What Is a Curvature Calculator?
A curvature calculator measures how sharply a curve bends at a given point. Curvature, denoted by the Greek letter kappa (κ), is large where a curve turns tightly and small where it is nearly straight. It is a key concept in calculus, differential geometry, physics, and engineering — from road and track design to the motion of particles. Enter a function or parametric curve and a point, and the calculator returns the curvature and the radius of curvature.
How to Use the Calculator
- Enter your curve — as a function y = f(x) or in parametric form.
- Enter the point at which to measure curvature.
- Calculate — see the curvature κ and the radius of curvature.
The Curvature Formula
For a function y = f(x), the curvature at a point is:
κ = |f''(x)| ÷ (1 + f'(x)²)^(3/2)
The radius of curvature is simply the reciprocal: R = 1 ÷ κ. A large radius means a gentle bend; a small radius means a sharp one.
Understanding Curvature
| Curvature κ | Meaning |
|---|---|
| Zero | Straight line (no bending) |
| Small | Gentle curve, large radius |
| Large | Sharp curve, small radius |
A circle of radius R has constant curvature 1/R everywhere. The radius of curvature at a point on any curve is the radius of the circle that best fits the curve there, called the osculating circle.
Where Curvature Is Used
- Road and railway design: ensuring safe, smooth curves.
- Physics: centripetal force depends on the radius of curvature of a path.
- Computer graphics and CAD: smoothing and modeling curves.
- Optics: lens and mirror surfaces are described by curvature.
Frequently Asked Questions
What is curvature?
Curvature measures how sharply a curve bends at a point. It is zero for a straight line and larger where the curve turns more tightly.
What is the formula for curvature?
For y = f(x), curvature is κ = |f''(x)| ÷ (1 + f'(x)²)^(3/2). The radius of curvature is the reciprocal, R = 1/κ.
What is the radius of curvature?
It is the radius of the circle that best approximates the curve at a point. A small radius means a sharp bend; a large radius means a gentle one.
What is the curvature of a circle?
A circle of radius R has constant curvature 1/R at every point — the smaller the circle, the greater the curvature.
Is this curvature calculator free?
Yes — it is completely free, requires no signup, and returns both curvature and radius of curvature.