Parametrized Curves

Picture a fly buzzing around a room. At every instant it has a definite position, and as time ticks forward that position sweeps out a path through space. Differential geometry begins by writing that idea down honestly: a parametrized curve is a vector-valued function of one variable, a rule that eats a single number t — read it as time — and spits out a point in space.

\mathbf{r}(t) = \big(x(t),\ y(t),\ z(t)\big), \qquad t \in [a, b].

Each coordinate is an ordinary function of t. Feed in a moment of time and you get back the fly's location at that moment. Let t run over its whole interval and the outputs trace a one-dimensional thread through the plane or through 3-space. That thread — the set of points visited — is the image (or trace) of the curve. But hold that word lightly: as we'll insist below, the curve is the function \mathbf{r}, not just the smudge it leaves behind.

Velocity: the tangent vector

Differentiate a parametrized curve coordinate by coordinate and you get its velocity, the vector that points the way the fly is heading:

\mathbf{r}'(t) = \big(x'(t),\ y'(t),\ z'(t)\big).

This is exactly the limit of a secant vector as the two sample times close together,

\mathbf{r}'(t) = \lim_{h \to 0} \frac{\mathbf{r}(t + h) - \mathbf{r}(t)}{h},

so it is tangent to the curve at \mathbf{r}(t): it lies along the path, in the direction of increasing t. Two pieces of information are bundled inside it — a direction (which way) and a length (how fast).

That length is the speed,

v(t) = \lVert \mathbf{r}'(t) \rVert = \sqrt{x'(t)^2 + y'(t)^2 + z'(t)^2}.

Strip the speed out and keep only the direction, and you have the unit tangent vector,

\mathbf{T}(t) = \frac{\mathbf{r}'(t)}{\lVert \mathbf{r}'(t) \rVert}.

The unit tangent is the workhorse of the whole subject: it records the pure heading of the curve, with the speed divided out, and it is the first of the three vectors of the moving frame we'll build later. Dividing by \lVert \mathbf{r}'(t) \rVert only makes sense when that length is not zero — which brings us to the one hygiene condition every good curve must satisfy.

A parametrized curve \mathbf{r}(t) is called regular when

its velocity never vanishes: \mathbf{r}'(t) \neq \mathbf{0} for every t;
consequently the unit tangent \mathbf{T} = \mathbf{r}'/\lVert \mathbf{r}' \rVert is defined everywhere — the curve has a well-defined direction at each point;
regularity is a property of the parametrization, not of the underlying set of points: a smooth-looking track can be traced by a non-regular parametrization that momentarily stops.

A vanishing velocity is where a curve is allowed to misbehave. Consider \mathbf{r}(t) = (t^3, t^2). Its image has a sharp cusp — a genuine corner — at the origin, even though x(t) and y(t) are perfectly smooth polynomials. What went wrong? At t = 0 the velocity \mathbf{r}'(0) = (0, 0) is the zero vector, so there is no tangent direction to speak of, and the point is free to change heading abruptly. Regularity outlaws exactly this. Where the velocity is non-zero, the curve is locally a smooth, corner-free arc with a well-defined tangent line.

Worked example 1 — the circle

The simplest closed curve is a circle of radius R, traced anticlockwise:

\mathbf{r}(t) = (R\cos t,\ R\sin t), \qquad t \in [0, 2\pi].

Differentiate each coordinate to get the velocity, then take its length:

\mathbf{r}'(t) = (-R\sin t,\ R\cos t), \qquad \lVert \mathbf{r}'(t) \rVert = \sqrt{R^2\sin^2 t + R^2\cos^2 t} = R.

The speed is the constant R — the fly circles at a steady rate — and the unit tangent is

\mathbf{T}(t) = (-\sin t,\ \cos t).

Notice \mathbf{T}(t) \cdot \mathbf{r}(t) = -R\sin t\cos t + R\sin t\cos t = 0: the velocity is always perpendicular to the radius, as your intuition about spinning things demands.

Worked example 2 — the helix

Lift the circle out of the plane by letting the third coordinate climb steadily. The result is the circular helix, the shape of a spring or a spiral staircase:

\mathbf{r}(t) = (\cos t,\ \sin t,\ c\,t),

where the constant c sets how quickly it rises (the pitch). Differentiating,

\mathbf{r}'(t) = (-\sin t,\ \cos t,\ c), \qquad \lVert \mathbf{r}'(t) \rVert = \sqrt{\sin^2 t + \cos^2 t + c^2} = \sqrt{1 + c^2}.

Again the speed is constant — \sqrt{1 + c^2}, independent of t — so the helix, like the circle, is traced at a uniform rate. Its unit tangent leans a fixed amount out of the horizontal:

\mathbf{T}(t) = \frac{1}{\sqrt{1 + c^2}}\,(-\sin t,\ \cos t,\ c).

Later, to build the curve's moving frame, we'll need a vector perpendicular to the plane spanned by \mathbf{r}' and the acceleration \mathbf{r}''. That perpendicular is manufactured with the cross product \mathbf{r}'(t) \times \mathbf{r}''(t) — the very combination that will define the binormal and drive the practical curvature formula on later pages. Keep it in your back pocket.

A curve confined to a plane can bend, but it can't twist. The helix is the cleanest example of genuine three-dimensionality: project it down onto the xy-plane and you recover a plain circle, yet the real helix corkscrews upward, never closing on itself. That extra degree of freedom — how sharply the curve pulls out of the plane of its own bending — is a real, measurable quantity called torsion, and it is invisible to any flat curve (whose torsion is exactly zero). DNA's double helix, the thread of a screw, and the coil of a telephone cord all live in this space-curve world. The circle and the helix will be our two running test cases precisely because one is flat and one twists.

The single most common beginner's error is to conflate a curve with its image — the set of points it passes through. They are not the same thing. Consider three functions:

\mathbf{r}_1(t) = (\cos t,\ \sin t), \quad \mathbf{r}_2(t) = (\cos 2t,\ \sin 2t), \quad \mathbf{r}_3(t) = (\cos t,\ -\sin t).

All three have the same image — the unit circle. But they are different curves: \mathbf{r}_2 races round twice as fast (speed 2, not 1), and \mathbf{r}_3 runs the opposite way, clockwise. A parametrized curve is the map t \mapsto \mathbf{r}(t), carrying with it a direction of travel and a speed schedule — not merely the trace left on the page. Two parametrizations that draw the same picture can have different velocities, different speeds, and different orientations. When someone says "the curve," ask: which parametrization? The picture alone doesn't pin it down.

Watch the tangent ride along

Below is the ellipse \mathbf{r}(t) = (2\cos t,\ \sin t). Drag the slider to move the sample time t: the dot sits at \mathbf{r}(t) and the arrow is the velocity \mathbf{r}'(t) = (-2\sin t,\ \cos t), always tangent to the curve and pointing the way of increasing t. Notice how the arrow grows longer where the curve is travelled faster (near the tight ends of the ellipse) and shorter where it dawdles — the length of the arrow is the speed.