Visualization of Elliptic Point Addition 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.

See also:

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 think this is on-topic for cryptography, as this is a basic elliptic-curve related question that anyone interested in ECC will come across. Moreover, it is easily answered.

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.

Therefore if you want to continue the drawing of this curve, you have to extend the rightmost part by making it more and more vertical. If you also extend your line, you will see that they will intersect.