Curvature
Curvature is a measure of how a curve or surface bends locally, or, for a function, how its slope changes across its input space.
Geometry for Programmers
Curvature is a quantity describing how rapidly a geometric curve or surface bends, with signed curvature carrying orientation-dependent sign.
- How it is used
- Uses curvature to measure and control bend in smooth curves, spline constraints, contour smoothing, and minimal surfaces.
- How deep it goes
- Explains curve curvature through fitting circles, tangents, and second derivatives, then sketches mean and principal surface curvatures.
- Examples
- Illustrates curvature with lines, circles, a sine graph, constrained polynomials, pixel-derived contours, and tri-periodic minimal surfaces.
- What you can do after
- Apply it by constraining polynomial derivatives and smoothing fitted image contours with cubic curve segments.
Sutskever's List
Curvature describes how the slope of an objective function changes across parameter space.
- How it is used
- Frames curvature as information that helps explain stalled neural-network training and motivate second-order optimization.
- How deep it goes
- Defines curvature as changing slope and explains its effect in narrow valleys, without deriving a second-order method.
- Examples
- Uses long, narrow valleys to contrast gradient descent’s oscillations and small steps with curvature-aware progress.
- What you can do after
- Understand why curvature-aware methods can navigate narrow valleys more effectively than gradient descent.
Related concepts
- second derivativemeasured by · A second derivative is the derivative of a function’s first derivative with respect to the same variable, describing how its rate of change varies.
- radiuscontrasts with · A radius is the distance from the center of a circle or geometric region to its boundary in a specified direction.
- projective transformationcontrasts with · A projective transformation is a mapping of projective-space points represented by homogeneous linear coordinates that preserves incidence but may not preserve parallelism.
- principal curvatureincludes · A principal curvature is an extremal signed curvature of a smooth surface at a point, obtained from normal sections in principal directions.
- second-order optimizationused by · Second-order optimization is an optimization approach that uses first-derivative information together with curvature information, such as a Hessian or its approximation.
- mean curvatureincludes · Mean curvature is half the sum of the two principal curvatures at a surface point.
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