Difficulty: Intermediate | Prerequisites: MATH 215 §13.1 (vector-valued functions), basic differentiation, trigonometric identities.
This material sits at the heart of multivariable calculus: you are learning to describe curves in 2D and 3D using parametric equations, then extracting geometric information (tangent directions, arc length) from those descriptions. If single-variable parametrisation felt comfortable, this extends the same idea into higher dimensions. If you skipped §13.1, go back and review vector-valued functions first.
Parametric curves let you describe paths in space using a single parameter (usually t). The derivative of a parametrisation gives the velocity (tangent) vector. Knowing how to build parametrisations for curves defined by geometric constraints (intersections of surfaces, helices, spirals) is the central skill here, and the velocity vector is the bridge to tangent lines, tangent planes, and arc length.
Parametrisation (parametric equation)
A vector-valued function r(t) = ⟨x(t), y(t)⟩ (2D) or r(t) = ⟨x(t), y(t), z(t)⟩ (3D) that traces out a curve as the parameter t varies over an interval. Think of it as giving the position of a moving particle at each moment t.
Velocity vector (tangent vector)
r′(t) = ⟨x′(t), y′(t), z′(t)⟩. The derivative of the position vector with respect to the parameter. It points in the direction the curve is heading and its magnitude gives the speed. In simple terms, this is the instantaneous direction and rate of travel along the curve.
Bernoulli spiral (logarithmic spiral)
A plane curve of the form r(t) = ⟨eᵗ cos t, eᵗ sin t⟩. The distance from the origin grows exponentially while the curve winds around it. A defining property is that the angle between the position vector and the tangent vector is constant.
Helix
A space curve that winds around a fixed axis at constant radius while advancing steadily along that axis. The standard parametrisation for a helix along the z-axis is r(t) = ⟨R cos t, R sin t, ct⟩, where R is the radius and c controls the vertical climb per revolution. Think of it as the path traced by a point on a turning screw.
Arc length
The total distance travelled along a curve from t = a to t = b, computed as L = ∫ₐᵇ |r′(t)| dt. This integral sums up the speed over time to give the physical length of the path.
Smooth curve
A curve whose parametrisation has a continuous derivative r′(t) that is never the zero vector on the interval. In simple terms, the curve has no sharp corners and no places where the particle stops.
When a curve is defined as the intersection of two surfaces, parametrise by choosing one coordinate as the free parameter and solving the surface equations for the others.
Two cylinders intersecting (e.g. x² + y² = 9 and x² + z² = 9, with z ≥ 0, y ≥ 0): subtract the two equations to get y² = z², so z = y in the given region. Set x = 3 cos t, y = 3 sin t (from the first cylinder), then z = 3 sin t. The parameter t runs from 0 to π/2 to keep y ≥ 0 and z ≥ 0.
The strategy: use one surface to set up the parametric coordinates, then substitute into the other surface to determine the remaining coordinate.
A helix of radius R, height h, and n full turns along the z-axis:
r(t) = ⟨R cos(2πnt/h · t)⟩ is wrong if you rush. Think step by step.
n full turns means the angle argument must go from 0 to 2πn. If t runs from 0 to 1, use θ = 2πnt. If t runs from 0 to 2πn, use θ = t directly.
The z-coordinate must go from 0 to h over the same interval. If t ∈ [0, 2πn], then z = ht/(2πn).
Full parametrisation: r(t) = ⟨R cos t, R sin t, ht/(2πn)⟩ for t ∈ [0, 2πn].
Check: at t = 0, you get (R, 0, 0). At t = 2πn, you get (R, 0, h). Each increase of 2π in t completes one full turn.
For r(t) = ⟨eᵗ cos t, eᵗ sin t⟩:
r′(t) = ⟨eᵗ(cos t − sin t), eᵗ(sin t + cos t)⟩.
The angle φ between r(t) and r′(t) is found via cos φ = (r · r′)/(|r| |r′|).
Computing: r · r′ = e²ᵗ(cos t(cos t − sin t) + sin t(sin t + cos t)) = e²ᵗ(cos²t − cos t sin t + sin²t + sin t cos t) = e²ᵗ.
|r| = eᵗ, |r′| = eᵗ√2. So cos φ = e²ᵗ/(eᵗ · eᵗ√2) = 1/√2. Therefore φ = π/4 ≈ 0.79 radians, independent of t.
If r(t) traces a curve that lies in a single plane, then r′(t) must always be parallel to that plane. Two non-parallel velocity vectors at different values of t determine the plane's orientation.
Find r′(t) and check whether r′(t) × r′(s) is a constant vector (same direction for all t, s). That constant vector is the plane's normal.
Alternatively, pick three points on the curve (three values of t), find the plane through them, and verify all points satisfy the equation.
For r(t) = ⟨t² − 1, 6 + 3t − t², 4 − 6t⟩: r′(t) = ⟨2t, 3 − 2t, −6⟩. Take t = 0: r′(0) = ⟨0, 3, −6⟩. Take t = 1: r′(1) = ⟨2, 1, −6⟩. The cross product ⟨0, 3, −6⟩ × ⟨2, 1, −6⟩ = ⟨−12, −12, −6⟩, which simplifies to the normal ⟨2, 2, 1⟩. Use a point on the curve (e.g. r(0) = ⟨−1, 6, 4⟩) to write the plane equation: 2(x + 1) + 2(y − 6) + 1(z − 4) = 0, i.e. 2x + 2y + z = 14.
