The ninth point

The eight coral dots are the given points, and the ivory curve is the cubic through them and Q: a cubic's equation has ten coefficients, so nine points usually fix one. Eight points put only eight conditions on the ten coefficients, so two independent equations are left, F and G (the corner's 10 − 8 = 2); mixed in any proportion, F + tG = 0, they make a whole family, drawn as thin ice-blue lines. Drag Q and the ivory curve sweeps across the picture, yet it always passes through the same point in the gold ring: the ninth point, which every curve of the ice-blue family passes through as well. Now drag the eight: the ninth point moves with them, decided by where they are; it can leave the picture, and then a gold arrow at the edge points to it. When the eight are placed too specially (four or five of them on one line, say), the ninth point is not fixed, and the corner says so. The corner counts how many of the nine points lie on the ivory curve and measures how far the ninth is from it: whenever the ninth point is fixed, 0.000000.