Why doesn’t the curve always pass through the middle control points?

Only the endpoints are interpolated (for a standard Bézier). Interior controls attract and shape the curve but generally do not lie on it; they define the Bernstein blend, not a polyline through every vertex.

What is special about de Casteljau versus the Bernstein formula?

Both define the same curve. de Casteljau is numerically stable and geometric: each step is a linear interpolation you can draw, which makes the construction intuitive and robust.

What happens if I place all control points on a straight line?

The Bézier curve collapses to that line segment (still parameterized by t). The scaffolding interpolations stay on the line, illustrating that Bézier evaluation preserves affine combinations.

Bézier & de Casteljau

Drag control points; live recursive linear-interpolation scaffolding evaluates B(t).

About this model

A Bézier curve of degree n is determined by control points P₀…Pₙ and evaluated with the Bernstein basis, or equivalently by de Casteljau’s algorithm: recursive linear interpolation B(t) built from nested segments (1−t)A + tB. This simulator lets you drag control points while the scaffolding of intermediate interpolated points is drawn live, showing how the curve is carved from the control polygon. The parameter t ∈ [0,1] traces from P₀ to Pₙ; the curve stays in the convex hull of the controls. The page focuses on evaluation geometry — not CAGD continuity between spans, rational NURBS weights, or curve fitting. Move points and scrub t to see de Casteljau’s pyramid construct B(t).

Who it's for: Computer graphics, CAGD, computational geometry, and intermediate programming / math visualization courses.

Key terms

Bézier curve
de Casteljau algorithm
Control points
Bernstein polynomials
Linear interpolation
Convex hull

How it works

Bézier curves are everywhere in graphics, fonts and CAD. The de Casteljau algorithm evaluates B(t) by repeated linear interpolation: take each pair of adjacent control points, slide a fraction *t* along the segment, and you get a new — shorter — control polygon. Iterate until a single point remains. It is numerically stable, geometric, and reveals beautiful scaffolding — drag any Pᵢ to reshape the curve in real time.

Key equations

B(t) = Σᵢ C(n,i) (1−t)ⁿ⁻ⁱ tⁱ Pᵢ

de Casteljau: Pᵢ⁾ʳ⁾ = (1−t)Pᵢ⁾ʳ⁻¹⁾ + t Pᵢ₊₁⁾ʳ⁻¹⁾

B(t) = P₀⁾ⁿ⁾, cost: O(n²)

Frequently asked questions

Why doesn’t the curve always pass through the middle control points?: Only the endpoints are interpolated (for a standard Bézier). Interior controls attract and shape the curve but generally do not lie on it; they define the Bernstein blend, not a polyline through every vertex.
What is special about de Casteljau versus the Bernstein formula?: Both define the same curve. de Casteljau is numerically stable and geometric: each step is a linear interpolation you can draw, which makes the construction intuitive and robust.
What happens if I place all control points on a straight line?: The Bézier curve collapses to that line segment (still parameterized by t). The scaffolding interpolations stay on the line, illustrating that Bézier evaluation preserves affine combinations.

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Bézier & de Casteljau

About this model

How it works

Key equations

Frequently asked questions

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About this model

How it works

Key equations

Frequently asked questions