Difficulty: Intermediate | Prerequisites: Vectors, dot and cross products, basic differentiation
Space curves are paths in 3D described by vector-valued functions r(t). This unit covers how to measure the curvature of these paths, how to decompose acceleration into tangential and normal components, and how to compute arc length. You will also need to find where a space curve intersects a plane or another curve, and to apply Newton's second law (F = ma) in vector form.
Vector-valued function / space curve
A function r(t) = ⟨x(t), y(t), z(t)⟩ that traces a curve in 3D as the parameter t varies. Think of it as a particle's position at time t.
Unit tangent vector T(t)
The normalised derivative of r(t): T(t) = r'(t) / |r'(t)|. It points in the direction of motion and has length 1.
Unit normal vector N(t)
N(t) = T'(t) / |T'(t)|. It points toward the centre of the curve's turning circle at each point. In simple terms, it tells you which way the curve is bending.
Curvature (κ)
A scalar measuring how sharply a curve bends at a given point. A straight line has curvature zero; a tight circle has high curvature.
Arc length
The total distance travelled along a curve from t = a to t = b, computed as the integral of |r'(t)| dt.
Reparametrisation by arc length
Rewriting r(t) so that the parameter s measures distance along the curve from a fixed starting point, with |r'(s)| = 1 (unit speed).
Tangential acceleration (aT)
The component of acceleration along the direction of motion. It measures how the speed changes. aT = (a · v) / |v|.
Normal acceleration (aN)
The component of acceleration perpendicular to the direction of motion. It measures how the direction changes. aN = |a × v| / |v|.
Given r(t) = ⟨x(t), y(t), z(t)⟩ and a plane equation such as 2x − y + z = 7, substitute the components of r(t) into the plane equation and solve for t.
Worked example from the cheat sheet:
r(t) = ⟨t² + 1, t − 1, 3t⟩ and plane 2x − y + z = 7.
Substituting: 2(t² + 1) − (t − 1) + 3t = 7
Simplifying: 2t² + 2 − t + 1 + 3t = 7, so 2t² + 2t + 3 = 7, giving 2t² + 2t − 4 = 0
Factoring: t² + t − 2 = 0, so (t + 2)(t − 1) = 0, giving t = −2 or t = 1
This yields two intersection points, one for each value of t.
The cheat sheet also identifies which point is closer to (2, 0, 5) and which is closer to (9, −3, −6) by computing the distance from the plane.
Given r₁(t) and r₂(s) (use different parameters), set their components equal and solve the resulting system of three equations in two unknowns. If a consistent solution exists, the curves intersect at those parameter values.
Worked example from the cheat sheet:
r₁(t) = ⟨t + 2, 6t + 3, 10(π + t)⟩ and r₂(s) = ⟨s − 1, s² − 10, 3s − 14⟩.
x: t + 2 = s − 1, so t = s − 3
y: 6t + 3 = s² − 10. Substituting t = s − 3: 6(s − 3) + 3 = s² − 10, giving s² − 6s + 5 = 0, so s = 5 or s = 1.
z: check cos(πt) = 3s − 14 for each solution to confirm consistency.
The cheat sheet finds s = 5, t = 2 works (with a check against the z-component), and the curves intersect at (4, 15, ...).
There are three forms of the curvature formula. Know all three and recognise when each is most convenient.
General 3D formula (cross-product form):
κ = |r'(t) × r''(t)| / |r'(t)|³
This is the workhorse. It works for any space curve r(t). Compute the cross product of the first and second derivatives, take its magnitude, and divide by the cube of the speed.
2D formula for y = f(x):
κ = |f''(x)| / (1 + (f'(x))²)^(3/2)
Use this when the curve is given as an explicit function of x rather than as a parametric curve.
Via the unit tangent vector:
κ = |T'(t)| / |r'(t)|
This follows directly from the definition but is usually less efficient to compute than the cross-product form.
T(t) = r'(t) / |r'(t)| points along the curve.
N(t) = T'(t) / |T'(t)| points toward the centre of curvature.
