concept vertical line in category algorithms

This is an excerpt from Manning's book Algorithms and Data Structures in Action MEAP V14. Login to get full access to this book.

8.4.2   Moving to Higher Dimensions

Now that we have understood the mechanism for real numbers, what about ℝ²? What about points in a Euclidean bidimensional space? What about ℂ (the set of complex numbers).

Figure 9.2 Partitioning points in the 2-D Cartesian space by cycling through the directions along which we split. For the first split (left) we choose point R and draw a vertical line (parallel to y axis, x coordinate is constant) passing through it. We have such created 2 half-spaces, on the left (yellow) and right (blue) of this line, grouping points W, P, O, Q, U on one side, and S, T on the other. Point R is the pivot of this partitioning.
Next, we choose point W in the yellow partition: this time, we draw a horizontal line (parallel to x axis, y coordinate is constant): it splits the yellow partition into 2 new partitions, one in the top-left area of the plane (red), containing P, O and U, and one in the bottom-left area (green), with just Q.
If we further split the red area at point P, we need to use again a vertical line, as shown in the right-most part of the figure.