Making continuity visible: piecewise cubic Bézier reconstruction of the apple logo in Wolfram Mathematica

Continuity conditions between adjacent Bézier segments are simple to state and difficult to picture. A student can write P3 = Q0 for positional continuity and Q1 = 2P3 − P2 for tangent continuity long before recognising which of two drawings satisfies the second. This article reports a curve fitting activity built around that gap. The outline of the Apple logo was reconstructed in Wolfram Mathematica from fourteen piecewise cubic Bézier segments, twelve for the body and two for the leaf, using control points read by hand from the boundary rather than produced by automatic vectorisation. The same set of boundary points was then used twice, once with positional continuity alone and once with tangent continuity imposed at every joint that should be smooth. The two outlines differ by a single algebraic constraint, yet the first shows a visible corner on the left shoulder of the apple while the second does not. Joints carrying sharp features, the leaf tips and the bite, were deliberately left outside the tangent condition, which turns smoothness into a design decision rather than a default. The activity needs no CAD system and makes the geometric meaning of each Bernstein property directly observable.