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 t1 ≤ t ≤ t2. 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.