Bezier Curves and de Casteljau
Drag four dots and a smooth curve bends to follow. The trick is repeated averaging: split each line at the same fraction, then split the splits, until one point is left to trace.
Bezier Curves and de Casteljau
A pen tool gives you a handful of points and somehow a smooth arc threads between them. There is no spline magic underneath: pick a fraction t, slide along every control segment by that same fraction, then treat those new points as control points and slide again. Keep collapsing until one point is left. That single point is where the curve is at t, and sweeping t from start to finish draws the whole thing.
The geometric construction and the algebraic Bernstein form land on the identical point. de Casteljau collapses the points by repeated interpolation; the Bernstein form writes the result directly as one blend, where each control point is weighted by a polynomial in t. The weights always sum to one, which is why the curve can never escape the shaded hull of the points.
The curve passes through the two endpoints, P0 and P3, and only through those. The interior points are pulls, not destinations: the curve leans toward them and leaves each endpoint heading straight at its neighbour, which is why the dashed control polygon is tangent to the curve at the ends. Drag an interior point far out and the arc bulges that way without ever touching it.
Because every step is an average of points already inside the polygon, the result stays inside their convex hull, the shaded region. This is exactly the curve a font outline stores and a pen tool draws: a cubic per segment, four points each, stitched end to end into letterforms and icons.
The plotted curve samples de Casteljau at 90 values of t. The readout point matches the Bernstein evaluation to floating-point precision, and stays within the convex hull of the 4 control points by construction.