Curvature (basic differential geometry)

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

What is it?

How fast a curve turns ( $\kappa = 1/\text{radius}$ of the best-fitting circle) or how a surface bends (mean and Gaussian curvature). Graphics and CAD use it to judge smoothness, smooth meshes and design roads and rails.

Formulas

\kappa = \frac{|y''|}{\big(1 + y'^2\big)^{3/2}}, \qquad \kappa = \frac{\norm{r' \times r''}}{\norm{r'}^3}

Where it shows up in computing

Mesh processing (discrete differential geometry)★★★★★fundamentalComputer graphics
Discrete mean curvature (from the cotangent Laplacian) drives smoothing, remeshing and feature detection.
Bézier curves and splines★★★★★frequentComputer graphics
Curvature-continuous ( $G^2$ ) joins and curvature combs are how type designers and CAD check fairness.
Trajectory optimization and MPC★★★★★frequentRobotics and control
Vehicle paths must respect a maximum curvature (minimum turning radius).

Where is it used?

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

3D Computer graphics

Mesh processing (discrete differential geometry)★★★★★
Bézier curves and splines★★★★★

⚙ Robotics and control

Trajectory optimization and MPC★★★★★

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