section_path: Definition/Hesse normal form
section_name: Hesse normal form
paragraph_text: Hesse normal form
  
  The normal form (also called the Hesse normal form, after the German mathematician Ludwig Otto Hesse), is based on the normal segment for a given line, which is defined to be the line segment drawn from the origin perpendicular to the line. This segment joins the origin with the closest point on the line to the origin. The normal form of the equation of a straight line on the plane is given by: x cos φ + y sin φ − p = 0, {\displaystyle x\cos \varphi +y\sin \varphi -p=0,} where φ {\displaystyle \varphi } is the angle of inclination of the normal segment (the oriented angle from the unit vector of the x -axis to this segment), and p is the (positive) length of the normal segment. The normal form can be derived from the standard form a x + b y = c {\displaystyle ax+by=c} by dividing all of the coefficients by c | c | a 2 + b 2. {\displaystyle {\frac {c}{|c|}}{\sqrt {a^{2}+b^{2}}}.}
  
  Unlike the slope-intercept and intercept forms, this form can represent any line but also requires only two finite parameters, φ {\displaystyle \varphi } and p, to be specified. If p > 0, then φ {\displaystyle \varphi } is uniquely defined modulo 2 π. On the other hand, if the line is through the origin (c = p = 0), one drops the c / | c | term to compute sin φ {\displaystyle \sin \varphi } and cos φ {\displaystyle \cos \varphi }, and it follows that φ {\displaystyle \varphi } is only defined modulo π.