Difficulty: Intermediate to Advanced | Prerequisites: Planes and surfaces (Sections 12.4 to 12.6), dot product, cross product
Parametric curves are how you describe motion through 3D space: a particle's position as a function of time, the curve where two surfaces meet, or the line of sight between two observers. This section also brings together distance problems in 3D, from the distance between two moving objects to whether a line of sight is blocked by a sphere. If you can write a vector equation of a line and minimise a distance function, you have the tools for all of these.
A vector-valued function r(t) = (x(t), y(t), z(t)) traces a curve in 3D as t varies. Lines are the simplest case: r(t) = P + t*d. To find a curve where two surfaces intersect, parametrise one surface and substitute into the other. Distance between moving objects is minimised by calculus on |r1(t) - r2(t)|^2; distance between skew lines uses the cross product of their direction vectors.
Vector-valued function (vector function)
A function r(t) = (x(t), y(t), z(t)) that assigns a position vector to each value of the parameter t. Think of it as a GPS tracker recording a particle's position over time.
Parametric curve
The set of all points traced out by r(t) as t ranges over some interval. The curve is the path itself, independent of how fast it is traversed.
Direction vector
A vector d that points along a line, determining its orientation. Any non-zero scalar multiple of d also works as a direction vector for the same line.
Parametric equation of a line
r(t) = P + td, where P is a point on the line and d is the direction vector. In component form: x = x0 + ta, y = y0 + tb, z = z0 + tc.
Trajectory
The path traced by a moving object. Two objects may have paths that cross (the trajectories intersect) without the objects ever being at the crossing point at the same time.
Skew lines
Two lines in 3D that are neither parallel nor intersecting. They pass each other without meeting. In simple terms, they are "tilted past each other" in different planes.
Line of sight
The straight line segment connecting two points (observers). If no object blocks this segment, the observers can see each other.
Equidistant locus (from three points)
The set of all points equidistant from three given points. In 3D, this is a line (the intersection of two perpendicular-bisector planes).
When two surfaces intersect, they share a curve. To parametrise it, use a natural parameter from one surface and substitute into the other.
Example: the cylinder x^2 + y^2 = 64 and the paraboloid x^2 + y^2 + 18z = 0. On the cylinder, set x = 8cos(t), y = 8sin(t). Then x^2 + y^2 = 64, so 64 + 18z = 0, giving z = -64/18 = -32/9. The curve is r(t) = (8cos(t), 8sin(t), -32/9), a circle at height z = -32/9.
Given r(t), eliminate the parameter to find a surface containing the curve.
Example: r(t) = (5t sin(3t), 6t cos(3t), t^2). Let z = t^2, so t = sqrt(z). Then x = 5sqrt(z) sin(3sqrt(z)) and y = 6sqrt(z) cos(3sqrt(z)). Compute x^2/25 + y^2/36 = z(sin^2(3t) + cos^2(3t)) = z. So the curve lies on the elliptic paraboloid x^2/25 + y^2/36 = z.
Example: r(t) = (3 + 5sin(t^4), 2 - 6sin(t^4), 3 + 7sin(t^4)). Let u = sin(t^4). Then (x - 3)/5 = u, (y - 2)/(-6) = u, (z - 3)/7 = u. These give linear relations: (x - 3)/5 = (2 - y)/6 = (z - 3)/7. This is a line in 3D, so the curve lives on any plane containing that line.
A line through point P in direction d: r(t) = P + t*d.
Example: r1(t) = (2, -1, 3) + t(4, 2, 8), r2(t) = (-1, -3, 1) + t(1, 4, -2).
Two lines are parallel if their direction vectors are scalar multiples. They intersect if there is a single t (and possibly a different s) giving the same point.
To find the set of points equidistant from (2,1,1), (-1,-1,10), and (1,3,-4), set the squared distances equal pairwise.
Equating distance-squared from P = (x,y,z) to the first two points gives one plane equation. Equating the first and third gives another. The intersection of these two planes is the desired line.
