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Curve — Definition, Examples & Types

Curve

A word used to indicate any path, whether actually curved or straight, closed or open. A curve can be on a plane or in three-dimensional space (or n-dimensional space, for that matter). Lines, circles, arcs, parabolas, polygons, and helixes are all types of curves.

Note: Typically curves are thought of as the set of all geometric figures that can be parametrized using a single parameter. This is not in fact accurate, but it is a useful way to conceptualize curves. The exceptions to this rule require some cleverness, or at least some exposure to space-filling curves.

 

See also

Surface, solid, arc length of a curve

Key Formula

r⃗(t)=(x(t),  y(t)),a≤t≤b\vec{r}(t) = \big(x(t),\; y(t)\big), \quad a \le t \le b
Where:
  • r⃗(t)\vec{r}(t) = Position on the curve at parameter value t
  • x(t)x(t) = The x-coordinate as a function of t
  • y(t)y(t) = The y-coordinate as a function of t
  • tt = The parameter that traces out the curve as it varies from a to b

Worked Example

Problem: A circle of radius 3 centered at the origin can be described as a curve. Write a parametric representation and identify three specific points on the curve.
Step 1: Choose a parametrization. A circle of radius 3 can be traced by letting the parameter t represent the angle measured from the positive x-axis.
r⃗(t)=(3cos⁡t,  3sin⁡t),0≤t<2π\vec{r}(t) = (3\cos t,\; 3\sin t), \quad 0 \le t < 2\pi
Step 2: Find the point at t = 0. Substitute into the parametric equations.
r⃗(0)=(3cos⁡0,  3sin⁡0)=(3,  0)\vec{r}(0) = (3\cos 0,\; 3\sin 0) = (3,\; 0)
Step 3: Find the point at t = π/2.
r⃗ ⁣(π2)=(3cos⁡π2,  3sin⁡π2)=(0,  3)\vec{r}\!\left(\tfrac{\pi}{2}\right) = \left(3\cos\tfrac{\pi}{2},\; 3\sin\tfrac{\pi}{2}\right) = (0,\; 3)
Step 4: Find the point at t = π.
r⃗(π)=(3cos⁡π,  3sin⁡π)=(−3,  0)\vec{r}(\pi) = (3\cos\pi,\; 3\sin\pi) = (-3,\; 0)
Answer: The circle of radius 3 is the curve r(t) = (3 cos t, 3 sin t). As t goes from 0 to 2π, the curve traces the entire circle, passing through (3, 0), (0, 3), (−3, 0), and back.

Another Example

Problem: A straight line segment from the point (1, 2) to the point (5, 10) is also a curve. Write it in parametric form and find the midpoint.
Step 1: Use the linear parametrization that starts at (1, 2) when t = 0 and ends at (5, 10) when t = 1.
r⃗(t)=(1+4t,  2+8t),0≤t≤1\vec{r}(t) = (1 + 4t,\; 2 + 8t), \quad 0 \le t \le 1
Step 2: Find the midpoint by evaluating at t = 0.5.
r⃗(0.5)=(1+4(0.5),  2+8(0.5))=(3,  6)\vec{r}(0.5) = (1 + 4(0.5),\; 2 + 8(0.5)) = (3,\; 6)
Answer: The line segment is the curve r(t) = (1 + 4t, 2 + 8t) for t in [0, 1], and its midpoint is (3, 6). This shows that even a straight path qualifies as a curve.

Frequently Asked Questions

Is a straight line considered a curve?
Yes. In mathematics, a straight line is a special case of a curve — one with zero curvature everywhere. The word 'curve' refers to any continuous path, not just paths that bend.
What is the difference between a curve and a function?
A function assigns exactly one output to each input, so its graph must pass the vertical line test. A curve has no such restriction. A circle, for example, is a curve but is not the graph of a single function y = f(x) because it fails the vertical line test. However, every function's graph is a curve.

Curve vs. Surface

A curve is a one-dimensional object: it is traced by varying a single parameter. A surface is a two-dimensional object: it requires two parameters to describe. For example, a circle is a curve (one parameter: the angle), while the outer shell of a sphere is a surface (two parameters: latitude and longitude).

Why It Matters

Curves appear everywhere in math and science — from the trajectory of a thrown ball (a parabola) to the orbit of a planet (an ellipse). Parametrizing curves lets you compute tangent lines, arc lengths, and areas under paths. In higher-level courses such as multivariable calculus and physics, working with curves is essential for understanding motion, force fields, and line integrals.

Common Mistakes

Mistake: Thinking a curve must be 'curvy' — that is, assuming straight lines and line segments are not curves.
Correction: In mathematics, any continuous path counts as a curve, including perfectly straight ones. A line is simply a curve with zero curvature.
Mistake: Believing every curve can be written as y = f(x).
Correction: Many curves, such as circles and figure-eights, fail the vertical line test and cannot be expressed as a single function y = f(x). Use parametric equations or implicit equations instead.

Related Terms

  • Parametrize — Method of describing a curve with a parameter
  • Line — Simplest type of curve — zero curvature
  • Circle — Closed curve with constant curvature
  • Parabola — Open curve defined by a quadratic equation
  • Arc Length of a Curve — Measures the total length along a curve
  • Surface — Two-dimensional analog of a curve
  • Helix — A curve that spirals through 3D space
  • Plane — Flat space in which many curves lie