# 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**