Difficulty: Intermediate to Advanced | Prerequisites: §13.1-13.2 (vector-valued functions, tangent vectors), integration techniques (by parts, trig substitution), hyperbolic functions.
Once you can parametrise a curve and compute its tangent vector, the next questions are: how long is the curve, and how sharply does it bend? Arc length and curvature answer those. This unit also introduces a gallery of named curves (cycloids, involutes, catenaries, Cornu spirals) that appear throughout physics and engineering. Each one illustrates a different technique: reparametrisation by arc length, computing curvature with the cross-product formula, and connecting geometric properties (like linearly increasing curvature) to real-world design. You should be comfortable with the §13.1-13.2 material on tangent vectors and differentiation rules before starting here.
Arc length is the integral of the speed |r'(t)|. Reparametrising a curve by arc length means using distance along the curve as the parameter, which simplifies curvature to κ = |dT/ds|. Curvature measures bending; special curves like cycloids, involutes, and Cornu spirals each have distinctive curvature properties that make them useful in engineering.
Arc length
The total distance along a curve from r(a) to r(b), computed as the integral of the speed: L = ∫ₐᵇ |r'(t)| dt.
Think of it as laying a string along the curve and measuring it.
Arc length function
s(t) = ∫ₐᵗ |r'(u)| du. It gives the distance travelled along the curve from the starting point up to parameter value t.
Arc length parametrisation
A parametrisation r(s) where the parameter s is arc length itself. In such a parametrisation, |r'(s)| = 1, meaning the curve is traced at unit speed.
In simple terms, each unit increase in the parameter corresponds to moving exactly one unit of distance along the curve.
Curvature (κ)
A scalar measuring how sharply a curve bends at a given point: κ = |dT/ds|, where T is the unit tangent vector and s is arc length.
Think of it as the reciprocal of the radius of the best-fitting circle at that point. High curvature means a tight bend; zero curvature means a straight line.
Cycloid
The curve traced by a point on the rim of a wheel of radius R rolling along a flat surface. Its parametrisation is c(t) = ⟨R(t - sin t), R(1 - cos t)⟩.
Involute (of a circle)
The curve traced by the end of a taut string being unwound from a circle of radius R. The string is always tangent to the circle at the point where it leaves the spool.
Catenary
The curve y = cosh(x) - 1 (or a shifted version). In the context of pursuit curves, it describes the path of a receiver running towards a moving ball.
Cornu spiral (clothoid)
The curve r(t) = ⟨S(t), C(t)⟩ where S and C are the Fresnel sine and cosine integrals. Its curvature increases linearly with arc length.
In simple terms, it is a spiral that starts straight and curls tighter at a steady rate.
Fresnel integrals
S(x) = ∫₀ˣ sin(u²) du and C(x) = ∫₀ˣ cos(u²) du. They do not have closed-form expressions but are well-tabulated and appear in optics and road design.
Hyperbolic functions
cosh(x) = (eˣ + e⁻ˣ)/2 and sinh(x) = (eˣ - e⁻ˣ)/2. Key identity: cosh²(x) - sinh²(x) = 1. The derivative of sinh is cosh, and the derivative of cosh is sinh (no sign change, unlike trig).
Computing arc length
For a curve r(t), a ≤ t ≤ b:
L = \int_a^b |\mathbf{r}'(t)|\, dt = \int_a^b \sqrt{[f'(t)]^2 + [g'(t)]^2 + [h'(t)]^2}\, dtThe integrand |r'(t)| is the speed. The integral accumulates distance.
The arc length function
Define s(t) = ∫ₐᵗ |r'(u)| du. This is monotonically increasing (assuming r'(t) ≠ 0), so it has an inverse t = t(s). Substituting back gives the arc length parametrisation r(s) = r(t(s)).
Why arc length parametrisation matters
When a curve is parametrised by arc length, |r'(s)| = 1. This simplifies many formulas, especially curvature: κ = |r''(s)|. In practice, you rarely compute a full reparametrisation (the integrals are often intractable), but understanding it is essential for the theory.
How to check if a parametrisation is by arc length
Compute |r'(t)|. If it equals 1 for all t, the curve is already parametrised by arc length.
Worked example (Cornu spiral): r(t) = ⟨S(t), C(t)⟩ where S'(t) = sin(t²) and C'(t) = cos(t²). Then r'(t) = ⟨sin(t²), cos(t²)⟩ and |r'(t)| = √(sin²(t²) + cos²(t²)) = 1. So t is already arc length.
