Is there any way of point square in elliptic curve cryptography?

Question

From elliptic curve cryptography we can evaluate

1. Point addition P+ Q
2. Point doubling P+P

But is there any way we can determine point square?

P^2

Answer 1

When we're doing point doubling then we're multiplying with 2, a vector. Multiplication with a vector is defined, multiplication with another point is not.

So this begs the question what kind of operation would be represented by squaring, as that's $P\times P$. So in general the answer is no; you'd first have to define what is meant with point multiplication.