Level Sets
- a level set of a real-valued function 𝑓 of 𝑛 real variables is a set where the function takes on a given constant value 𝑐:
- 𝐿𝑐(𝑓) = {(𝑥1, …, 𝑥𝑛) | 𝑓(𝑥1, …, 𝑥𝑛) = 𝑐}
- level curve - contour line - isoline - is a level set whose function 𝑓 has two independent variables
- level surface - isosurface - is a level set whose function 𝑓 has three independent variables
- level hypersurface - is a level set whose function 𝑓 has 4 or more independent variables
- a level set is a special case of a fiber
Level Sets - Examples
Consider the 2D Euclidean distance metric:
A level set 𝐿𝑟(𝑑) of the Euclidean distance metric, consists of all points that lie at a distance 𝑟 from the origin.
For example:
- (3,4) ∊ 𝐿5(𝑑)
Level Sets - Use Cases
- used to visualize linear functionals
Resources
Level Sets Examples - Where 𝑓 is a Linear Functional
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Level Sets Examples - Where 𝑓 is a Non-Linear Function
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