Difficulty: Intermediate | Prerequisites: Parametric curves, derivatives of vector functions, dot product, cross product, magnitude of a vector.
Once you can describe curves in 3D with parametric equations, the next questions are: how long is the curve, how sharply does it bend, and how does acceleration break down along and perpendicular to the path? Arc length, curvature, and the tangential/normal decomposition of acceleration answer these. They build directly on the vector derivative r'(t) and use dot and cross products extensively. These ideas are central to physics (forces on a moving particle) and will reappear when you study line integrals later in the course.
Arc length measures the total distance along a curve by integrating the speed |r'(t)|. Curvature measures how fast the curve changes direction per unit arc length. Acceleration splits into a tangential component (speeding up or slowing down) and a normal component (changing direction), using the unit tangent and unit normal vectors.
Arc length
The total distance travelled along a curve r(t) from t = a to t = b, computed as s = ∫ₐᵇ |r'(t)| dt. Think of it as straightening the curve out and measuring it with a ruler.
Unit tangent vector, T(t)
The direction the curve is heading at each point, normalised to length 1: T(t) = r'(t) / |r'(t)|. In simple terms, it points "forward" along the curve.
Unit normal vector, N(t)
The direction the curve is turning towards, perpendicular to T: N(t) = T'(t) / |T'(t)|. Think of it as pointing towards the centre of the circle the curve is locally following.
Curvature, κ(t)
A measure of how sharply the curve bends at a given point: κ = |dT/ds|, the rate of change of the unit tangent with respect to arc length. A straight line has κ = 0; a tight circle has large κ. The reciprocal 1/κ is the radius of curvature.
Tangential acceleration, aT
The component of acceleration along the direction of motion. It measures how fast the speed is changing: aT = d|v|/dt = (r'(t) · r''(t)) / |r'(t)|. In simple terms, this is the "gas pedal" part of acceleration.
Normal acceleration, aN
The component of acceleration perpendicular to the direction of motion. It measures how fast the direction is changing: aN = κ|v|² = |r'(t) × r''(t)| / |r'(t)|. Think of it as the "steering wheel" part of acceleration.
Computing arc length
Given r(t) = ⟨x(t), y(t), z(t)⟩, the arc length from t = a to t = b is:
s = ∫ₐᵇ |r'(t)| dt
where |r'(t)| = √(x'(t)² + y'(t)² + z'(t)²) is the speed.
The quantity |r'(t)| dt is a tiny piece of distance, and integrating adds them up. If r(t) traces a curve in 2D (say z(t) = 0), this reduces to the arc length formula from Calc II.
Tangent unit vector
T(t) = r'(t) / |r'(t)|
This is just the velocity vector divided by its own magnitude, giving a unit vector that points in the direction of travel.
For a function y = f(x)
When the curve is given as y = f(x) rather than a parametric form, the relevant expressions become:
r(x) = ⟨x, f(x)⟩, so r'(x) = ⟨1, f'(x)⟩
|r'(x)| = √(1 + f'(x)²)
The cross product |r' × r''| simplifies so that curvature formulas involve |f''(x)| in the numerator and (1 + f'(x)²)^(3/2) in the denominator.
Curvature measures how rapidly the direction of the curve changes per unit of distance travelled.
Definition
κ(t) = |dT/ds| = |T'(t)| / |r'(t)|
This is the magnitude of the rate of change of the unit tangent vector with respect to arc length.
Computational formula (using cross product)
κ(t) = |r'(t) × r''(t)| / |r'(t)|³
This is usually the most practical formula because it avoids computing T(t) and differentiating it.
For y = f(x)
κ(x) = |f''(x)| / (1 + f'(x)²)^(3/2)
This follows from the cross product formula when you parametrise the curve as r(x) = ⟨x, f(x)⟩.
Radius of curvature
R = 1/κ. A large radius of curvature means the curve is nearly straight; a small radius means it bends sharply. The radius of curvature is the radius of the "osculating circle," the circle that best fits the curve at that point.
The acceleration vector a = r''(t) can be decomposed as:
a = aT T + aN N
where T is the unit tangent and N is the unit normal.
Tangential component
aT = (r'(t) · r''(t)) / |r'(t)| = d|v|/dt
This equals the derivative of the speed. It tells you whether the particle is speeding up (aT > 0) or slowing down (aT < 0).
Alternate way to remember: aT = component of a along v, which is just the scalar projection of a onto v.
Normal component
aN = |r'(t) × r''(t)| / |r'(t)| = κ|v|²
This is always non-negative. It measures how strongly the curve is pulling the particle away from a straight-line path.
The relationship aN = κ|v|² is worth remembering: at a given speed, more curvature means more normal acceleration. At a given curvature, more speed means dramatically more normal acceleration (because of the square).