Definition
Curvature κ measures the rate at which the unit tangent vector T changes direction with respect to arc length:
\kappa = \left|\frac{d\mathbf{T}}{ds}\right|Three useful formulas for κ
Formula 1 (arc length parametrisation): If r(s) is parametrised by arc length, κ = |r''(s)|.
Formula 2 (general parameter): κ = |T'(t)| / |r'(t)|.
Formula 3 (cross-product formula, often the most practical):
\kappa = \frac{|\mathbf{r}'(t) \times \mathbf{r}''(t)|}{|\mathbf{r}'(t)|^3}For plane curves r(t) = ⟨x(t), y(t)⟩, the cross product reduces to a determinant:
\kappa = \frac{|x'y'' - y'x''|}{(x'^2 + y'^2)^{3/2}}Choosing the right formula
If the curve is already parametrised by arc length (|r'| = 1), Formula 1 is immediate. Otherwise, Formula 3 usually involves less work than Formula 2 because you do not need to compute T(t) = r'(t)/|r'(t)| and then differentiate the quotient.
Curvature of the circle involute
The circle involute r(θ) = R⟨cos θ + θ sin θ, sin θ - θ cos θ⟩ has |r'(θ)| = Rθ. After computing r''(θ) and applying the plane-curve formula, the curvature works out to κ = 1/(Rθ). So curvature decreases as the involute spirals outward.
Curvature of the Cornu spiral
Since the Cornu spiral is parametrised by arc length (s = t), its curvature is κ = |r''(t)|. We have r'(t) = ⟨sin(t²), cos(t²)⟩ and r''(t) = ⟨2t cos(t²), -2t sin(t²)⟩, giving κ = 2t. So for t ≥ 0, curvature is a linear function of arc length.
Setup
A wheel of radius R rolls along a flat surface without slipping. Track the point on the rim that starts at the bottom of the wheel (touching the ground).
Parametrisation
With the contact point starting at the origin, one arch of the cycloid is:
\vec{c}(t) = \langle R(t - \sin t),\; R(1 - \cos t) \rangle, \quad 0 \le t \le 2\piHere t is the angle the wheel has rotated through.
Derivation sketch: The centre of the wheel is at (Rt, R). The tracked point is offset from the centre by R in the direction that depends on the rotation angle: the horizontal offset is -R sin t and the vertical offset is -R cos t.
Arc length of one arch
Differentiate: c'(t) = ⟨R(1 - cos t), R sin t⟩.
|c'(t)| = R√((1 - cos t)² + sin²t) = R√(2 - 2 cos t) = 2R|sin(t/2)| = 2R sin(t/2) for 0 ≤ t ≤ 2π.
Integrating:
L = \int_0^{2\pi} 2R\sin\frac{t}{2}\, dt = 2R\left[-2\cos\frac{t}{2}\right]_0^{2\pi} = 2R(2 + 2) = 8RThe length of one arch of a cycloid is 8R, which is four times the diameter of the wheel. This is a clean, memorable result.
Setup
Unwind a taut string from a spool of radius R. The free end of the string traces the involute. The string is always tangent to the circle at the point where it leaves.
Length of the unwound thread
At angle θ, the arc of circle that has been unwound has length Rθ. So the purple segment (the taut string) has length Rθ.
Parametrisation
The vector from the origin to the point on the circle at angle θ is R⟨cos θ, sin θ⟩. The unwound string extends from that point in a direction perpendicular to the radius (tangent to the circle), rotated 90° clockwise, which is ⟨sin θ, -cos θ⟩. The string's length is Rθ, so:
\mathbf{r}(\theta) = R\langle \cos\theta + \theta\sin\theta,\; \sin\theta - \theta\cos\theta \rangleArc length from θ = 0 to θ = b
r'(θ) = R⟨θ cos θ, θ sin θ⟩ = Rθ⟨cos θ, sin θ⟩, so |r'(θ)| = Rθ.
L = \int_0^b R\theta\, d\theta = \frac{Rb^2}{2}Arc length parametrisation
From s = Rθ²/2, solve for θ = √(2s/R). Substitute back into r(θ).
