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: where is the angle of inclination of the normal segment (the oriented angle from the unit vector of the -axis to this segment), and is the (positive) length of the normal segment. The normal form can be derived from the standard form by dividing all of the coefficients by Unlike the slope-intercept and intercept forms, this form can represent any line but also requires only two finite parameters, and , to be specified. If , then is uniquely defined modulo . On the other hand, if the line is through the origin (), one drops the term to compute and , and it follows that is only defined modulo . Other representations Vectors The vector equation of the line through points A and B is given by (where λ is a scalar). If a is vector OA and b is vector OB, then the equation of the line can be written: . A ray starting at point A is described by limiting λ. One ray is obtained if λ ≥ 0, and the opposite ray comes from λ ≤ 0. Polar coordinates In a Cartesian plane, polar coordinates are related to Cartesian coordinates by the parametric equations: In polar coordinates, the equation of a line not passing through the origin—the point with coordinates —can be written with and Here, is the (positive) length of the line segment perpendicular to the line and delimited by the origin and the line, and is the (oriented) angle from the -axis to this segment. It may be useful to express the equation in terms of the angle between the -axis and the line. In this case, the equation becomes with and These equations can be derived from the normal form of the line equation by setting and and then applying the angle difference identity for sine or cosine. These equations can also be proven geometrically by applying right triangle definitions of sine and cosine to the right triangle that has a point of the line and the origin as vertices, and the line