Plane Partitioning

A line (hyperplane) separates the plane (space) into two parts, and within each part the function $\mathbf{m}^{\top}\mathbf{x}$ has the same sign. This observation makes it easy to determine whether a set of points all lie on the same side of a line/hyperplane.

For example, in the case of a line, depending on the orientation of the generating vector, it is possible to determine in which of the two half-planes (left or right) a generic point lies by examining $s = a x_i + b y_i + c$: when $s<0$, the point lies to the left of the line; when $s>0$, the point lies to the right; and finally, when $s=0$, the point lies on the line.

This observation, namely that the equation of a plane provides an efficient way to determine which half-plane contains a generic point, will be used in the chapter on classifiers: the simple equation of a line or plane can be used as a classifier if the category space, generated by suitable $n$ measurable image features, is separable by a linear surface.