I have been given a textbook which defines the addition of two points on an elliptic curve and the doubling of a point on an elliptic curve. This textbook explains elliptic curves in projective space where $z=1$ and it expresses the points on the curve as affine points of the form $[x,y,1]$ but it never really explains why we want to do this.
I have looked over many solutions of which only explain the addition of two points $P$ and $Q$ as the following:
We find the line between the points, find the point of intersection on the curve and finally reflect it over the x axis.
The textbook I have been given says that for adding two points P and Q we use the group operation $P\bigoplus Q=O*(P*Q)$ where $O$ is a "point at infinity" $[0,1,0].$ We first find the slope of the line between the points and then we plug it in to the following equations:
and the point should be denoted as $[x,y,1]$.
This textbook has confused me beyond belief.
To my understanding, a projective plane essentially is a plane that is parallel to another plane that lies flat on the x,y axis. This plane is perpendicular to the z axis (points satisfy $z=1$), it contains affine points of the original plane and the point at infinity $O$. If one were to construct a 3 dimensional line crossing the origin it would eventually pass through the coordinates $[x_z,y_z,z]$, so the point $(1,2)$ would be found on the projective curve as $[1,2,1]$.
Is there any specific reason why we want to express points of the elliptic curve in projective space? Does it make adding two points or doubling a point easier?How might I explain to someone why I chose to represent it using projective space if they were to ask me why not just use the algorithm shown in the picture?