Smoothness
Smoothness is the property of changing gradually, so nearby inputs produce similar outputs without discontinuities or abrupt changes.
Geometry for Programmers
Smoothness is the degree to which a mathematical function or geometric object lacks discontinuities, sharp joins, or abrupt changes in direction.
- How it is used
- Guides the design and evaluation of continuous, smooth, and curvy piecewise or parametric curves.
- How deep it goes
- Explains continuity orders, derivative matching, curvature, cubic construction, and tangent normalization, stopping short of formal proofs.
- Examples
- Uses piecewise parabolas, joined line-like pieces, cubic contour segments, and normalized or exaggerated tangent vectors.
- What you can do after
- Build smooth piecewise curves by matching endpoint values and derivatives, then control tangent magnitudes to avoid contour artifacts.
Bayesian Optimization in Action
Smoothness is the property that nearby or similar inputs produce correlated and therefore similar function values.
- How it is used
- Models correlated objective values in Gaussian processes and guides covariance-function choices for optimization.
- How deep it goes
- Explains nearby-point correlation, posterior behavior, covariance control, length scales, ARD, and Matérn smoothness without deriving kernels.
- Examples
- Uses observations at 3 with nearby points 2.9 and 3.1, plus conditioned GP prediction plots and a Forrester-function dataset.
- What you can do after
- Apply covariance functions and length-scale choices to control GP smoothness and interpret local uncertainty.
Read how Bayesian Optimization in Action explains smoothness →
Related concepts
- covariance functionconfigured by · A covariance function, also called a kernel, specifies the covariance between function values at pairs of inputs in a Gaussian process.
- length scalecontrasts with · A length scale is a positive parameter that determines the distance over which a function’s correlations or similarities substantially change.
- order of continuitymeasured by · The order of continuity is the highest derivative order through which a function and its derivatives remain continuous.
- derivativeuses · A derivative is a mathematical quantity that measures the instantaneous rate of change of a function with respect to an input variable.
- Gaussian processused by · A Gaussian process is a probability distribution over functions in which every finite collection of function values has a joint Gaussian distribution.
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