A Bézier curve is a parametric polynomial curve controlled by a small set of points.

For three control points, a quadratic Bézier curve is

$$ \mathbf{B}(t)=(1-t)^2\mathbf{P}_0+2(1-t)t\mathbf{P}_1+t^2\mathbf{P}_2, \qquad 0\le t\le1. $$

The curve starts at $\mathbf{P}_0$ and ends at $\mathbf{P}_2$. The middle control point usually does not lie on the curve; instead it pulls the curve and determines the endpoint tangent directions.

Higher-degree Bézier curves follow the same idea. They can also be evaluated by de Casteljau's algorithm, which repeatedly linearly interpolates between neighboring control points.

Because the interpolation weights are nonnegative and sum to one, a Bézier curve stays inside the convex hull of its control points.

Quadratic and cubic Bézier segments are widely used for vector paths, font outlines and geometric modeling because they are compact, smooth and easy to transform.