Elliptic curves are a family of curves which have the formula:
$$ y^2 = x^3 + ax + b $$
Elliptic Curves form an abelian group under addition.
Identity elements: The identity elements of ellipric curves is the point at infinity.
as long as we do not pick a perfectly vertical line, if we intersect two points in an elliptic curve, then we will also intersect a 3rd point on the elliptic curve.
Inverse: The inverse of an elliptic curve point is the negative of the y value of the pair.
One can derive how to compute P₃ = (x₃, y₃) where P₃ = P₁ ⊕ P₂ using the following formula.
$$ \lambda = \frac{y_2-y_1}{x_2-x_1} \newline x_3 = \lambda^2-x_1-x_2;\space y_3=\lambda\left(x_1-x_2\right) -y_1 $$
It has associativity:
$$ \left(a+b\right)P +cP = aP+\left(b+c\right)P $$