The draughtsman's spline

Drag the ten points up and down on the picture. The gold curve is the natural cubic spline through them: cubic pieces joined so that position, slope and bend all match at each seam, with no bend at the two end points. The ice-blue comb draws the curvature at each place, along the normal; the line through the tips of its teeth is unbroken, though it can turn a corner at a seam, and falls to zero at the ends. The coral curve is the polynomial of degree 9 through the same points: it tends to swing out at the ends, more so as you move the points, and its total bending, top right, is often tens or hundreds of times the spline's. Drag the slider that adds a bump: it lays on the spline a wave that is zero at every point, so the curve still passes through them all, but the comb goes wild at once. Whichever way you drag from 0, the larger the bump, the larger the total bending in the corner, always more than the spline's, by exactly the bump's own share (the third row works it out again from the formula). The equation at the top of the corner holds for every curve through these points, and the extra share is never negative, so of all the curves through them, the spline bends least. The total bending here is ∫ s″², a different measure from the comb: the comb shows curvature, the readout adds up the square of the second derivative; it is close to the energy of a bent strip of wood wherever its slope is small.