Why is it better to add and double points on an elliptic curve using projective space?
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?


Because it is a lot more efficient.

If you take a look at the operations involved in performing an elliptic curve operation on a prime curve, they are:


Modular addition and subtraction
Modular multiplication
Modular inversion, that is, given $a$, find $b$ such that $a \cdot b \equiv 1 \pmod p$


It turns out that modular addition and subtraction is fast, modular multiplication is somewhat slower, and modular inverse is dreadfully slow.

And, if we do everything using affine coordinates, it turns out that we need to do a modular inversion for every point addition or doubling.

In contrast, if we do things using projective coordinates, it turns out that we don't need to compute any inverses at all (we do increase the number of modular additions and multiplications we need, however the time involved there is much less than doing a single modular inversion, and so it is a win).

Now, at the end, we'll need to convert back to affine coordinates; that involves a single modular inversion; however when we do a point multiplication, that means that we replaced several hundred modular inversions with just one - a definite win.