A **plane curve** is any curve
which can be drawn on the plane. Some curves are fairly simple,
like a circle, and will have fairly simple algebraic equations. Some are
very complex, like your signature, and may be very difficult to describe
with an equation. Plane curves
were studied intensively from the seventeenth through the nineteenth
centuries. Current mathematical research and coursework is often
much more generalized and abstract than the items you will find on this web site.

The equation for any given plane curve will depend on the coordinate
system used. Three common **coordinate systems**
and their generic plane curve equations include:

- rectangular, $f(x,y)=0$
- polar, $f(r,\theta)=0$
- parametric, $x=f(t), y=g(t)$

Plane curves will also arise from the **level
curves**, or contour curves, of functions of two independent
variables. Functions of two variables will produce
three-dimensional graphs of ordered triples, $(x,y,z)$,
where $z=f(x,y)$.

The simplest plane curves arise from **algebraic
equations**. An equation is algebraic when the only
operations involved are addition, subtraction, multiplication, division,
extraction of roots, and raising to integral or fractional powers. Algebraic
functions of one variable can be written as polynomial equations in two
variables. They are often classified as linear (lines), quadratic
(conic sections), or higher plane curves (using polynomials of degree 3 or
more).

Equations which are not algebraic are called **transcendental
equations**. The simplest transcendental equations will
involve
exponential, logarithmic, trigonometric, or hyperbolic functions (all
of which
are related in the complex number system). Other examples of
transcendental functions would include the Gamma function, Bessel
functions, and
the Error function of statistics.

Plane curves can exhibit many different interesting **features**.
Some of these features include:

- multiple branches
- asymptotes, either linear or nonlinear
- isolated points
- nodes (where a curve crosses itself)
- cusps (where a curve is tangent to itself)

Some plane curves are **degenerate curves**.
Degeneracy
occurs
when a curve is both a simpler type and a limiting case of a
family of curves. Algebraically, this often occurs when some
coefficient
or parameter vanishes. Two examples include:

- a single point as a degenerate circle (when the radius of a circle shrinks to zero)
- two intersecting lines as a degenerate hyperbola (when the constant term of the algebraic equation reaches the value where the equation becomes factorable as the difference of two squares)

An advanced study of plane curves would introduce the **projective
plane** and **homogeneous coordinates**.
The
projective
plane is essentially the Euclidean plane with a "line at
infinity" added. Lines which are parallel in the Euclidean plane
have
a point of intersection on the line at infinity. Homogeneous
coordinates
allow us to describe all points in the projective plane with a family
of ordered
triples of real numbers (infinity is never used as a coordinate).
By using
the projective plane, many different-looking curves can be seen to be
related
(for example, the circle and the hyperbola). But on this web
site, we will
generally not make these connections. To learn more, you should
take a
course in algebraic geometry.