Curvature
Using κ = |T'(θ)| / |r'(θ)| or the cross-product formula: T(θ) = ⟨cos θ, sin θ⟩, so T'(θ) = ⟨-sin θ, cos θ⟩ and |T'(θ)| = 1. Therefore κ = 1/(Rθ). The curvature decreases inversely with θ: the involute is tightest near the spool and flattens as it spirals out.
The catenary curve
The curve y = cosh(x) - 1 is the shape a hanging chain or cable takes under its own weight. In the homework problem, it describes the path of a receiver who always runs directly towards a moving ball. The parametrisation is r(x) = ⟨x, cosh(x) - 1, 0⟩.
Key hyperbolic identities
d/dx sinh(x) = cosh(x)
d/dx cosh(x) = sinh(x)
cosh²(x) - sinh²(x) = 1
1 + sinh²(x) = cosh²(x)
Arc length of the catenary
r'(x) = ⟨1, sinh(x), 0⟩ so |r'(x)| = √(1 + sinh²(x)) = cosh(x). The arc length from 0 to a is ∫₀ᵃ cosh(x) dx = sinh(a).
Projectile motion in the yz-plane
The football travels in the yz-plane with initial speed v₀ = 50 ft/s at angle α above the ground. Acceleration is ⟨0, 0, -g⟩ with g = 32 ft/s².
The position as a function of time: y(t) = v₀ cos(α) · t and z(t) = v₀ sin(α) · t - (1/2)g t².
The ball must travel 20 yards = 60 feet in the y-direction and return to z = 0 (ground level) at the catch point. Setting z = 0 gives t = 2v₀ sin(α)/g, and substituting into y(t) = 60 gives the equation to solve for α.
Receiver speed
The receiver's arc length along the catenary equals sinh(a) where a = ln(21 + √440). The receiver must cover this distance in the same time the ball is in the air. Speed = sinh(a) / t_flight.
Definition
The Cornu spiral (also called a clothoid or Euler spiral) is r(t) = ⟨S(t), C(t)⟩ where S(x) = ∫₀ˣ sin(u²) du and C(x) = ∫₀ˣ cos(u²) du.
Arc length parametrisation
As shown above, |r'(t)| = 1, so the parameter t is already arc length. No reparametrisation needed.
Curvature is linear in arc length
Since t = s (arc length), κ = |r''(t)| = 2t = 2s. The curvature increases linearly from 0. This means the spiral starts out nearly straight and curls more tightly at a constant rate.
Why this matters for highway design
On a straight road, curvature is 0. On a circular exit ramp, curvature is constant (1/R). If you transition abruptly from straight to circular, a driver must turn the steering wheel instantaneously, which is uncomfortable and dangerous at speed.
A Cornu spiral provides a smooth transition: its curvature increases linearly from 0 to 1/R, allowing the driver to turn the wheel at a steady rate. This is why highway engineers use clothoid transition curves (also called spiral easements) at the entry and exit of every curved section of road.
Arc length:
L = \int_a^b |\mathbf{r}'(t)|\, dtCurvature (cross-product formula):
\kappa = \frac{|\mathbf{r}'(t) \times \mathbf{r}''(t)|}{|\mathbf{r}'(t)|^3}Curvature (plane curve):
\kappa = \frac{|x'y'' - y'x''|}{(x'^2 + y'^2)^{3/2}}Cycloid parametrisation:
\vec{c}(t) = \langle R(t - \sin t),\; R(1 - \cos t) \rangleCycloid arch length: L = 8R.
Circle involute:
\mathbf{r}(\theta) = R\langle \cos\theta + \theta\sin\theta,\; \sin\theta - \theta\cos\theta \rangleInvolute arc length from 0 to b: L = Rb²/2. Involute curvature: κ = 1/(Rθ).
Cornu spiral curvature: κ = 2s (linear in arc length).
Hyperbolic identities: cosh²x - sinh²x = 1, d/dx cosh x = sinh x, d/dx sinh x = cosh x.
Cycloids appear in the brachistochrone problem (the curve of fastest descent under gravity) and in the design of gear teeth. Circle involutes are the standard tooth profile for spur gears because the contact angle between meshing involute gears remains constant. Cornu spirals are used as transition curves on highways and railways, providing a smooth change in curvature from straight to curved sections. Catenaries describe the shape of hanging cables and chains, and appear in the design of suspension bridges and power lines.
