Algebraic Solution for Distinct Points

This section provides an algebraic solution for calculating the addition operation of two distinct points on an elliptic curve.

Case 3: P and Q are two distinct points.

```Let m be the slope of the straight line passing through P,
the line can be expressed as the following equation:
y = m(x - xP) + yP

The other intersection, -R, of the line and the curve must satisfy
the line equation and the curve equation:
-yR = m (xR  - xP) + yP        (1)
(-yR)2 = (xR)3 + a(xR) + b     (2)

Note that -yR in equation (2) can be replaced by the right
hand side of equation (1):
(m(xR - xP) + yP)2 = (xR)3 + a(xR) + b

Regrouping terms will result a cubic equation for xR:
(xR)3 - m2(xR)2 + (a - 2m(yP) + 2m2(xP))(xR) + b - (m(xP)-yP)2 = 0

We know that a cubis equation has 3 roots and their sum is the negation
of the coefficient of the second term based on Vieta's first formula,
see Vieta's formulas at wikipedia.org:
xP + xQ + xR = m2              (7)

Rearrange terms of equations (1) and (7) gives us the simpliest way
to calculate, R:
xR = m2 - xP - xQ              (8)
yR = m(xP - xR) - yP           (9)

The slope of the line, m, can be determined by the other point, Q,
on the line:
yP - yQ
m = ---------                (10)
xP - xQ
```

Now we can calculate R = P + Q with equations (8), (9) and (10) with any given distinct points, P and Q.

By the way, proof of Vieta's formulas can be found in "Cubic Equations" by Arkajyoti Banerjee and Ayush G Rai at https://brilliant.org/wiki/cubic-equations/