The Division of the Plane

The line (hyperplane) separates the plane (space) into two parts, and within each of these parts, the function $\mathbf{m}^{\top}\mathbf{x}$ assumes the same sign. Through this consideration, it is easy to determine whether a set of points lies entirely on the same side of a line/hyperplane or not.

For example, in the case of the line, depending on how the generating vector is oriented, one can understand in which of the two half-planes (left, right) a generic point falls by studying $s = a x_i + b y_i + c$: when $s<0$ the point is to the left of the line, $s>0$ the point is to the right, and finally when $s=0$ the point is on the line.

This consideration, namely that the equation of a plane efficiently determines in which half-space a generic point falls, will be used in the chapter on classifiers: the simple equation of a line or a plane can be used as a classifier if the category space, generated by appropriate $n$ measurable features of the image, is separable by a linear surface.