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If I plot a slightly non-vertical line that's not a tangent, it seems that it would intersect an elliptic curve at only 2 points? Is this correct?

If so it throws me off my understanding the explanations that say that you always intersect 3 points (with multiplicity at tangents).

With an exactly vertical line, it seems that the second point is an inverse while the third point is the O; i. e. P + -P + O = O

However if the line is slightly non-vertical it gives P + -Q + O = O which seems fishy because P has multiple inverses? What am I understanding wrong?

Update: As correctly pointed out in the answer and comment the third point does exist but it just plots way outside the plot area. I managed to plot it on Wolfram Alpha and the two attached images show what happens with a quite diagonal line and with a quite vertical line, for the same elliptic curve. Here is the image of a diagonal line a secant with three points of intersection clearly in sight

and here is the almost vertical one.

a nearly vertical secant with the third point of intersection very far out

Also, another way I convinced myself is that for any non-vertical line, y grows linearly with x whereas the elliptical curve grows super-linearly with x, and hence faster than the line. So eventually the curve will overtake the line.

The arc of the moral universe is long, but it bends towards justice. -Theodore Parker

user2297550
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  • What Bernd (+1) says below (the third point of intersection may be outside the region you plotted). Also, a non-vertical line won't pass thru the point at infinity ($=0$). Switch to another affine chart to see that. – Jyrki Lahtonen Dec 22 '17 at 07:19
  • I really liked this explanation: http://web.math.ucsb.edu/~padraic/ucsb_2014_15/ccs_discrete_f2014/ccs_discrete_f2014_lecture9.pdf which confirmed my thought that the tangent at the point of inflection intersects exactly one point on the curve, but not the "point at infinity" (I think). – user2297550 Dec 23 '17 at 12:41

2 Answers2

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Hint: If you write down the equation for the elliptic curve (at least the affine part) and the line explicitely and then solve for intersection you will find the third solution.

Bernd
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By eliminating $y$, you get a cubic polynomial. By the fundamental theorem of algebra, cubics have one or three roots (degenerate cases excluded).