usually used for a vast generalization of elliptic geometry. See [[Riemannian geometry]].

Elliptic orbit (3281 bytes)
1: ... [[astrodynamics]] or [[celestial mechanics]] a [[elliptic orbit]] is an [[orbit]] with the [[eccentricity]]...
3: [[Specific orbital energy|Specific energy]] of an elliptical orbit is negative.
6: ...] (<math>v\,</math>) of a body traveling along '''elliptic orbit''' can be computed as:
17: ... (<math>T\,\!</math>) of a body traveling along [[elliptic orbit]] can be computed as:
27: ...c orbital energy]] (<math>\epsilon\,</math>) of [[elliptic orbit]] is negative and the [[orbital energy cons...

Elliptic function (3530 bytes)
1: ...riodic function|periodic]] in two directions. The elliptic functions can be seen as analogs of the [[trigono...
3: Formally, an elliptic function is a [[meromorphic function]] ''f'' defi...
8: ...terms of these. The [[Jacobi's elliptic functions|elliptic functions]] introduced by [[Carl Jacobi]], and th...
10: ...ptic functions are the [[inverse function]]s of [[elliptic integral]]s, which is how they were introduced hi...
12: ...d ''b'' are called ''fundamental periods''. Every elliptic function has a pair of fundamental periods, but t...

Elliptic operator (1813 bytes)
1: In [[mathematics]], an '''Elliptic operator''' is a major type of [[differential op...
2: An important example of an elliptic operator is the [[Laplacian]]. Equations of the ...
6: ...on]] or the [[Schrodinger equation]] also involve elliptic operators (on the [[LHS]], say) as well as a time...
14: ...ial_{x_k} </math>. Such an operator is called ''elliptic'' [[iff]] for every ''x''
32: is an elliptic operator.

Weierstrass's elliptic functions (9756 bytes)
1: ...s]], [[Weierstrass]] introduced some particular [[elliptic function]]s that have become the basis for the mo...
7: Then '''Weierstrass's elliptic function''' is an elliptic function with periods <math>2\omega_1</math> and ...
24: Further development of the theory of elliptic functions shows that the condition on Weierstrass...
71: The Weierstrass elliptic functions have several properties that may be pro...
113: ...> and letting &tau; vary leads into the area of [[elliptic modular function]]s.

Jacobi's elliptic functions (10118 bytes)
1: ... now seen: that can be said for the [[Weierstrass elliptic functions]]. They are not, however, outmoded. The...
4: ...led the [[quarter period]]s. The twelve Jacobian elliptic functions are then pq, where p and q are one of t...
6: The Jacobian elliptic functions are then the unique doubly-periodic, me...
11: ...e Jacobian elliptic functions are then the unique elliptic functions that satisfy the above properties.
13: ==Definition as inverses of elliptic integrals==

Hasse's theorem on elliptic curves (549 bytes)
1: ...If ''N'' is the number of points on an [[elliptic curve]] ''E'' over a finite field of ''q'' elements, th...
7: [[category: algebraic curves]]

Curve (10841 bytes)
3: ... line]]. A large number of other [[List of curves|curves]] have been studied in [[geometry]].
5: ...urve]]''), or [[graph of a function]] ([[Phillips curve]]).
10: ...ion makes it possible to talk about closed simple curve).
12: ...simple closed curve''' is also called a '''Jordan curve'''.
14: ... plane. These definitions also apply to algebraic curves (see below).

Cubic curve (2088 bytes)
1: ...[[mathematics]], a '''cubic curve''' is a [[plane curve]] ''C'' defined by a cubic equation
9: ...s through ''P''. Therefore we can find some cubic curve through any nine given points.
11: ...ne by drawing the curve. The real points of cubic curves were studied by [[Newton]]; they fall into one o...
13: ...ierstrass form. For example, there are many cubic curves that have no such point, when ''K'' is the [[rat...

Horseshoe Curve (2181 bytes)
1: ...jpg|300px|right|thumb|Satellite view of Horseshoe Curve, west of Altoona, Pennsylvania. The uphill grade...
3: ...ghtening to 609 ft (186 m) on the south side. The curve was built in response to the difficulty of constr...
5: ...ail removed one of the tracks in [[1981]] and the Curve remains in this three-track configuration. It ha...
7: ...ring [[World War II]] would have blown it up. The curve was designated as a [[National Historic Landmark]...
9: ...a's [[minor league baseball]] team, the [[Altoona Curve]].

Modular curve (2634 bytes)
1: ...[[Riemann surface]], or corresponding [[algebraic curve]], constructed as
6: ...t is usually, more strictly, meant by a ''modular curve''.
8: ...s with a given basis for the ''N''-torsion. These curves have been studied in great detail.
10: ...ion field''' is a [[function field]] of a modular curve (or, occasionally, of some other [[moduli space]]...
12: ...s [[correspondence]]s connecting pairs of modular curves.

Rotation curve (506 bytes)
1: ...:newtonianfig2.png|thumb|300px|right|The rotation curve of a [[galaxy]]; that is, the [[orbital velocity]...
2: In [[astrophysics]], the '''rotation curve''' of a [[galaxy]] is a [[graph of a function|gra...

Forgetting curve (2827 bytes)
1: ...[[graph of a function|graph]] of the forgetting [[curve]] shows that humans tend to halve their memory of...
16: ...the first to describe the shape of the forgetting curve.
18: ... vivid or traumatic memories. The flatness of the curve is not necessarily evidence for the decrease in t...
20: .... This means that, in this period, the forgetting curve "falls" by 10%.

Altoona Curve (581 bytes)
1: [[Image:AltoonaCurve_100.png|right|Altoona Curve]]
3: ...he [[Pittsburgh Pirates]] major-league club. The Curve play in [[Blair County Ballpark]], located in Alt...
6: * [http://www.altoonacurve.com/ Altoona Curve official site]

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