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> Pick two different random points with different x value on the curve, connect these two points with a straight line, let’s say A A and B B . Then you will no

> Pick two different random points with different x value on the curve, connect these two points with a straight line, let’s say A A and B B . Then you will notice the line touches the curve at a third point. I seem to always get hung up on this part of the explanation. Looking at the graph, I can see points along the curve, where a line would only intersect with 2 points on the curve. What do you do in that case? Is this a matter of, yes those points are there, but they are rare enough that we just pick another set of random points and try again, or is there another solution to the issue?

deleted 5y ago
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eat_veggies 5y ago
Yep, there are vertical lines that intersect the curve at only two points. In that case there is a special "infinity" point (also known as zero). See page 21 in this presentation which I think explains it a little bit better: https://www.math.brown.edu/johsilve/Presentations/WyomingEllipticCurve.pdf https://www.math.brown.edu/johsilve/Presentations/WyomingEll...
dlubarov 5y ago
It holds for any two points with distinct x coordinates. Note that the third point might be outside the range of coordinates shown in the article's graphs. Particularly if the line is nearly vertical, you may need to zoom out to see the third point.
- hh3k0 5y ago
  Sure seems to me that he is either unaware of or struggles with the point at infinity, so I'll add a link for him in my reply to your comment: https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication#Point_at_infinity https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplic....
loup-vaillant 5y ago
When it intersect with only 2 points, one of those points is intersected "twice": the line is tangent to the curve. You could also see the tangent as the limit when you intersect with two points, and then draw one of those points towards the other, closer and closer. In practice, this means that P+P doesn't compute exactly the same way as P+Q where P and Q are different points. But in practice it really does mean the same thing.
SamBam 5y ago
> Looking at the graph, I can see points along the curve, where a line would only intersect with 2 points on the curve. I always start out under this impression too, but then I think some more and realize that there are only two conditions where this is possible: 1. The line is vertical 2. The line is the tangent of the curve at one of the points 1 Is illegal by the rules of picking points, and for 2 I believe the tangent counts as two points, and in any case, for any arbitrary point A, there will be only three other points that will allow the line to be a tangent (one when where A is on the tangent and up to two where B is on the tangent, I believe). So once you've picked an arbitrary point, and you don't move vertically, there will be no more than three lines that don't follow the three-point rule, and every other possible line will follow the rule.
johbjo 5y ago
As long as the points are not above/below each other, it will always intersect. We can see in one of the linked notes that vertical points are defined as each other's inverses, and adding them results in a type of "zero-element".

deleted 5y ago

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eat_veggies 5y ago

Yep, there are vertical lines that intersect the curve at only two points. In that case there is a special "infinity" point (also known as zero). See page 21 in this presentation which I think explains it a little bit better: https://www.math.brown.edu/johsilve/Presentations/WyomingEllipticCurve.pdf https://www.math.brown.edu/johsilve/Presentations/WyomingEll...

dlubarov 5y ago

It holds for any two points with distinct x coordinates. Note that the third point might be outside the range of coordinates shown in the article's graphs. Particularly if the line is nearly vertical, you may need to zoom out to see the third point.

hh3k0 5y ago
Sure seems to me that he is either unaware of or struggles with the point at infinity, so I'll add a link for him in my reply to your comment: https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication#Point_at_infinity https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplic....

loup-vaillant 5y ago

When it intersect with only 2 points, one of those points is intersected "twice": the line is tangent to the curve. You could also see the tangent as the limit when you intersect with two points, and then draw one of those points towards the other, closer and closer. In practice, this means that P+P doesn't compute exactly the same way as P+Q where P and Q are different points. But in practice it really does mean the same thing.

SamBam 5y ago

> Looking at the graph, I can see points along the curve, where a line would only intersect with 2 points on the curve. I always start out under this impression too, but then I think some more and realize that there are only two conditions where this is possible: 1. The line is vertical 2. The line is the tangent of the curve at one of the points 1 Is illegal by the rules of picking points, and for 2 I believe the tangent counts as two points, and in any case, for any arbitrary point A, there will be only three other points that will allow the line to be a tangent (one when where A is on the tangent and up to two where B is on the tangent, I believe). So once you've picked an arbitrary point, and you don't move vertically, there will be no more than three lines that don't follow the three-point rule, and every other possible line will follow the rule.

johbjo 5y ago

As long as the points are not above/below each other, it will always intersect. We can see in one of the linked notes that vertical points are defined as each other's inverses, and adding them results in a type of "zero-element".