Why we first stripped away the speed
In the previous guide you learned to walk a curve at unit speed, reparametrizing so that the parameter s measures honest distance travelled and the velocity has length exactly 1. That was not housekeeping for its own sake — it was the move that makes bending visible. When you drive a winding road, two completely different things change your direction: stepping on the gas, and turning the wheel. Speed up on a straight stretch and your velocity vector grows longer but never swings sideways; the road is straight, yet the velocity is still changing. Only by killing the speed change can we isolate the turning.
At unit speed the velocity is just the unit tangent vector T(s) — an arrow of fixed length 1 that always points the way you face. Because its length can never change, the only thing T can do as you advance is rotate. So the rate at which T turns, the derivative T'(s), captures pure turning and nothing else. Its magnitude is exactly the curvature, written kappa(s) = |T'(s)|. Length out of the picture, T'(s) measures how fast the tangent swings per unit of distance walked — degrees of turn per metre, if you like.
The straight line and the circle: the two anchors
Any honest definition should agree with the cases you already trust. Take a straight line walked at unit speed: the tangent T never turns, it points the same way forever, so T'(s) is the zero vector and kappa = 0 everywhere. A line has zero curvature — exactly what 'doesn't bend' ought to mean. This is the floor of the whole subject; nothing curves less than a straight line, and curvature is never negative, because it is the length of an arrow.
Now the other anchor: a circle of radius r. Walk it at unit speed and the tangent makes one full turn — 2 pi radians — over the whole trip, and the whole trip has length 2 pi r, the circumference. Turning per distance is therefore (2 pi radians) / (2 pi r) = 1 / r, the same at every point because a circle bends evenly. So a circle of radius r has curvature kappa = 1 / r. Read that relationship slowly: a big circle (large r) is nearly straight and barely curves, giving small kappa; a tiny circle (small r) whips around fast and has huge kappa. Curvature is the reciprocal of size.
Two anchors that pin down kappa: straight line, radius = infinity : kappa = 0 circle of radius r : kappa = 1 / r big circle (r large) -> kappa small (nearly straight) small circle (r small) -> kappa large (turns hard) Units: kappa is 1/length, e.g. per-metre. A line is the r -> infinity limit, where 1/r falls all the way to 0.
The osculating circle: curvature you can draw
If a circle of radius r has curvature 1/r, we can run the idea backwards. At any point of a general curve, measure the curvature kappa there, and ask: which circle bends at exactly that rate? The one with radius R = 1 / kappa. Lay that circle against the curve at the point, matching not just the position but the direction and the bending, and it grips the curve more snugly than any other circle could. This is the osculating circle — from the Latin for 'kissing'. Its radius R = 1/kappa is the radius of curvature, and its centre, sitting a distance R off to the side the curve is turning toward, is the centre of curvature.
This is the picture to keep in your head, because it turns an abstract derivative into something you can sketch. Where a curve runs nearly straight, the kissing circle is enormous (R large, kappa small); where the curve whips around a tight bend, the kissing circle shrinks to hug it (R small, kappa large). Watch the centre of curvature trace its own path as you slide along the curve, and you have drawn the evolute — the curve swept out by all the centres of curvature, a hint of how much structure a single number per point already carries.
Which way it bends: the normal and a worked example
Curvature kappa tells you how hard a curve turns but never which way — it is a length, always non-negative, blind to left versus right. The missing direction lives in T'(s) itself, before you take its length. Here a small piece of magic helps: because T has constant length 1, its derivative T'(s) is always perpendicular to T. (Differentiate T . T = 1 and you get 2 T . T' = 0, so T . T' = 0.) That perpendicular direction, made into a unit arrow, is the principal normal N(s). It points squarely toward the inside of the bend — toward the centre of curvature. Together they give the tidy statement T'(s) = kappa(s) N(s): the tangent turns toward the inside, at a rate set by kappa.
- Start with a unit-speed curve. Take the circle of radius r, written r(s) = (r cos(s/r), r sin(s/r)); the s/r inside keeps the speed exactly 1.
- Differentiate once to get the unit tangent: T(s) = (-sin(s/r), cos(s/r)). Check its length: sin^2 + cos^2 = 1, good — unit speed confirmed.
- Differentiate again: T'(s) = (-(1/r) cos(s/r), -(1/r) sin(s/r)). Pull out the common factor 1/r.
- Its length is kappa = |T'(s)| = (1/r) * sqrt(cos^2 + sin^2) = 1/r. The curvature of a circle really is 1/r, computed straight from the definition.
- The leftover unit direction is N(s) = (-cos(s/r), -sin(s/r)), which points from the rim straight back to the centre (0, 0) — the principal normal aims at the centre of curvature, as promised.
That worked example is worth more than its arithmetic: it shows the whole machine turning over on the simplest curved case and landing on the answers your intuition already trusted. The same two derivatives — first to get T, second to get kappa and N — run on any unit-speed curve, plane or spatial. In three dimensions a third direction completes the frame: the binormal B = T x N, perpendicular to both, whose rate of change measures how the curve twists out of its own plane. That twisting is torsion, and it is exactly where the next guide picks up.
Signed curvature, the any-parameter formula, and honest limits
For a curve living flat in the plane, you can do better than a non-negative kappa. Fix an orientation — say, agree that left turns count as positive — and you get the signed curvature, which carries a plus or minus to record whether the curve veers left or right at each point. It flips sign exactly at an inflection, where the curve passes from bending one way to the other and momentarily straightens. The plain kappa is just the absolute value of the signed version; the sign is the extra honesty available only in the plane, and it is the seed of the turning number you will meet at the end of this rung.
In practice you rarely have a curve handed to you already at unit speed — reparametrizing is often ugly or impossible in closed form. So there is a formula that reads curvature off any regular parametrization without first slowing to unit pace. For a plane curve (x(t), y(t)) it is kappa = |x' y'' - y' x''| / (x'^2 + y'^2)^(3/2), and in space the matching version is kappa = |r' x r''| / |r'|^3, with the cross product doing the work. The (3/2) power and the cross product are precisely the bookkeeping that divides out the speed you happened to choose, so that what survives is the road's own bending — the same kappa the clean s-formula would give.
Two honest boundaries before we move on. First, every formula here assumed a regular curve with a genuine, nonzero velocity; at a sharp corner or a cusp, where the velocity stalls to zero, the tangent jumps and curvature is simply undefined — corners are not curves of huge curvature, they are points where the smooth theory does not apply. Second, where kappa = 0 (an inflection, or a stretch of straight line), the principal normal N is not pinned down at all, because there is no inside-of-a-bend to point at; you only recover a definite N where the curve is genuinely turning. Naming these gaps is not a weakness of the theory — it is the theory being precise about where it speaks.