Every letter on this screen is a few points plus one interpolation repeated. As t runs from 0 to 1, de Casteljau’s algorithm interpolates linearly along each edge of the control polygon, then interpolates again on the results, layer calling layer until a single point remains, and that point traces the curve. The coefficients of a cubic Bézier are 1, 3, 3, 1, exactly one row of Pascal’s triangle. The curve always stays inside the convex hull of its control points, while t is not proportional to arc length, so the moving point speeds up and slows down with the curvature.
▸ Placing the control points P₀…Pₙ…
▸ Building the control polygon
▸ Loading de Casteljau's algorithm (nested interpolation)
control points · interpolation · Bernstein · spline
Interpolation & values
✒️ Bézier curve
de Casteljau · t sweeps 0→1
Curve degree—
Number of control points—
Parameter t—
Point B(t) · x—
Point B(t) · y—
Interpolation stages—
Tip
A Bézier curve is "pulled" toward the control points but only passes through P₀ and Pₙ. The de Casteljau algorithm repeats linear interpolation stage by stage until one point remains — exactly the point on the curve at parameter t. Drag the control points to reshape it.
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Drag the control points to change the curve · drag the t bar to set it by hand · change the Scenario (degree 1/2/3 · spline · high degree · font letter)
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Bernstein weights Bᵢ(t) vs t (the "blending" curves)B₀B₁B₂B₃
About Bézier Curves
Every letter on this screen is a few points plus one interpolation repeated. As t runs from 0 to 1, de Casteljau’s algorithm interpolates linearly along each edge of the control polygon, then interpolates again on the results, layer calling layer until a single point remains, and that point traces the curve. The coefficients of a cubic Bézier are 1, 3, 3, 1, exactly one row of Pascal’s triangle. The curve always stays inside the convex hull of its control points, while t is not proportional to arc length, so the moving point speeds up and slows down with the curvature.