Definition:Algebraic Curve

Definition

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

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

• Definition:Algebraic Variety: an example of which is an algebraic curve

• Results about algebraic curves can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): algebraic curve
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): curve
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): algebraic curve
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): curve