Definition:Algebraic Curve

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Definition

An algebraic curve is a curve which is the locus of a set of polynomial equations.


Planar

A planar algebraic curve is an algebraic curve whose polynomial equations representing it are in two variables.


Degree

The degree of an algebraic curve is defined as the highest degree of the polynomial equations defining it.


Examples

Straight Line

The straight line is an example of an algebraic curve.

From General Equation of Straight Line in Plane, a straight line can be expressed by an equation in the form:

$a x + b y + c = 0$


Circle

The circle is an example of an algebraic curve.

From Cartesian Equation of Circle, a circle can be expressed by an equation in the form:

$\paren {x - a}^2 + \paren {y - b}^2 = R^2$

For example:

$x^2 + y^2 = 4$

is the equation defining a circle whose center is at $\tuple {0, 0}$ and whose radius is $2$.


Conic Section

The conic section is an example of an algebraic curve.

From Equation of Conic Section, a conic section can be expressed by an equation in the form:

$a x^2 + b x y + c y^2 + d x + e y + f = 0$

for some $a, b, c, d, e, f \in \R$.


Elliptic Curve

The elliptic curve is an example of an algebraic curve.

By definition of the standard form of the elliptic curve, an elliptic curve can be expressed by an equation in the form:

$y^3 = x^3 + a x + b$

for some $a, b \in \R$.


Also see

  • Results about algebraic curves can be found here.


Sources

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