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ch04: the line through (x,y) and (x,-y) is not "tangent"

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On the elliptic curve, a line that goes through two different points
with the same `x` coordinates, but different `y` coordinates (they must
be `y` and `-y`) is not "tangent".
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Vasil Dimov 2021-02-25 07:23:18 +01:00
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commit 821580d2c7
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@ -186,7 +186,7 @@ There are a couple of special cases that explain the need for the "point at infi
If P~1~ and P~2~ are the same point, the line "between" P~1~ and P~2~ should extend to be the tangent on the curve at this point P~1~. This tangent will intersect the curve in exactly one new point. You can use techniques from calculus to determine the slope of the tangent line. These techniques curiously work, even though we are restricting our interest to points on the curve with two integer coordinates!
In some cases (i.e., if P~1~ and P~2~ have the same x values but different y values), the tangent line will be exactly vertical, in which case P~3~ = "point at infinity."
In some cases (i.e., if P~1~ and P~2~ have the same x values but different y values), the line between P~1~ and P~2~ will be exactly vertical, in which case P~3~ = "point at infinity."
If P~1~ is the "point at infinity," then P~1~ + P~2~ = P~2~. Similarly, if P~2~ is the point at infinity, then P~1~ + P~2~ = P~1~. This shows how the point at infinity plays the role of zero.