Bézier Curve — Definition, Formula & Examples

A Bézier curve is a smooth curve whose shape is determined by a set of control points. By adjusting these control points, you control how the curve bends between its start and end points.

A Bézier curve of degree $n$ is a parametric curve defined by $\mathbf{B}(t) = \sum_{i=0}^{n} \binom{n}{i}(1-t)^{n-i}\,t^{i}\,\mathbf{P}_i$ for $t \in [0,1]$ , where $\mathbf{P}_0, \mathbf{P}_1, \ldots, \mathbf{P}_n$ are the control points and $\binom{n}{i}$ are binomial coefficients. The curve interpolates $\mathbf{P}_0$ and $\mathbf{P}_n$ and is tangent to the control polygon at those endpoints.

Key Formula

\mathbf{B}(t) = (1-t)^2\,\mathbf{P}_0 + 2(1-t)\,t\,\mathbf{P}_1 + t^2\,\mathbf{P}_2

Where:

$t$ = Parameter ranging from 0 to 1 that traces the curve from start to end
$\mathbf{P}_0$ = Starting control point (the curve begins here)
$\mathbf{P}_1$ = Middle control point that influences the curve's bend
$\mathbf{P}_2$ = Ending control point (the curve ends here)

How It Works

You place control points in the plane, then evaluate the Bézier formula at values of

t

from 0 to 1 to trace out the curve. At

t = 0

the curve starts at the first control point, and at

t = 1

it ends at the last. The intermediate control points pull the curve toward themselves without the curve necessarily passing through them. A quadratic Bézier has 3 control points (degree 2), and a cubic Bézier has 4 (degree 3). Cubic Bézier curves are the standard in font design and vector graphics software.

Worked Example

Problem: Find the point on a quadratic Bézier curve at t = 0.5, given control points P₀ = (0, 0), P₁ = (2, 4), and P₂ = (4, 0).

Step 1: Write the quadratic Bézier formula and substitute t = 0.5.

\mathbf{B}(0.5) = (1-0.5)^2(0,0) + 2(1-0.5)(0.5)(2,4) + (0.5)^2(4,0)

Step 2: Evaluate each coefficient:

(0.5)^2 = 0.25

and

2(0.5)(0.5) = 0.5

\mathbf{B}(0.5) = 0.25(0,0) + 0.5(2,4) + 0.25(4,0)

Step 3: Multiply and add the components.

\mathbf{B}(0.5) = (0,0) + (1,2) + (1,0) = (2, 2)

Answer: At t = 0.5, the curve passes through the point (2, 2). Notice this is directly below P₁ = (2, 4) — the curve is pulled toward the middle control point but does not reach it.

Why It Matters

Bézier curves are the backbone of vector graphics tools like Adobe Illustrator and font rendering engines. In computer science and engineering courses, they introduce parametric equations, polynomial blending, and the connection between algebra and visual design.

Common Mistakes

Mistake: Assuming the curve passes through all control points.

Correction: A Bézier curve only guaranteed passes through the first and last control points. The intermediate control points influence the shape but generally are not on the curve.