Students often confuse the parameter t with arc length s. Unless |r'(t)| = 1, the parameter is not arc length, and κ ≠ |r''(t)| in general. Only use κ = |r''(t)| when you have confirmed the parametrisation is by arc length.
When computing cycloid arc length, students sometimes forget that √(2 - 2 cos t) simplifies to 2|sin(t/2)| via the half-angle identity. Without this simplification the integral looks intractable.
Students sometimes mix up the involute direction. The unwound string extends tangent to the circle, perpendicular to the radius. Getting this 90° rotation wrong gives the wrong parametrisation.
When working with hyperbolic functions, students sometimes apply trig identities (like sin²x + cos²x = 1) to sinh and cosh. The correct identity is cosh²x - sinh²x = 1, with a minus sign.
⚠️ Arc length integrals are frequently tested. Know the formula and be ready to simplify the integrand using trig or hyperbolic identities.
⚠️ "Show that r(t) parametrises the curve by arc length" means: compute |r'(t)| and show it equals 1.
⚠️ The cross-product curvature formula is the workhorse for exam problems. Memorise it and practise applying it to specific curves.
⚠️ The cycloid result L = 8R is a classic. Expect to either derive it or use it.
⚠️ For the Cornu spiral, the key result is that κ = 2t (linear in arc length). The explanation for highway design is a common short-answer or essay question.
True or false: if |r'(t)| = 3 for all t, the curve is parametrised by arc length. (False, arc length parametrisation requires |r'| = 1.)
Fill in the blank: the length of one arch of a cycloid traced by a wheel of radius R is ______. (8R.)
True or false: the curvature of a straight line is 1. (False, it is 0.)
Fill in the blank: the curvature of the Cornu spiral is κ = ______, a linear function of arc length. (2t, or equivalently 2s.)
True or false: cosh²(x) + sinh²(x) = 1. (False, the correct identity is cosh²(x) - sinh²(x) = 1.)
Q: Find the arc length of r(t) = ⟨3 cos t, 4t, 3 sin t⟩ from t = 0 to t = 2π.
A: r'(t) = ⟨-3 sin t, 4, 3 cos t⟩. |r'(t)| = √(9 sin²t + 16 + 9 cos²t) = √(9 + 16) = 5. So L = ∫₀²π 5 dt = 10π.
Q: Show that r(t) = ⟨cos t, sin t⟩ is parametrised by arc length.
A: r'(t) = ⟨-sin t, cos t⟩. |r'(t)| = √(sin²t + cos²t) = 1. Since |r'(t)| = 1 for all t, the curve is parametrised by arc length.
Q: Compute the curvature of y = x² at x = 0.
A: Parametrise as r(t) = ⟨t, t²⟩. r'(t) = ⟨1, 2t⟩, r''(t) = ⟨0, 2⟩. Using the plane-curve formula: κ = |1(2) - 2t(0)| / (1 + 4t²)^(3/2) = 2/(1 + 4t²)^(3/2). At t = 0, κ = 2.
Q: A wheel of radius 5 rolls along the ground. What is the length of one arch of the resulting cycloid?
A: L = 8R = 8(5) = 40.
Q: Explain in one sentence why Cornu spirals are used for highway exit ramps.
A: Their curvature increases linearly with distance, allowing a driver to turn the steering wheel at a constant rate when transitioning from a straight road to a circular curve.
Arc length and curvature build directly on the tangent-vector material from §13.1-13.2. The unit tangent T, principal normal N, and binormal B (the Frenet-Serret frame, §13.4) all depend on arc length parametrisation and curvature.
Curvature reappears in Chapter 16 when you study line integrals and in the study of surfaces (Chapters 14-15), where Gaussian curvature generalises the idea to two dimensions.
The projectile-motion setup (Problem 8) connects to the vector calculus of position, velocity, and acceleration from Chapter 13, tying the pure geometry of curves to Newtonian mechanics.
arc length, arc length parametrisation, reparametrisation, curvature, curvature formula, cross product curvature, unit tangent vector, cycloid, cycloid arc length, circle involute, involute of a circle, catenary, cosh, sinh, hyperbolic functions, Cornu spiral, clothoid, Euler spiral, Fresnel integrals, Fresnel sine integral, Fresnel cosine integral, transition curve, highway design curve, brachistochrone, MATH 215, Stewart Multivariable Calculus §13.3, §13.4, Calculus III