Curvature Calculator

What is Curvature?

Curvature (κ) is a fundamental concept in calculus and differential geometry that measures how sharply a curve bends at a given point. A straight line has zero curvature everywhere, while a circle has constant curvature equal to the reciprocal of its radius. The smaller the radius, the higher the curvature. Our curvature calculator helps you compute this value for any function or parametric curve instantly.

Understanding curvature is essential in fields like road design, computer graphics, optics, and physics. Engineers use curvature to design safe roads with appropriate turning radii, while computer graphics professionals need curvature continuity for smooth 3D surfaces. The osculating circle visualization in our tool shows the circle that best approximates your curve at the chosen point.

How to Use the Curvature Calculator

Using our curvature calculator is straightforward. First, select the curve type: explicit function y = f(x) or parametric curve defined by x(t) and y(t). Enter your mathematical expression using standard notation, then specify the point at which you want to calculate curvature. The tool instantly computes the curvature value, radius of curvature, and shows step-by-step derivative calculations. You can also click any of the quick examples to see how different functions behave.

Curvature Formulas

For an explicit function y = f(x), the curvature is calculated using the formula $\kappa = |f''(x)| / (1 + (f'(x))^2)^{3/2}$, where $f'(x)$ is the first derivative (slope) and $f''(x)$ is the second derivative. For a parametric curve defined by x(t) and y(t), the formula is $\kappa = |x' y'' - y' x''| / ((x')^2 + (y')^2)^{3/2}$, where primes denote derivatives with respect to parameter t. The radius of curvature is simply $R = 1/\kappa$.

Common Curvature Values

A straight line y = mx + b has curvature κ = 0 everywhere. A circle of radius r has constant curvature κ = 1/r. The parabola y = x² at x = 0 has curvature κ = 2 (radius R = 0.5). The sine function y = sin(x) has curvature κ = 0 at x = 0 (inflection point) and κ = 1 at x = π/2. The exponential curve y = eˣ at x = 0 has curvature κ ≈ 0.354 (radius R ≈ 2.828).

The Osculating Circle

The osculating circle (from Latin "osculare," meaning "to kiss") at a point on a curve is the circle that best approximates the curve near that point. It passes through the point, shares the same tangent direction, and matches the curvature. The center of this circle lies along the normal line on the concave side of the curve, at a distance equal to the radius of curvature. Our tool visualizes this circle with a green dashed line.

Applications of Curvature

In road and railway design, curvature determines minimum turning radius and safe speed limits. Computer-aided design (CAD) uses curvature continuity (G2) for smooth surfaces in cars and aircraft. In optics, lens surface curvature affects focal length through the lensmaker's equation. Physics relates curvature to centripetal acceleration: a = κv² for a particle moving along a curved path. In general relativity, spacetime curvature describes gravity itself.

You can also try our Arc Length Calculator for calculating the length of a curve, or our Average Rate of Change Calculator for finding the slope between two points on a function.