Projective Space

Given a coordinate system, n-dimensional real affine space is the set of points parameterized by the set of all n-component real column vectors $(x_{1}, \ldots, x_{n})^{\top}\in I\!\!R^n$.

Similarly, the points of real n-dimensional projective space ${I\!\!P}^{n}$ can be represented by n+1-component real column vectors $(x_{1}, \ldots, x_{n+1})^{\top}\in I\!\!R^{n+1}$, with the provisos that at least one coordinate must be non-zero and that the vectors $(x_{1}, \ldots, x_{n+1})^{\top}$ and $(\lambda x_{1}, \ldots, \lambda x_{n+1})^{\top}$ represent the same point of ${I\!\!P}^{n}$ for all $\lambda\not=0$. The x_i are called homogeneous coordinates for the projective point.