Mathematically. secp256k1 is an elliptic curve which is defined over a set of finite fields Fp when we say finite, we mean the value ranges from 1 to p -1 where p is a prime number that ranges between 2 ^ 256 - 2 ^ 32 - 977 this is a very solid prime number which does not create extra spaces or holes in the algorithm. The name secp256k1 is derived because it's a secure algorithm which has about 256 bits of data and k1 is a special family of curves.
secp256k1 is the elliptic curve used especially in Bitcoin for wallet transactions signing and verifications
The secp256k1 theorizes that a line which passes or cuts across two points on the curve P and Q must also touch another third point R on the elliptic curve, this is denoted by the formula y^2 = x^3 + 7 (mod p) where x and y are points on the curve, they are distinct points on the curve.
For the secp256k1 there are two points on the curve denoted as p and q and when these are added together p + q they form another point on the curve where p ≠ q which means they are distinct, when this operation occur, we call that point addition. where p = (x1, y1) and q = (x2, y2).
point doubling operation in an elliptic curve is done when both points on the curve are the same which is P = Q which satisfies the formula where
p = (xº, yº) q = (xª, yª)
x¡ = ¡2 - xº - xª but if xº = xª then x¡ = ¡2 - 2x¡ for p = q