Where does curvature collide?

In geometry, curvature measures how sharply a curve or surface bends away from being straight or flat at a point: a straight line has zero curvature, and a tight circle bends more than a wide one. Newton set out how to compute the radius of curvature of a plane curve in his method of fluxions (written 1671, published 1736), and Gauss's 1827 Disquisitiones generales circa superficies curvas extended the idea to surfaces, defining a curvature that stays the same however the surface is bent without being stretched. Here the referent is a purely mathematical property of shape, computed from derivatives, and it is not the bending of physical spacetime nor the shape of a bond market's interest rates.[1][2]

Unit

A number at a point, the reciprocal of the radius of the best-fitting circle

Recovery means

Not applicable

Measured by

Computed from the first and second derivatives of a curve, or from a surface's metric

In physics, curvature describes how gravity bends the geometry of spacetime itself, so that matter and energy warp the space and time around them and moving bodies follow the curved paths we observe as gravitational attraction. Einstein's 1916 Die Grundlage der allgemeinen Relativitätstheorie laid out the general theory of relativity, in which the curvature of spacetime is tied to the distribution of mass and energy by the field equations he had completed the previous year. The referent is a physical property of the four-dimensional geometry of the universe, revealed by its effect on clocks, light, and orbits, and it is not the abstract bending of a mathematician's curve nor a pattern in interest rates.[3]

Unit

The degree to which four-dimensional spacetime departs from flat, carried by the curvature tensor

Recovery means

Not applicable

Measured by

Inferred from the bending of light, the precession of orbits, gravitational time dilation, and gravitational waves

In finance, curvature names the middle of the three shapes an interest-rate or yield curve can take: after its overall level and its slope, the curvature factor captures the bowing of the curve, where medium-term rates sit relative to short and long ones. Litterman and Scheinkman's 1991 Common factors affecting bond returns identified level, slope, and curvature as the three factors that explain most of how bond returns move together. The referent is a statistical shape factor of a set of market interest rates, estimated from bond prices, and it has nothing to do with a bent geometric curve or the warping of spacetime beyond the shared word.[4]

Unit

A factor score for the bowing of the yield curve across maturities

Recovery means

Not applicable

Measured by

Extracted as the third component when principal component analysis is run on yields or returns across maturities