new line, which is considered incident with all the points at infinity (and no other points). This line is called the line at infinity. The extended structure is a projective plane and is called the extended Euclidean plane or the real projective plane. The process outlined above, used to obtain it, is called "projective completion" or projectivization. This plane can also be constructed by starting from R3 viewed as a vector space, see below. Projective Moulton plane The points of the Moulton plane are the points of the Euclidean plane, with coordinates in the usual way. To create the Moulton plane from the Euclidean plane some of the lines are redefined. That is, some of their point sets will be changed, but other lines will remain unchanged. Redefine all the lines with negative slopes so that they look like "bent" lines, meaning that these lines keep their points with negative x-coordinates, but the rest of their points are replaced with the points of the line with the same y-intercept but twice the slope wherever their x-coordinate is positive. The Moulton plane has parallel classes of lines and is an affine plane. It can be projectivized, as in the previous example, to obtain the projective Moulton plane. Desargues' theorem is not a valid theorem in either the Moulton plane or the projective Moulton plane. A finite example This example has just thirteen points and thirteen lines. We label the points P1, ..., P13 and the lines m1, ..., m13. The incidence relation (which points are on which lines) can be given by the following incidence matrix. The rows are labelled by the points and the columns are labelled by the lines. A 1 in row i and column j means that the point Pi is on the line mj, while a 0 (which we represent here by a blank cell for ease of reading) means that they are not incident. The matrix is in Paige–Wexler normal form. {| class="wikitable" style="text-align:center;" |- ! ! m1 ! m2 !! m3 !! m4 ! m5 !! m6 !! m7 ! m8 !! m9 !! m10 ! m11!! m12!! m13 |- style="border-bottom:2px solid #999;" ! P1 | bgcolor="#9cf"|1 || bgcolor="#9cf"|1 || bgcolor="#9cf"|1 || bgcolor="#9cf"|1 || || || || || || || || || |- ! P2 | bgcolor="#9cf"|1 || || || || bgcolor="#9cf"|1 || bgcolor="#9cf"|1 || bgcolor="#9cf"|1 || || || || || || |- !