# Plane Curve: Definition, Examples

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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 .

## 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) 
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. A variety of plane 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 .

## References

 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
 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
 Length of Curve and Surface Area.

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Stephanie Glen. "Plane Curve: Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/plane-curve/
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