These two vectors, together with B(t) = T(t) × N(t) (the binormal), form the TNB frame (also called the Frenet-Serret frame). The TNB frame is a moving coordinate system attached to the curve.
The arc length of r(t) from t = a to t = b is:
L = ∫ₐᵇ |r'(t)| dt
Equivalently, if r(t) = ⟨x(t), y(t), z(t)⟩, this is:
L = ∫ₐᵇ √(x'(t)² + y'(t)² + z'(t)²) dt
The idea is to replace the arbitrary parameter t with a parameter s that measures actual distance along the curve.
Step 1: Compute the arc length function s(t) = ∫₀ᵗ |r'(u)| du.
Step 2: Solve for t in terms of s.
Step 3: Substitute back into r(t) to get r(s).
When parametrised by arc length, the speed is always 1: |r'(s)| = 1.
Worked example from the cheat sheet:
r(t) = ⟨4 sin t, 3t + 3, 4 cos t⟩.
|r'(t)| = √((4 cos t)² + 3² + (−4 sin t)²) = √(16 + 9) = 5.
Arc length from 0 to t: s = ∫₀ᵗ 5 du = 5t.
So t = s/5, and the reparametrised curve is r(s/5) = ⟨4 sin(s/5), 3(s/5) + 3, 4 cos(s/5)⟩.
The acceleration vector a(t) = r''(t) can be decomposed into two components:
Tangential component aT = (a · v) / |v|, where v = r'(t). This measures how fast the speed is changing.
Normal component aN = |a × v| / |v|. This measures how fast the direction is changing.
Alternatively, aN = κ|v|², which connects normal acceleration directly to curvature.
Given position r(t), the force on a particle of mass m is:
F = m · a(t) = m · r''(t)
Worked example from the cheat sheet:
Given r(t) = ⟨12 + 4t − 5t², arctan t, ln(t² + 1)⟩:
v(t) = r'(t) = ⟨4 − 10t, 1/(1 + t²), 2t/(t² + 1)⟩
a(t) = r''(t) = ⟨−10, −2t/(1 + t²)², (2(1 + t²) − 4t²)/(1 + t²)²⟩
F = ma = m · a(t), computed component by component.
The cheat sheet lists these differentiation rules, which are tested regularly:
d/dt [f(t) · g(t)] = f'(t) · g(t) + f(t) · g'(t) (scalar product rule)
d/dt [f(t) · g(t)] = f'(t) · g(t) + f(t) · g'(t) (dot product rule for vectors)
d/dt [f(t) × g(t)] = f'(t) × g(t) + f(t) × g'(t) (cross product rule, order matters)
Chain rule: d/dt [r(f(t))] = r'(f(t)) · f'(t)
When given a(t) and initial conditions for velocity and position, integrate twice:
v(t) = ∫ a(t) dt + C₁ (use initial velocity to find C₁)
r(t) = ∫ v(t) dt + C₂ (use initial position to find C₂)
The cheat sheet works through a full example of this, finding constants of integration from given initial conditions v(0) and r(0).
Curvature (3D): κ = |r'(t) × r''(t)| / |r'(t)|³
Curvature (2D): κ = |f''(x)| / (1 + (f'(x))²)^(3/2)
Curvature (via T): κ = |T'(t)| / |r'(t)|
Unit tangent: T(t) = r'(t) / |r'(t)|
Unit normal: N(t) = T'(t) / |T'(t)|
Arc length: L = ∫ₐᵇ |r'(t)| dt
Tangential acceleration: aT = (a · v) / |v|
Normal acceleration: aN = |a × v| / |v|
Tangential projection: aT = r'' · r' / |r'|² (alternative form; gives aT as the scalar projection)
Speed: |r'(t)| = √(x'(t)² + y'(t)² + z'(t)²)
Students often confuse the parameter t with arc length s. The parameter t can be anything (time, angle, an arbitrary number). Only after reparametrising does the parameter equal distance along the curve.
