Every point of the Euclidean plane is defined by two coordinates, assumed to be taken with respect to a Cartesian coordinate system. A plane curve is then defined by a relation f (x, y) = 0 between the coordinates x and y of its points, where f is a function of two variables.
For example, a line in the plane has the equation ax + by + c = 0, where f (x, y) = ax + by + c is a first-degree polynomial.
It divides the plane into two half-planes defined by the inequalities f (x, y) > 0 and f (x, y) < 0. To determine them, the half-plane containing the origin of the coordinate system is the one with the inequality f (x, y) > 0 if c = f (0, 0) > 0, or the one with the inequality f (x, y) < 0 if c < 0.
The power of lines ------------------------
But another piece of information is hidden in the coefficients. By normalizing the equation f (x, y) = ax + by + c = 0 of the line (D) — that is, by dividing the three coefficients a, b and c by the nonzero real number a2+b2,\sqrt{a^2 +b^2}, we obtain an equation that can be written fD(x, y) = x sin(α) ‒ y cos(α) + d = 0, where α is the slope of the line and | fD(0, 0) | = | *d | is the distance from the origin to the line.