• 1 Symbolic representation
• 2 Smooth plane curve
• 3 Algebraic plane curve
• 4 Examples
• 5 See also
• 6 References
• 7 External links

Symbolic representation

A plane curve can often be represented in Cartesian coordinates by an implicit equation of the form ${\displaystyle f(x,y)=0}$ for some specific function f. If this equation can be solved explicitly for y or x – that is, rewritten as ${\displaystyle y=g(x)}$ or ${\displaystyle x=h(y)}$ for specific function g or h – then this provides an alternative, explicit, form of the representation. A plane curve can also often be represented in Cartesian coordinates by a parametric equation of the form ${\displaystyle (x,y)=(x(t),y(t))}$ for specific functions ${\displaystyle x(t)}$ and ${\displaystyle y(t).}$Cartesian coordinates by an implicit equation of the form ${\displaystyle f(x,y)=0}$ for some specific function f. If this equation can be solved explicitly for y or x – that is, rewritten as ${\displaystyle y=g(x)}$ or ${\displaystyle x=h(y)}$ for specific function g or h – then this provides an alternative, explicit, form of the representation. A plane curve can also often be represented in Cartesian coordinates by a parametric equation of the form ${\displaystyle (x,y)=(x(t),y(t))}$ for specific functions ${\displaystyle x(t)}$ and ${\displaystyle y(t).}$

Plane curves can sometimes also be represented in alternative coordinate systems, such as polar coordinates that express the location of each point in terms of an angle and a distance from the origin.

Smooth plane curve

A smooth plane curve is a curve in a real Euclidean plane R² and is a one-dimensional smooth manifold. This means that a smooth plane curve is a plane curve which "locally looks like a line", in the sense that near every point, it may be mapped to a line by a smooth function. Equivalently, a smooth plane curve can be given locally by an equation f(x, y) = 0, where f : R² → R is a smooth function, and the partial derivatives ∂f/∂x and ∂f/∂

Plane curves can sometimes also be represented in alternative coordinate systems, such as polar coordinates that express the location of each point in terms of an angle and a distance from the origin.

A smooth plane curve is a curve in a real Euclidean plane R² and is a one-dimensional smooth manifold. This means that a smooth plane curve is a plane curve which "locally looks like a line", in the sense that near every point, it may be mapped to a line by a smooth function. Equivalently, a smooth plane curve can be given locally by an equation f(x, y) = 0, where f : R² → R is a smooth function, and the partial derivatives ∂f/∂x and ∂f/∂y are never both 0 at a point of the curve.

Algebraic plane curveAn algebraic plane curve is a curve in an affine or projective plane given by one polynomial equation f(x, y) = 0 (or F(x, y, z) = 0, where F is a homogeneous polynomial, in the projective case.)

Algebraic curves have been studied extensively since the 18th century.

Every algebraic plane curve has a degree, the degree of the defining equation, which is equal, in case of an Algebraic curves have been studied extensively since the 18th century.

Every algebraic plane curve has a degree, the degree of the defining equation, which is equal, in case of an algebraically closed field, to the number of intersections of the curve with a line in general position. For example, the circle given by the equation x² + y² = 1 has degree 2.

The non-singular plane algebraic curves of degree 2 are called conic sections, and their projective completion are all isomorphic to the projective completion of the circle x² + y² = 1 (that is the projective curve of equation x² + y² – z²= 0). The plane curves of degree 3 are called cubic plane curves and, if they are non-singular, elliptic curves. Those of degree 4 are called quartic plane curves.

Numerous examples of plane curves are shown in Gallery of curves and listed at List of curves. The algebraic curves of degree 1 or 2 are shown here (an algebraic curve of degree less than 3 is always contained in a plane):

Name	Implicit equation	Parametric equation	As a