Where two curves meet

Drag the gold curve across the picture, turn it with the slider, and choose what each curve is: a line, an ellipse, a circle or a cubic. The coral dots are the intersections you can see. As you move it, two of them close in until they meet (at that touching moment the point counts twice), then part as a pair of small ice-blue rings: they have turned complex, with an i in their coordinates and no place in the real plane, and the picture combines the two coordinates into one number, x + iy, and marks them where that number sits in the plane. The two rings are joined by a dotted line, and the real line through each pair is drawn long, dashed and ice-blue. Turn two lines parallel (the ivory one sits at 24°, so turn the gold one to 24° too) and their intersection goes off to infinity; a degree short, it is still there, just far off the picture. Both circles pass through the same two points at infinity, so two of their intersections are always there. As long as the two curves share no stretch, the corner counts the visible ones, the complex ones and those at infinity, and the total stays at the product of the two degrees, m · n.