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Free curvature calculator & curve curvature calculator. Calculate curvature κ (kappa), radius of curvature ρ (rho), and analyze how curves bend. Our calculator uses differential geometry formulas to determine curvature at any point for explicit functions, parametric curves, and circles.
Last updated: February 2, 2026
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Enter functions like x², x³, sin(x), or cos(x)
Enter the x-coordinate where you want to calculate curvature
Curvature (κ):
0.178885
Radius of Curvature (ρ):
5.590170
First Derivative:
f'(1) = 2.000
Second Derivative:
f''(1) = 2.000
Formula:
κ = |2.000| / (1 + 2.000²)^(3/2) = 0.178885
Interpretation:
Moderate curvature - parabolic curve
Curvature Concepts:
Supported curves
Explicit, Parametric, Polar
Analyzes curvature κ for various curve representations and coordinate systems
Formula
ρ = 1/κ
Calculates the radius of the circle that best fits the curve at a point
Circle formula
κ = 1/r
For circles, curvature is constant and inversely proportional to radius
Formula
κ = |f''(x)| / (1 + (f'(x))²)^(3/2)
Uses first and second derivatives to calculate curvature at any point
Properties
Same tangent & curvature
The osculating circle shares the curve's tangent and curvature at contact
Analysis
Point-specific κ
Calculate how sharply a curve bends at any specified location
For function f(x) = x² at point x = 1:
Curvature (κ)
0.358569
Radius (ρ)
2.788675
Our curvature calculator analyzes curves using differential geometry principles. The calculation applies derivative analysis to determine how rapidly a curve changes direction at any given point. Curvature κ (kappa) quantifies this rate of change, while radius of curvature ρ (rho) represents the osculating circle radius.
κ = |f''(x)| / (1 + (f'(x))²)^(3/2)This formula calculates curvature for explicit functions y = f(x) using the first derivative f'(x) (slope) and second derivative f''(x) (rate of slope change). The result measures how sharply the curve bends at point x.
Shows osculating circles and curvature values at different points
Curvature is a fundamental concept in differential geometry that measures the deviation of a curve from being a straight line. The curvature formula derives from the rate of change of the unit tangent vector with respect to arc length. For practical calculations, we use derivatives with respect to x rather than arc length, which gives us the formula above.
Need help with other calculus tools? Check out our derivative calculator and concavity calculator.
Get Custom Calculator for Your PlatformResult: κ ≈ 0.358569, ρ ≈ 2.788675
Moderate curvature indicating a parabolic bend with osculating circle radius of 2.79 units.
Circle with radius r = 5
κ = 1/5 = 0.2 (constant everywhere)
f(x) = 2x + 1
κ = 0 (no curvature - straight)
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