Forgetting that the cross product in the curvature formula is order-sensitive: r'(t) × r''(t) is not the same as r''(t) × r'(t). The magnitude is the same, but getting the sign wrong in related calculations (like the binormal) can cause errors.
Treating aT and aN as vectors. They are scalars (magnitudes of the tangential and normal components). The full decomposition is a = aT · T + aN · N.
When integrating acceleration to recover position, forgetting that each integration introduces a vector constant of integration (three scalar constants per integration step, not one).
⚠️ Curvature computation (especially the cross-product formula) is a near-certain exam question. Practise until it is mechanical.
⚠️ The arc length integral and reparametrisation by arc length are frequently tested, often with a curve whose speed simplifies to a constant (making the integral clean).
⚠️ Decomposing acceleration into aT and aN is a standard problem. Know both formulas cold.
⚠️ Recovering position from acceleration with initial conditions (integrate twice, solve for constants) appears regularly.
⚠️ Finding where a space curve intersects a plane is a quick, targeted question. Substitute and solve.
True or false: The curvature of a straight line is 1. (False. It is 0.)
Fill in the blank: Arc length from a to b is ∫ₐᵇ ________ dt. (|r'(t)|)
True or false: When a curve is parametrised by arc length, the speed |r'(s)| equals 1. (True.)
Fill in the blank: The tangential component of acceleration measures how fast the ________ changes. (Speed.)
True or false: The cross product formula for curvature is κ = |r' · r''| / |r'|³. (False. It uses the cross product ×, not the dot product ·.)
Q: Compute the curvature of r(t) = ⟨t, t², 0⟩ at t = 0.
A: r'(t) = ⟨1, 2t, 0⟩, r''(t) = ⟨0, 2, 0⟩. At t = 0: r' = ⟨1, 0, 0⟩, r'' = ⟨0, 2, 0⟩. Cross product: ⟨0, 0, 2⟩, magnitude 2. |r'| = 1. κ = 2/1 = 2.
Q: A particle has acceleration a(t) = ⟨0, 0, −9.8⟩, initial velocity v(0) = ⟨1, 2, 3⟩, and initial position r(0) = ⟨0, 0, 0⟩. Find r(t).
A: v(t) = ⟨t + 0, 2t + 0, −9.8t + 3⟩ = ⟨t, 2t, 3 − 9.8t⟩ (using v(0) to find constants, which are all zero except the z-component). Wait: v(t) = ∫ a dt = ⟨0, 0, −9.8t⟩ + C₁. v(0) = C₁ = ⟨1, 2, 3⟩. So v(t) = ⟨1, 2, 3 − 9.8t⟩. Then r(t) = ⟨t, 2t, 3t − 4.9t²⟩.
Q: Find where r(t) = ⟨t, 2t, 3t⟩ meets the plane x + y + z = 6.
A: t + 2t + 3t = 6, so 6t = 6, t = 1. The intersection point is (1, 2, 3).
Q: What is the arc length of r(t) = ⟨3 cos t, 3 sin t, 4t⟩ from t = 0 to t = 2π?
A: |r'(t)| = √(9 sin²t + 9 cos²t + 16) = √25 = 5. L = ∫₀²π 5 dt = 10π.
Curvature connects directly to the geometry of surfaces (Gaussian curvature, principal curvatures) studied in differential geometry. The TNB frame reappears when computing torsion and when studying the motion of charged particles in physics. Arc length parametrisation is fundamental to line integrals in the next unit: you integrate a function along a curve with respect to arc length. The F = ma problems here are the vector-calculus version of the force problems from introductory physics, extended to three dimensions.
space curves, vector-valued functions, parametric curves, curvature, kappa, unit tangent vector, unit normal vector, binormal vector, TNB frame, Frenet-Serret frame, arc length, reparametrisation, arc length parametrisation, tangential acceleration, normal acceleration, acceleration decomposition, Newton's second law vectors, F = ma, product rule vectors, cross product rule, dot product rule, chain rule vector functions, Calculus III, MATH 126, midterm 1 review, multivariable calculus