Equations · equation
Curvature
Vectors & Geometry · 1673
- Equation
- κ = 1/R; for a curve y(x): κ = |y″| / (1 + y′²)^(3/2)
- Field
- Vectors & Geometry
- Symbols
- κ curvature · R the radius of the circle that best fits the curve at that point · y′, y″ the first and second derivatives
- Year
- 1673
Significance
How sharply a path bends: the reciprocal of the radius of the circle that hugs it. A straight line has zero, a tight corner a lot. Huygens needed it in 1673 to make a pendulum keep exact time; a road engineer needs it to bank a bend. Gauss showed a surface's curvature can be measured from within it, without ever leaving — and it is that intrinsic reading, generalised to four dimensions as the Riemann curvature tensor, that general relativity is written in. This is the one-dimensional ancestor of it, not the thing itself.
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