Velocity vector: r′(t) = ⟨x′(t), y′(t), z′(t)⟩
Speed: |r′(t)| = √(x′(t)² + y′(t)² + z′(t)²)
Arc length: L = ∫ₐᵇ |r′(t)| dt
Angle between two vectors: cos φ = (u · v) / (|u| |v|)
Standard helix: r(t) = ⟨R cos t, R sin t, ct⟩, with radius R and pitch 2πc
Arc length of one full turn of a helix: L₁ = 2π√(R² + c²), where c = h/(2πn)
Circle on a cylinder of radius R: x = R cos t, y = R sin t (or any two-coordinate version)
Helices model the structure of DNA, springs, drill bits, and screw threads. The arc length formula for a helix lets biologists estimate the physical length of a DNA strand from its radius, height, and number of turns (approximately 2.9 × 10⁸ turns for human DNA).
Logarithmic spirals (Bernoulli spirals) appear in nautilus shells, hurricane bands, and the arms of spiral galaxies. The constant-angle property means that growth in these systems is self-similar: the shape looks the same at every scale.
Students often confuse the parameter t with time. The parameter is just a label; it can represent time, angle, or any convenient variable. The curve itself does not change if you reparametrise it.
A common error when parametrising helices is getting the parameter range wrong. If you want n full turns, the angular argument must traverse a total of 2πn, not just 2π. Check by substituting the endpoint and confirming you return to the correct (x, y) position.
Students sometimes assume that r′(t) = 0 at some point means the curve stops. It means the parametrisation is not smooth there, but the curve itself might still be perfectly well-defined (a cusp, for instance).
When showing a curve lies in a plane, students often try to verify the plane equation for every t algebraically but forget to first find the normal vector. The cross product of two tangent vectors gives the normal directly.
⚠️ Parametrising intersections of surfaces is a staple exam question. Practise subtracting surface equations to eliminate variables, then choosing the right trig substitution.
⚠️ The constant-angle property of the Bernoulli spiral is proved using the dot product formula. Be ready to compute r · r′, |r|, and |r′| cleanly.
⚠️ Helix arc length calculations appear frequently. Know the formula L = 2πn√(R² + c²) where c = h/(2πn), and be able to derive it from the integral.
⚠️ Proving a curve lies in a plane requires finding a normal vector (via the cross product of two tangent vectors) and then verifying the plane equation. This tests both differentiation and cross product skills.
True or false: The velocity vector r′(t) is always perpendicular to the position vector r(t). (False. This is only true for curves on a sphere centred at the origin.)
Fill in the blank: A helix with radius R and n turns over height h has arc length L = ____. (L = √((2πnR)² + h²))
True or false: If two surfaces are both cylinders, their intersection is always a plane curve. (False. It can be a space curve, though portions may lie in a plane.)
Fill in the blank: The angle between r(t) and r′(t) for the Bernoulli spiral eᵗ(cos t, sin t) is ____ radians. (π/4 ≈ 0.79)
True or false: To show a parametric curve lies in a plane, it suffices to check that r′(t) is constant. (False. r′(t) need not be constant; its cross product with another tangent vector must yield a fixed normal direction.)
Q: Parametrise the curve of intersection of x² + y² = 4 and z = x + y, for y ≥ 0.
A: Set x = 2 cos t, y = 2 sin t, t ∈ [0, π]. Then z = 2 cos t + 2 sin t. So r(t) = ⟨2 cos t, 2 sin t, 2 cos t + 2 sin t⟩.
Q: A helix has radius 3, makes 5 full turns, and rises to height 20. Write the parametrisation and compute its arc length.
A: r(t) = ⟨3 cos t, 3 sin t, 20t/(10π)⟩ = ⟨3 cos t, 3 sin t, 2t/π⟩ for t ∈ [0, 10π]. Speed: |r′(t)| = √(9 + 4/π²). Arc length: L = 10π√(9 + 4/π²) ≈ 94.56.
Q: Show that r(t) = ⟨t, 2t + 1, 3 − t⟩ lies in a plane and find the plane's equation.
A: r′(t) = ⟨1, 2, −1⟩, which is constant. The curve is a straight line (trivially in any plane containing it with normal perpendicular to ⟨1, 2, −1⟩). For instance, one such plane: take the point r(0) = ⟨0, 1, 3⟩ and normal ⟨2, −1, 0⟩ (perpendicular to ⟨1, 2, −1⟩): 2x − y = −1, i.e. 2x − y + 1 = 0. (Note: a line lies in infinitely many planes.)
Q: For the Bernoulli spiral r(t) = ⟨eᵗ cos t, eᵗ sin t⟩, compute |r′(t)| in terms of |r(t)|.
A: |r(t)| = eᵗ, |r′(t)| = eᵗ√2. So |r′(t)| = √2 · |r(t)|. The speed is always √2 times the distance from the origin.
This material connects directly to §14.3 (tangent planes): the velocity vectors of parametric curves lying on a surface are tangent to that surface, and their cross product gives the normal to the tangent plane. That bridge from curves to surfaces is the key idea in Problem 5 of this homework set.
Arc length reappears in §13.3 when you study arc-length parametrisation and curvature. Curvature measures how quickly the tangent direction changes per unit arc length, so the speed |r′(t)| from this section is directly relevant.
The dot product and cross product calculations here are the same tools used throughout MATH 215, from line integrals (§16.2) to surface integrals (§16.6).
Parametric curves, parametrisation, vector-valued functions, velocity vector, tangent vector, Bernoulli spiral, logarithmic spiral, helix, coil, arc length, arc length integral, space curves, intersection of surfaces, intersection of cylinders, plane containing a curve, normal vector, cross product, dot product, MATH 215, Calculus III, §13.2, §13.3, Stewart Multivariable Calculus