signed curvature
Ordinary curvature is always a positive number (or zero): it tells you how sharply a curve bends, but not which way. For a curve drawn in a plane, though, there is extra information available — a bend can go to the left or to the right of your direction of travel. Signed curvature is the refinement that keeps this left-or-right information by attaching a plus or minus sign, so a road bending left gets one sign and a road bending right gets the other.
Here is how the sign is fixed. For a plane curve travelled in a chosen direction, set up the unit tangent T and then rotate it a quarter turn counterclockwise to get a unit normal n (this n is determined by orientation, unlike the principal normal which always points to the concave side). The signed curvature kappa_s is then defined by T'(s) = kappa_s n. If the curve turns toward n (to the left of travel), kappa_s is positive; if it turns the other way, kappa_s is negative. Its absolute value is the ordinary curvature, |kappa_s| = kappa. A convenient formula in coordinates is kappa_s = (x' y'' - y' x'') / (x'^2 + y'^2)^(3/2) — same as the ordinary curvature formula but without the absolute value, so the numerator's sign survives. At an inflection point, where the curve switches from bending one way to the other, kappa_s passes through zero and changes sign.
Signed curvature is exactly the right tool for the global theory of plane curves. Its total accumulation — the integral of kappa_s with respect to arc length around a closed curve — measures the net turning of the tangent and equals 2 pi times the turning number (rotation index), the content of the Hopf Umlaufsatz; for a simple closed convex curve this total is exactly 2 pi. Two honest cautions. First, the sign is a convention tied to orientation: reverse the direction you travel the curve, and every kappa_s flips sign, so the sign is meaningful only once you fix an orientation. Second, signed curvature is special to plane curves — space curves do not have it, because in three dimensions there is no canonical 'left versus right', which is one reason space curves need torsion as a separate quantity.
Trace a sine wave y = sin x from left to right. On the rising-then-cresting humps (above the x-axis near the peaks) the curve bends one way and kappa_s has one sign; on the troughs it bends the opposite way and kappa_s has the other sign. At each inflection point where it crosses the axis, kappa_s = 0 and the sign flips.
Signed curvature records which way the curve bends; it vanishes and flips sign at inflection points.
Signed curvature exists only for plane curves, and its sign depends on the chosen orientation — reversing the direction of travel flips every sign. Space curves have no signed curvature; in 3D there is no canonical left-versus-right, which is why torsion enters instead.