Section 12.10

Curvature

Quantifying the "tightness" of the turn at any point on a curve.

1

Introduction

Curvature () measures how quickly the tangent vector changes direction.
- A straight line has zero curvature.
- A small circle has high curvature (tight turn).
- A large circle has low curvature (gentle turn).

Interactive: Osculating Circle

The Osculating Circle ("kissing circle") best approximates the curve at Point P.
Drag to see the circle change size. Sharper turn = Smaller circle = High Curvature.

2

Key Formulas

General Vector Formula

The cross-product form is usually easier to calculate.

Plane Curve y=f(x)

3

Worked Examples

Example 1: Curvature of a Line (Level 1)

Show that the curvature of a straight line is 0.

Let .

  • (Constant vector)
  • (Zero vector)
  • Using the formula:.

Example 2: Curvature of a Circle (Level 2)

Find the curvature of a circle with radius : .

Derivatives


Magnitudes

Cross Product

.
Magnitude = .

Result

.

Makes sense: Large radius = Small curvature.

Example 3: Curvature of a Parabola (Level 3)

Find the curvature of at the origin ().

Use the function formula: .

  • At :
4

Practice Quiz

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