Although the previous geometric descriptions of elliptic curves provides an excellent method of illustrating elliptic curve arithmetic, it is not a practical way to implement arithmetic computations. Algebraic formulae are constructed to efficiently compute the geometric arithmetic.
When P = (xP,yP) and Q = (xQ,yQ) are not negative of each other,
P + Q = R where
s = (yP - yQ) / (xP - xQ)
xR = s2 - xP - xQ and yR = -yP + s(xP - xR)
Note that s is the slope of the line through P and Q.
When yP is not 0,
2P = R where
s = (3xP2 + a) / (2yP )
xR = s2 - 2xP and yR = -yP + s(xP - xR)
Recall that a is one of the parameters chosen with the elliptic curve and that s is the tangent on the point P.