Arc length: s = ∫ₐᵇ |r'(t)| dt
Unit tangent: T(t) = r'(t) / |r'(t)|
Curvature (general): κ = |T'(t)| / |r'(t)| = |r'(t) × r''(t)| / |r'(t)|³
Curvature (y = f(x)): κ = |f''(x)| / (1 + f'(x)²)^(3/2)
Radius of curvature: R = 1/κ
Tangential acceleration: aT = (r' · r'') / |r'| = |v|' (derivative of speed)
Normal acceleration: aN = |r' × r''| / |r'| = κ|v|²
Full decomposition: a = aT T + aN N
Curvature determines how much a car needs to slow down on a bend: highway engineers design curves with curvature that keeps normal acceleration within safe limits at the posted speed.
Arc length parametrisation is used in robotics and CNC machining, where a tool must move at constant speed along a path. The tangential/normal decomposition of acceleration appears throughout orbital mechanics and anywhere forces act on a moving body.
Students often confuse speed |v| with velocity v. Speed is a scalar (always non-negative). Velocity is a vector. The arc length integral uses speed, not velocity.
Curvature is not the same as the second derivative. For y = f(x), curvature uses f''(x) but also involves (1 + f'(x)²)^(3/2) in the denominator. A parabola has a changing curvature even though f''(x) is constant.
Students sometimes believe aN can be negative. It cannot. Normal acceleration is defined as a magnitude and is always ≥ 0.
The cross product formula for curvature has |r'(t)|³ (cubed) in the denominator. Students frequently write |r'(t)|² by mistake, which gives the wrong units and the wrong answer.
⚠️ The cross product curvature formula κ = |r' × r''| / |r'|³ is the most commonly tested version. Know it cold.
⚠️ Be prepared to find aT and aN given r(t). This usually appears as: compute r'(t) and r''(t), then use the dot product formula for aT and the cross product formula for aN.
⚠️ For y = f(x) curvature problems, the formula κ = |f''(x)| / (1 + f'(x)²)^(3/2) is often faster than converting to a parametric form.
⚠️ Arc length computations can produce integrals that are difficult to evaluate. If the integral simplifies cleanly, the problem is likely testing whether you set it up correctly.
True or false: Arc length depends on the parametrisation of the curve.
Fill in the blank: Curvature is the rate of change of the unit tangent vector with respect to ______.
True or false: The normal component of acceleration can be negative.
Fill in the blank: The cross product curvature formula has |r'(t)| raised to the power ______ in the denominator.
True or false: If a particle moves along a straight line at varying speed, its normal acceleration is zero.
Q: Find the arc length of r(t) = ⟨3t, 4t⟩ from t = 0 to t = 2.
A: r'(t) = ⟨3, 4⟩, so |r'(t)| = 5. Arc length = ∫₀² 5 dt = 10.
Q: Find the curvature of y = x² at x = 0.
A: f'(x) = 2x, f''(x) = 2. At x = 0: κ = |2| / (1 + 0)^(3/2) = 2.
Q: Given r(t) = ⟨cos t, sin t, t⟩, find the unit tangent vector T(t).
A: r'(t) = ⟨-sin t, cos t, 1⟩. |r'(t)| = √(sin²t + cos²t + 1) = √2. So T(t) = (1/√2)⟨-sin t, cos t, 1⟩.
Q: For r(t) = ⟨t, t², 0⟩, find aT and aN at t = 0.
A: r'(t) = ⟨1, 2t, 0⟩, r''(t) = ⟨0, 2, 0⟩. At t = 0: r'(0) = ⟨1, 0, 0⟩, r''(0) = ⟨0, 2, 0⟩. aT = (r' · r'') / |r'| = 0/1 = 0. r' × r'' = ⟨0, 0, 2⟩, so |r' × r''| = 2. aN = 2/1 = 2.
Q: What is the radius of curvature of a circle of radius 5?
A: A circle of radius R has curvature κ = 1/R, so κ = 1/5. The radius of curvature is 1/κ = 5, confirming that the osculating circle is the circle itself.
Arc length parametrisation underpins line integrals, which you will encounter when studying work done by a force field along a curve. Curvature connects to the second derivative test in multivariable calculus through the shape of level curves. The tangential/normal decomposition of acceleration is the same framework used in physics for centripetal acceleration and uniform circular motion.
Arc length, arc length parametrisation, unit tangent vector, T(t), unit normal vector, N(t), curvature, kappa, κ, radius of curvature, osculating circle, tangential acceleration, normal acceleration, centripetal acceleration, speed, velocity, cross product curvature formula, MATH 126, Calculus III, multivariable calculus, UW MATH 126, vector-valued functions, TNB frame, Frenet frame