There is a slight bug on the interaction. When you set P=Q or for example you can't get the one P at the top and Q at the bottom. The lines disappear.

Basically you need the "infinite/zero" point to compensate for a situation when you have two points completely perpendicular to the x-axis. AKA it is not intersecting a third point. So it intersects this special "infinite" point.

And conceptually why you need this "infinite" point is that without it you can't add points together properly.

Say for counter argument instead of doing this "flip or mirror" across the x-axis (in the interaction it is the red dot appearing). And instead the red dot just appears on the same side as the two points being added on the curve - without the flipping.

If P1+P2 = Q instead of this Q' that is flipped. And P2+Q = P1

If you try and add P1+P2+Q you would get either Q+Q or P1+P1 depending on if you did (P1+P2)+Q or adding up P1+(P2+Q) which are not equal.

so you need this red dot flipping thing happening in the interaction. However, if you have this flipping that means P1+P2 = Q' which is the mirror flip of Q.

So Q'+Q need to equal this special infinite/zero point to ensure associativity works.