Plane curve

In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves.

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/∂y are never both 0 at a point of the curve.

Algebraic plane curve

An 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 were 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 four are called quartic plane curves.

Examples

Name	Implicit equation	Parametric equation	As a function	graph
Straight line	$a x+b y=c$	$(x_0 + \alpha t,y_0+\beta t)$	$y=m x+c$
Circle	$x^2+y^2=r^2$	$(r \cos t, r \sin t)$
Parabola	$y-x^2=0$	$(t,t^2)$	$y=x^2$	100px
Ellipse	$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$	$(a \cos t, b \sin t)$		100px
Hyperbola	$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$	$(a \cosh t, b \sinh t)$