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Plane Curve — Definition, Formula & Examples

A plane curve is a curve that lies entirely within a single two-dimensional plane. Circles, parabolas, ellipses, and sine waves are all examples of plane curves.

A plane curve is a continuous function γ:I→R2\gamma: I \to \mathbb{R}^2, where II is an interval in R\mathbb{R}, mapping each parameter value to a point (x,y)(x, y) in the Euclidean plane. Equivalently, it is the image of such a mapping, described explicitly as y=f(x)y = f(x), implicitly as F(x,y)=0F(x, y) = 0, or parametrically as (x(t), y(t))(x(t),\, y(t)).

How It Works

A plane curve can be specified in three common ways. An explicit equation like y=x2y = x^2 directly gives yy as a function of xx. An implicit equation like x2+y2=1x^2 + y^2 = 1 defines the curve as the set of all (x,y)(x, y) satisfying the relation. A parametric representation like x=cos⁡t,  y=sin⁡tx = \cos t,\; y = \sin t traces the curve as the parameter tt varies over an interval. Each representation suits different contexts: parametric forms handle self-intersecting curves and motion problems naturally, while implicit forms are standard for conic sections.

Worked Example

Problem: Verify that the parametric equations x=3cos⁡tx = 3\cos t, y=3sin⁡ty = 3\sin t for t∈[0,2π)t \in [0, 2\pi) describe a plane curve and identify it.
Step 1: Compute x2+y2x^2 + y^2 using the parametric equations.
x2+y2=(3cos⁡t)2+(3sin⁡t)2=9cos⁡2t+9sin⁡2tx^2 + y^2 = (3\cos t)^2 + (3\sin t)^2 = 9\cos^2 t + 9\sin^2 t
Step 2: Apply the Pythagorean identity cos⁡2t+sin⁡2t=1\cos^2 t + \sin^2 t = 1.
x2+y2=9(cos⁡2t+sin⁡2t)=9x^2 + y^2 = 9(\cos^2 t + \sin^2 t) = 9
Step 3: The result is the implicit equation of a circle. Since all points (x,y)(x, y) lie in R2\mathbb{R}^2, this is a plane curve.
x2+y2=9x^2 + y^2 = 9
Answer: The parametric equations describe a plane curve: a circle of radius 3 centered at the origin.

Why It Matters

Plane curves are the foundation of analytic geometry and single-variable calculus. Arc length, curvature, tangent lines, and area computations all assume you are working with a curve in a plane. In engineering and physics, plane curves model projectile trajectories, orbital paths, and cross-sectional profiles of surfaces.

Common Mistakes

Mistake: Assuming every curve is a plane curve.
Correction: A helix, such as (cos⁡t, sin⁡t, t)(\cos t,\, \sin t,\, t), extends into three dimensions and is not a plane curve. A curve must lie entirely in a single plane to qualify.