I'm reading about elliptic curves and how drawing a line between two points on the curve will always intersect with a third point. It seems pretty easy, though, to draw a line such that it only intersects at two points (see image below). Am I choosing the points incorrectly? Are they supposed to be chosen on opposite sides of the y-axis or something?

example of line intersecting two points on EC


closed as off-topic by Lord Shark the Unknown, Lee David Chung Lin, The Count, TomGrubb, Aqua Jul 7 at 6:02

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    $\begingroup$ Just continue the diagram a bit north-east. $\endgroup$ – Lord Shark the Unknown Jul 6 at 18:45
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    $\begingroup$ The elliptic curve is defined by $f(x,y) =y^2-x^3-Ax-B=0$ if $(a,b),(a+c,b+d)$ are two points on the curve and $c \ne 0$ then the cubic polynomial $h(t) = f(a+ct,b+dt)$ always has a 3rd root $T$ and $(a+cT,b+dT)$ is a 3rd point of the curve on the line $\endgroup$ – reuns Jul 6 at 18:51
  • $\begingroup$ What about for a vertical line then? In like regular english, does it intersect at three points still? If so, what does that look like? $\endgroup$ – Alex Gausman Jul 7 at 6:01
  • $\begingroup$ Nevermind, just realized that infinity is the third point for elliptic curves, and (for the elliptic curve cryptography) inf = -inf sort of how 0 = -0 for decimal numbers. Thanks for the responses! $\endgroup$ – Alex Gausman Jul 7 at 6:08