Normal equation

normalized equation

The equation of a line in a plane of the form

$$x\cos\alpha+y\sin\alpha-p=0,$$

where $x$ and $y$ are rectangular Cartesian coordinates in the plane, $\cos\alpha$ and $\sin\alpha$ are the coordinates of the unit vector $\{\cos\alpha,\sin\alpha\}$ perpendicular to the line, and $p\geq0$ is the distance of the coordinate origin from the line. An equation of a line of the form

$$Ax+By+C=0$$

reduces to normal form after multiplication by the normalizing factor $\lambda$ of absolute value $(A^2+B^2)^{-1/2}$ and of sign opposite to that of $C$ (if $C=0$, the sign of $\lambda$ is arbitrary).

Similarly, the equation of a plane

$$Ax+By+Cz+D=0$ reduces to the normal form $$x\cos\alpha+y\cos\beta+z\cos\gamma-p=0,$$

where $\cos\alpha$, $\cos\beta$ and $\cos\gamma$ are the direction cosines of a vector perpendicular to the plane, after multiplication by the normalizing factor $\gamma$ of absolute value $(A^2+B^2+C^2)^{-1/2}$ and of sign opposite to that of $D$.