= Projective geometry
Projective geometry studies <projective spaces> and incidence relations such as which points lie on which lines, with transformations induced by invertible <linear maps>. Over a <field> $K$, points of $\mathbb P^n(K)$ are one-dimensional <vector subspaces> of $K^{n+1}$, represented by nonzero homogeneous coordinates $[x_0:\cdots:x_n]$ up to a common nonzero scalar.
The chart $x_0\ne0$ gives affine coordinates $(x_1/x_0,\ldots,x_n/x_0)$. Points with $x_0=0$ form a hyperplane at infinity. In the projective plane, distinct lines meet in one point: parallel affine lines acquire their shared direction point at infinity. This statement concerns the projective plane, not arbitrary pairs of lines in higher dimension.
A <projective transformation> preserves incidence and the <cross-ratio>, but generally not Euclidean lengths or angles. This makes the subject natural for perspective and for homogeneous equations in <algebraic geometry>.
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