Short answers, pulled from the story.
Euclid defined a line as a breadthless length that lies evenly with respect to the points on itself. This definition relied on physical intuition rather than rigorous proof and dominated geometry for over two millennia.
Mathematicians like David Hilbert began to reconstruct geometry from the ground up in the 19th century. They replaced intuitive descriptions with strict axioms to ensure logical consistency.
In three-dimensional space, a line emerges as the intersection of two distinct planes. It exists only where those planes cross and is the common solution to two linear equations.
The standard form of this equation is ax plus by equals c. This formula allows for the representation of vertical lines when the coefficient b is zero.
In elliptic geometry, lines are represented as great circles on a sphere and any two lines will eventually intersect. This contradicts the Euclidean notion of parallel lines that never meet.