Difficulty: Intermediate | Prerequisites: parametric curves, derivatives of vector-valued functions, cross product

Curvature quantifies how sharply a curve bends at each point. It is a scalar (always non-negative) that you can compute from a parametric description of the curve. This concept bridges the gap between the geometry of curves and the calculus of vector-valued functions. You should already be comfortable differentiating vector-valued functions and computing cross products before tackling curvature.

TL;DR

Curvature kappa measures how fast a curve is changing direction at a given point. A straight line has curvature 0 everywhere. A circle of radius R has constant curvature 1/R. For general parametric curves, you compute curvature using the cross product of the first and second derivatives.

Key Terms

Curvature (kappa)

A non-negative scalar that measures how "bent" a smooth curve is at a given point. Defined as the magnitude of the rate of change of the unit tangent vector with respect to arc length.

Think of it as: the sharper the turn, the higher the curvature.

Unit tangent vector T(t)

T(t) = r'(t) / |r'(t)|. Points in the direction the curve is heading at parameter value t.

In simple terms, the direction of travel, normalised to length 1.

Radius of curvature

R = 1/kappa. The radius of the osculating circle (the circle that best approximates the curve at that point).

In simple terms, if the curve were bending at a constant rate, this is the radius of the circle it would trace.

Core Content

Definition

• Curvature kappa(t) is always greater than or equal to 0

• It measures how rapidly the direction of the curve is changing

Special Cases

• Lines: curvature = 0 everywhere (no bending at all)

• Circles of radius R: curvature = 1/R everywhere (constant bending, and smaller circles bend more sharply)

Two Formulas for Computing Curvature

Formula 1 (using the unit tangent vector):

kappa(t) = |T'(t)| / |r'(t)|

This is the definition restated: the rate of change of the unit tangent, divided by the speed.

Formula 2 (using the cross product, usually easier):

kappa(t) = |r'(t) x r''(t)| / |r'(t)|^3

This formula avoids computing T(t) and its derivative directly. You compute r'(t) and r''(t), take their cross product, find its magnitude, and divide by |r'(t)| cubed. This is typically the faster route on exams.

When to Use Which

• If the problem already gives you T(t) or asks you to find it, Formula 1 may be natural

• In most other situations, Formula 2 (the cross product version) is quicker and less error-prone

Formulas

\kappa(t) = \frac{1}{|\vec{v}|}\left|\frac{d\vec{T}}{dt}\right| = \frac{|\vec{T}'|}{|\vec{r}'|}

\kappa(t) = \frac{|\vec{r}' \times \vec{r}''|}{|\vec{r}'|^3}

\text{Circle of radius } R\text{: } \kappa = \frac{1}{R}

\text{Line: } \kappa = 0

Real-World Applications

Curvature determines how tightly a road or railway can turn at a given speed. Civil engineers use curvature to design safe curves with appropriate banking angles. In optics, the curvature of a lens surface governs how strongly it focuses or disperses light.

Common Misconceptions

• Students sometimes think curvature can be negative. It cannot. Curvature is defined as a magnitude and is always >= 0.

• Confusing curvature with slope: a curve can have a steep slope but zero curvature (a straight line going steeply uphill). Curvature is about how much the direction is changing, not how steep the path is.

• Forgetting to cube |r'(t)| in the denominator of Formula 2. This is a common arithmetic slip on exams.

Why It Matters / Exam Flags

• ⚠️ The cross product formula kappa = |r' x r''| / |r'|^3 is the one you will most likely use on the exam. Practise it until the steps are automatic.

• ⚠️ Know the two benchmark cases: lines (kappa = 0) and circles (kappa = 1/R).

• ⚠️ Be able to compute r'(t) and r''(t) quickly for polynomial and trigonometric parametrisations.

Quick Self-Test

1. True or false: A straight line has curvature 1.

   • False. A straight line has curvature 0.

1. Fill in the blank: A circle of radius 5 has curvature ___.

   • 1/5.

1. True or false: Curvature can be negative.

   • False. Curvature is always >= 0.

1. Fill in the blank: The easier computational formula for curvature is kappa = |r' x r''| / |r'|___.

   • Cubed (to the power of 3).

Practice Q&A

Q: Find the curvature of the circle r(t) = <3cos(t), 3sin(t), 0>.

A: r'(t) = <-3sin(t), 3cos(t), 0>, so |r'(t)| = 3. r''(t) = <-3cos(t), -3sin(t), 0>. r' x r'' = <0, 0, 9>. |r' x r''| = 9. Curvature = 9/27 = 1/3. This matches kappa = 1/R with R = 3.

Q: What is the curvature of r(t) =

A: r'(t) = <1, 2, 3>, r''(t) = <0, 0, 0>. r' x r'' = <0, 0, 0>. |r' x r''| = 0. Curvature = 0. This makes sense because the curve is a straight line.

Q: Compute the curvature of r(t) =

A: r'(t) = <1, 2t, 0>, r''(t) = <0, 2, 0>. At t = 0: r'(0) = <1, 0, 0>, r''(0) = <0, 2, 0>. r' x r'' = <0, 0, 2>. |r' x r''| = 2. |r'(0)| = 1. Curvature = 2/1 = 2.

Connections to Other Topics

Curvature feeds directly into the decomposition of acceleration into tangential and normal components (covered in the next set of notes on velocity and acceleration vectors). The normal component of acceleration equals kappa times v squared. Curvature also appears in the study of the osculating plane and the Frenet-Serret formulas, which give a complete local description of curve geometry.

Related Terms / Tags

curvature, kappa, unit tangent vector, radius of curvature, osculating circle, parametric curve, cross product formula, bending, arc length, smooth curve, MATH 126, Calculus III, vector-valued functions