Bézier curves and splines

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

What is it?

Polynomial curves controlled by a few points, glued with continuity conditions on derivatives ( $C^1$ : same tangent, $C^2$ : same curvature). Fonts, SVG paths, CAD surfaces and animation curves are built from them.

Formulas

$B(t) = (1-t)^3P_0 + 3(1-t)^2t\,P_1 + 3(1-t)t^2P_2 + t^3P_3, \quad t \in [0,1]$; $B'(0) = 3(P_1 - P_0), \qquad B'(1) = 3(P_3 - P_2)$
tangents at the ends: how $C^1$ joins are built

The mathematics behind it

Polynomial functions★★★★★fundamental
Bézier curves and splines are polynomial pieces written in the Bernstein basis.
Continuity★★★★★frequent
Joining spline pieces with $C^0$ , $C^1$ or $C^2$ continuity decides how smooth the curve looks.
Higher-order derivatives★★★★★frequent
Matching second derivatives at joints ( $C^2$ ) removes visible kinks in fonts and CAD curves.
Interpolation★★★★★frequent
De Casteljau's algorithm evaluates a Bézier curve by repeated linear interpolation.
Curvature (basic differential geometry)★★★★★frequent
Curvature-continuous ( $G^2$ ) joins and curvature combs are how type designers and CAD check fairness.

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