Bézier Curves

Every letter on your screen is drawn with Bézier curves. Four control points and one parameter t trace a smooth path from start to finish.

t = 0.00 speed = 17.49

How it works

Parametric curves and arc length

In one sentence

A cubic Bézier curve $\mathbf B(t) = (1-t)^3\mathbf P_0 + 3(1-t)^2t\,\mathbf P_1 + 3(1-t)t^2\mathbf P_2 + t^3\mathbf P_3$ , $0 \le t \le 1$ , is a weighted average of four control points whose weights always sum to $1$ .

General formula

{\htmlData{sym=0}{\mathbf{B}(t)}} = (1-{\htmlData{sym=5}{t}})^3{\htmlData{sym=1}{\mathbf{P}_0}} + 3(1-{\htmlData{sym=5}{t}})^2{\htmlData{sym=5}{t}}\,{\htmlData{sym=2}{\mathbf{P}_1}} + 3(1-{\htmlData{sym=5}{t}}){\htmlData{sym=5}{t}}^2\,{\htmlData{sym=3}{\mathbf{P}_2}} + {\htmlData{sym=5}{t}}^3{\htmlData{sym=4}{\mathbf{P}_3}}

At each $t$ the weights say how hard each control point pulls: the curve starts at $\mathbf P_0$ , ends at $\mathbf P_3$ , and leans toward $\mathbf P_1$ and $\mathbf P_2$ .

What the symbols mean

$\mathbf{B}(t)$
Point on the curve
Where the curve is at parameter t.
$\mathbf{P}_0$
Start point
Where the curve begins.
$\mathbf{P}_1$
First control point
Pulls the curve toward it near the start. The curve doesn't pass through it.
$\mathbf{P}_2$
Second control point
Pulls the curve toward it near the end.
$\mathbf{P}_3$
End point
Where the curve finishes.
$t$
Parameter
Runs from 0 at the start to 1 at the end.

As $t$ runs from $0$ to $1$ , the point blends the control points with polynomial weights.
De Casteljau's construction reaches the same point by repeated linear interpolation.
The derivative $\mathbf{B}'(t)$ is the velocity along the curve, tangent to it.
Arc length is $\int_0^1\lVert\mathbf{B}'(t)\rVert\,dt$ , a Calculus II integral computed numerically.

Learn the math

Arc LengthCalc II · Lesson 11, Applications of Integrals Parametric Equations and CurvesCalc II · Lesson 16, Parametric Equations and Polar Coordinates Tangents with Parametric EquationsCalc II · Lesson 17, Parametric Equations and Polar Coordinates Vector FunctionsCalc III · Lesson 10, Three-Dimensional Space

Learn the math

Arc LengthCalc II · Lesson 11, Applications of Integrals

Parametric Equations and CurvesCalc II · Lesson 16, Parametric Equations and Polar Coordinates

Tangents with Parametric EquationsCalc II · Lesson 17, Parametric Equations and Polar Coordinates

Vector FunctionsCalc III · Lesson 10, Three-Dimensional Space