Calculus×Computing
ES

Surfaces and level sets

Level UniversityDifficulty ★★★★★Concept⌖ Open in the map

What is it?

Three ways to describe a surface: a graph z=f(x,y)z = f(x,y), a level set F(x,y,z)=cF(x,y,z) = c (implicit), or a parametrization r(u,v)r(u,v). Contour maps of loss functions and the isosurfaces of medical imaging are level sets.

Formulas

{(x,y):f(x,y)=c},{x∈ℝ3:F(x)=0},r(u,v)∈ℝ3\{(x, y) : f(x, y) = c\}, \qquad \{x \in \R^3 : F(x) = 0\}, \qquad r(u, v) \in \R^3

Where it shows up in computing

  • Signed distance fields and ray marching★★★★★fundamentalComputer graphics

    An SDF represents a shape as the zero level set of a distance function.

  • Surface normals★★★★★fundamentalComputer graphics

    Normals of parametric surfaces come from ru×rvr_u \times r_v; of implicit ones from ∇F\nabla F.

Where it shows up in AI

  • Loss landscape★★★★★frequentAI and machine learning

    Contour plots of a loss along two directions are the standard way to visualize its landscape.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

3D Computer graphics

⚛ Physics and simulation

ℒ AI and machine learning

⚙ Robotics and control

What depends on it

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.

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