Given r1(t) and r2(t) as positions at time t, the distance at time t is |r1(t) - r2(t)|.
Minimise |r1(t) - r2(t)|^2 (avoiding the square root): differentiate with respect to t, set equal to zero, solve for t.
For r1 - r2 = (3 + 3t, 2 - 2t, 2 + 10t), minimise 9 + 18t + 9t^2 + 4 - 8t + 4t^2 + 4 + 40t + 100t^2 = 113t^2 + 50t + 17. Set the derivative 226t + 50 = 0 to get t = -25/113. The minimum distance is sqrt(17 - 625/113).
The trajectories are the paths in space, ignoring time. Two non-parallel, non-intersecting lines are skew.
The shortest distance between skew lines r1 = P1 + sd1 and r2 = P2 + td2 is |(P2 - P1) . (d1 x d2)| / |d1 x d2|.
This uses the fact that the shortest connecting segment is perpendicular to both direction vectors, and d1 x d2 gives that perpendicular direction.
The line of sight from A = (0, -2, 1) to B = (1, 0, 2) is r(t) = (1-t)A + tB = (t, -2 + 2t, 1 + t) for t in [0, 1].
To check if a sphere (centre C, radius R) blocks the line, substitute r(t) into |r(t) - C|^2 = R^2. If the resulting quadratic in t has solutions in [0, 1], the line is blocked.
Example: sphere centred at (0, -1, 1) with radius 2/3. Compute |r(t) - (0, -1, 1)|^2 = t^2 + (2t - 1)^2 + t^2 = 6t^2 - 4t + 1. Set equal to 4/9: 6t^2 - 4t + 1 = 4/9, i.e. 6t^2 - 4t + 5/9 = 0. Discriminant: 16 - 120/9 = 16 - 40/3 = 8/3 > 0, so two solutions exist in [0,1] (check), and the sphere blocks the view.
\mathbf{r}(t) = \mathbf{P} + t\,\mathbf{d}Vector equation of a line through P with direction d.
|\mathbf{r}_1(t) - \mathbf{r}_2(t)|^2Minimise this (as a function of t) to find when two objects are closest. Differentiate and set equal to zero.
D = \frac{|(\mathbf{P}_2 - \mathbf{P}_1) \cdot (\mathbf{d}_1 \times \mathbf{d}_2)|}{|\mathbf{d}_1 \times \mathbf{d}_2|}Shortest distance between two skew lines. The numerator is the absolute value of the scalar triple product; the denominator is the magnitude of the cross product of the direction vectors.
|\mathbf{r}(t) - \mathbf{C}|^2 = R^2Substitute the parametric line into the sphere equation to check whether the line intersects the sphere. Solve the resulting quadratic in t.
Parametric curves describe any trajectory: a drone's flight path, a satellite orbit, a robot arm's motion. The distance-between-lines formula is used in air traffic control to compute closest approach between aircraft. The line-of-sight-plus-sphere problem models radar occlusion: can a sensor "see" a target, or is a planet (modelled as a sphere) in the way?
Students confuse "the jets collide" with "the trajectories intersect." Collision requires both jets at the same point at the same time (same t). Trajectory intersection only requires the paths to cross, possibly at different times.
When minimising distance, students sometimes minimise |r1(t) - r2(t)| directly. It is much easier to minimise the square of the distance (no square root to differentiate). The minimum occurs at the same t.
Students forget the absolute value in the skew-lines distance formula. The scalar triple product can be negative depending on orientation; the distance is always positive.
When checking sphere intersection with a line segment (not an infinite line), students solve the quadratic but forget to check that the solutions for t lie in [0, 1]. A solution at t = 3 means the sphere is on the line's extension, not between the two points.
⚠️ Finding a parametric curve from the intersection of two surfaces is a standard exam problem. Know the cylinder trick: set x = r cos(t), y = r sin(t), then solve for z.
⚠️ The distinction between "closest distance between jets at the same time" and "closest distance between their trajectories" is a favourite exam question because it tests whether you understand the role of the parameter t.
