Curvature measures how sharply a curve turns at a particular point. A straight line has zero curvature because its direction never changes, while a tight bend has large curvature. In calculus, curvature connects geometry with derivatives by describing how fast the tangent direction changes as you move along a curve.
It matters in physics, engineering, computer graphics, and road design because turning rate affects motion, force, and shape.
Key Facts
- Curvature measures the rate at which the unit tangent vector changes with arc length: κ = |dT/ds|.
- For a plane curve y = f(x), curvature is κ = |y''| / (1 + (y')^2)^(3/2).
- For a parametric curve r(t) = <x(t), y(t)>, curvature is κ = |x'y'' - y'x''| / ((x')^2 + (y')^2)^(3/2).
- The radius of curvature is R = 1/κ, so larger curvature means a smaller turning radius.
- The osculating circle has radius R and matches the curve's tangent and curvature at the point of contact.
- A line has κ = 0, while a circle of radius r has constant curvature κ = 1/r.
Vocabulary
- Curvature
- Curvature is a measure of how quickly a curve changes direction at a point.
- Unit tangent vector
- The unit tangent vector is a length 1 vector that points in the direction of motion along a curve.
- Arc length
- Arc length is the distance measured along a curve rather than straight across it.
- Radius of curvature
- The radius of curvature is the radius of the circle that best fits the curve at a point.
- Osculating circle
- The osculating circle is the circle that touches a curve at a point and has the same tangent direction and curvature there.
Common Mistakes to Avoid
- Using y'' alone as curvature. This is wrong because curvature also depends on the slope y' through the factor (1 + (y')^2)^(3/2).
- Forgetting the absolute value in the curvature formula. Curvature is usually a nonnegative size of turning, while the sign belongs to signed curvature or orientation.
- Confusing radius of curvature with curvature. They are reciprocals, so a small radius means large curvature and a large radius means small curvature.
- Assuming the osculating circle matches the whole curve. It only gives the best local circular approximation near one point, not a global model of the curve.
Practice Questions
- 1 Find the curvature of y = x^2 at x = 0 using κ = |y''| / (1 + (y')^2)^(3/2).
- 2 A curve has curvature κ = 0.25 m^-1 at a point. What is its radius of curvature, and what does that radius represent?
- 3 Two roads have the same slope at a point, but Road A bends gently and Road B bends sharply. Explain which road has larger curvature and how its osculating circle compares in size.