# Visualization of Elliptic Point Addition [closed]

In an introductory text, I found that point addition for elliptic curves can be made plausible as follows:

Given two Points, P, Q, the sum is defined as the point on the curve I get by connecting P with Q with a straight line. This line intersects the curve on a third point. After reflecting y I get the sum P+Q.

However, taking the points R and Q on the attached figure $y^2=x^3-\frac{29}{10}x +1$, where is the third point?

Should it be the point at infinity? That cannot be from my point of view, because $P+Q=\infty$ would imply $P = -Q$ which is not the case.

http://demonstrations.wolfram.com/AdditionOfPointsOnAnEllipticCurveOverTheReals/ The Snapshot 1 is the curve I talk about. The following is stated below:

For two points P and Q on an elliptic curve, the addition is defined as follows. Draw the line through P and Q to intersect the curve in a third point; then reflect that point in the axis. • I'm voting to close this question as off-topic because it belongs to Math.SE Nov 21 '16 at 9:21
• It is also unclear; what are the equation of the curve and the coordinates of the points? Nov 21 '16 at 9:27
• I added the equation and a linkto wolfram, where point addition is explained. Nov 21 '16 at 9:41
• Math are the foundations of cryptography. Elliptic curve arithmetics are especially relevant. This may also belong to Math.SE, but definitely has its place here. Nov 21 '16 at 14:25
• In your drawing, you have a tangent at $P$, and in that construction this would actually represent $P + P = R$. Not sure what that has to do with $R+Q$ at all.
– tylo
Nov 21 '16 at 15:37

Although on the picture it seems that there is no third intersection point in the plane, there is one. Note that not the whole curve can be fit into this picture, since at some point we cut off the rest of the $x$ and $y$-axis. What remains unknown is what happens when $x$ becomes larger. Well, what happens is that slope of $E$ keeps increasing (resp. decreasing), going towards infinity (resp negative infinity). You can see this by considering the function $y(x)=\sqrt{x^3-\frac{29}{10}x+1}$, and always taking the positive (resp. negative) root. The derivative will go to infinity (resp. negative infinity) as $x$ goes to infinity.
• The picture is a bit confusing. The question is about the line $QR$, where $Q$ is the green dot and $R=(x_3,y_3)$. This intersects $E$ in $Q$, $R$ and some point on $E$ which has negative $y$-coordinate (and nowhere else). The line going through $P$ does not intersect $E$ in any other point than shown on the picture. This is a so-called tangent line at $P$. We think of it as intersecting twice at $P$, and once at the other point we see in the picture (and nowhere else) Nov 22 '16 at 10:41