⚠️ The skew-lines distance formula (scalar triple product over cross product magnitude) is compact but easy to misapply. Practise identifying P1, P2, d1, d2 from the problem.
⚠️ Sphere-line intersection (does the line of sight get blocked?) tests your ability to substitute a parametric equation into a surface equation and solve a quadratic. Check that solutions fall in the correct parameter range.
The line through (1, 0, -1) in the direction (2, 3, 4) has vector equation r(t) = ______. (Answer: (1 + 2t, 3t, -1 + 4t).)
True or false: if two lines in 3D are not parallel and do not intersect, they are called skew. (True.)
To find when two jets are closest, you minimise ______ with respect to t. (Answer: |r1(t) - r2(t)|^2.)
The shortest distance between skew lines uses the ______ of the direction vectors in the denominator. (Answer: cross product.)
True or false: if a quadratic in t from a line-sphere substitution has two real roots, both necessarily fall in [0, 1]. (False: you must check the interval.)
Q: Find parametric equations for the curve of intersection of the cylinder x^2 + y^2 = 64 and the paraboloid x^2 + y^2 + 18z = 0.
A: Parametrise the cylinder as x = 8cos(t), y = 8sin(t). Then x^2 + y^2 = 64, so 64 + 18z = 0, giving z = -32/9. The curve is r(t) = (8cos(t), 8sin(t), -32/9) for t in [0, 2pi). It is a horizontal circle at height -32/9.
Q: Two jets fly as r1(t) = (2 + 4t, -1 + 2t, 3 + 8t) and r2(t) = (-1 + t, -3 + 4t, 1 - 2t). Find the time at which they are closest and the minimum distance.
A: r1 - r2 = (3 + 3t, 2 - 2t, 2 + 10t). Squared distance = (3 + 3t)^2 + (2 - 2t)^2 + (2 + 10t)^2 = 113t^2 + 50t + 17. Derivative = 226t + 50 = 0 gives t = -25/113. Minimum distance = sqrt(17 - 625/113) = sqrt(1296/113) = 36/sqrt(113).
Q: Find the shortest distance between the trajectories (as lines in space) of the two jets above.
A: d1 = (4, 2, 8), d2 = (1, 4, -2). Cross product d1 x d2 = (2*(-2) - 84, 81 - 4*(-2), 44 - 21) = (-36, 16, 14). Magnitude = sqrt(1296 + 256 + 196) = sqrt(1748) = 2sqrt(437). P2 - P1 = (-3, -2, -2). Dot with (-36, 16, 14): 108 - 32 - 28 = 48. Distance = 48 / (2sqrt(437)) = 24/sqrt(437).
Q: Two people are at A = (0, -2, 1) and B = (1, 0, 2). A sphere of radius 2/3 is centred at (0, -1, 1). Does it block the line of sight?
A: Parametrise: r(t) = (1-t)(0, -2, 1) + t(1, 0, 2) = (t, -2 + 2t, 1 + t). Distance squared to centre: t^2 + (-1 + 2t)^2 + t^2 = 6t^2 - 4t + 1. Set equal to 4/9: 6t^2 - 4t + 5/9 = 0. Discriminant = 16 - 40/3 = 8/3 > 0, so two real roots. Both roots lie in (0, 1), so the sphere blocks the line of sight.
Parametric curves are the foundation for Chapter 13: arc length, curvature, and the TNB frame all build on the vector function r(t). The skew-lines distance formula relies on the cross product and scalar triple product from Section 12.4. The line-sphere intersection technique generalises to ray-surface intersection, which is central to computer graphics (ray tracing). The equidistant-locus idea (finding the line equidistant from three points) connects back to the perpendicular-bisector planes from the Planes notes.
Tags: parametric curve, vector-valued function, vector equation of a line, direction vector, parametric line, trajectory, skew lines, distance between skew lines, scalar triple product, cross product, closest approach, line of sight, sphere intersection, ray tracing, equidistant locus, MATH 215, Calculus III, Stewart Section 13.1