# Bézier Curves

A Bézier curve is a type of spline: a curve defined by control points. These curves are used often in computer graphics, computer-assisted design, and typography. Pierre Bézier worked as a designer for Renault in the mid-20th century, and invented these curves to help model automobile bodies. Bézier curves can be defined algebraically by parametric polynomial equations. Here, though, we explore their geometric construction.

A linear, or first-order, Bézier curve, is the simplest, and is what all higher-order curves are built from. Begin with a parameter, t, which varies from 0 to 1. We represent t as a point on a line segment. However far t is along the segment, construct a point at that same distance ratio from P0 to P1. That point will trace out the Bézier curve from P0 to P1, in this case a line segment. Drag t in the applet below to construct the first-order Bézier curve from P0 to P1.

 Sorry, this page requires a Java-compatible web browser. Linear Bézier curve from P0 to P1

To construct a quadratic, or second-order, Bézier curve, begin with the same parameter, t, represented as a point on a line segment. Use the distance ratio defined by t to simultaneously construct a point at that same distance ratio from P0 to P1, and one from P1 to P2. Call these points Q0 and Q1. Now construct a point at the same distance ratio (defined by t) from Q0 to Q1. This is the orange point in the applet below. That point will trace out the second-order Bézier curve defined by P0, P1, and P2.

Drag t in the applet below to construct the second-order Bézier curve.