A **plane curve **is a curve in a two-dimensional plane. In other words, the points of the curve are all on the same plane. In comparison, a space curve’s points do not necessarily all lie on a single plane [1].

## Plane Curve Equations

A **plane curve** can be defined with the parametric equation (which means that both x and y are defined as functions of a parameter, usually *t*) [2]

x = x(t), y = y(t),

Where coordinates (x, y) are expressed as functions of t on the closed interval t_{1} ≤ t ≤ t_{2}. x(t) and y(t) are continuous functions, with a sufficient number of continuous derivatives; if there are *r* continuous derivatives, then the curve is class *r*. In vector notation, a parametric plane curve can be specified by a vector-valued function **r** = **r**(t).

The equation for a plane curve can also be expressed as rectangular coordinates f(x, y) = 0, or polar coordinates f(r, θ) = 0.

Plane curves can show a variety of **interesting features**, including:

- Asymptotes,
- Cusps (where a curve is tangent to itself),
- Acnodes (isolated points),
- Nodes (where the curve intersects itself).

## Types of Plane Curve

The simplest plane curves arise from algebraic equations (those involving addition, subtraction, multiplication, division, roots, and raising to powers). If the curve can be described by a polynomial equation, they are called algebraic curves.

**Simple plane curves**are non intersecting. In other words, they do not cross their own paths. If a curve intersects itself, then it’s*not simple*.- A
**closed plane curve**has no endpoints; it completely encloses an area. For example, a circle or ellipse; the Lamé curve is closed when*n*in its Cartesian equation is a positive integer. If a curve has endpoints (like a parabola), then it is an*open curve*.

A **smooth plane curve** is given by a pair of parametric equations on the closed interval [a, b]; derivatives for x(t) and y(t) exists and are continuous on [a, b] *and *the first derivatives are not simultaneously zero on that interval [3].

See also:

## References

[1] Montoya, D. & Naves, D. On Plane and Space Curves. Retrieved January 15, 2022 from: https://web.ma.utexas.edu/users/drp/files/Spring2020Projects/DRP_Final_Project%20-%20Daniel%20Naves.pdf

[2] Patrikalakis, N. et al. (2009). 1.1.1 Plane Curves. Retrieved January 15, 2022 from: https://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/node5.html

[3] Length of Curve and Surface Area.