Hi Guys , last article we spoke about addition , one of the most important “**invented**” operations on eliptic curve arithmetic . There’s been great feedback highlihghting that the rule **“if a line crosses two points , it will cross a third point”** might not be **absolute**.

That’s is true , there’s a few exceptions and today I will try to explain one , what would happen if point **P** and **Q** share the same position (x and y wise):

**So how do you add two points (P and Q ) that share the same location?** , well this is called “Point Doubling” also referred as adding a point to itself.

This is nothing more than another invented operation and as we saw in EC addition it is represented by another mathematical identity , as well as a different “**Slope**” equation.

#### Point Doubling Slope:

To calculate the slope when **P == Q** we apply the following equation:

s = (3 \* Px \*\* 2 +a ) / (2 \* Py)

![](https://hackernoon.com/hn-images/1*Epi5zzMWbd1fz8umqoMk4Q.png)

Slope in point doubling

That will give you the value of the slope , remember that the slope plus some other domain parameters are curcial to then caculate addition or in this case point doubling.

#### Point Doubling:

Point doubling or “adding a point to itself” follows a very simple equation , and of course it requires the slope and the domain parameters that we’ve seen in previous articles.

But basically it goes like this:

![](https://hackernoon.com/hn-images/1*ltuQVYQYfmfu7-KGYYYOVQ.png)

Point Doubling in action

RX = S \*\* 2 - 2 \* PX  
RY = -1 \* (PY + S \* ( RX - PX))

And again the same catch , R becomes the negative reflection of R , so it looks something like:

![](https://hackernoon.com/hn-images/1*3PJR7OO8DL55Lc6lefizDA.png)

Reflection of R

#### Conclusion:

So point doubing is the second invented arithmetic operation we’ve talked about when it comes to Elliptic Curves , and some of the same rules apply than with addition:

*   by knowing X you know Y
*   R is the reflection of -R over the X axis

Some gisted code so you can have a play:

#### PS: Thanks for all the comments and questions

Different

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