Standardlinjer

Standardlinjer is a term used in analytic geometry to refer to lines described by a standard linear equation of the form Ax + By = C, where A, B, and C are real numbers with not both A and B equal to zero. This form provides a canonical way to represent a straight line in the plane and is widely used in mathematics, computer science, and applied fields that involve geometric constraints.

In the standard form, the vector (A, B) is normal to the line, meaning it points perpendicular to the line’s direction. The equation can be manipulated by multiplying through by a nonzero scalar without changing the geometric set of points it represents. If B ≠ 0, it can be rewritten as a slope-intercept equation y = (-A/B)x + (C/B); if A ≠ 0, it can be written as x = (C - By)/A for vertical lines. The distance from a point (x0, y0) to the line is given by |Ax0 + By0 − C| / sqrt(A^2 + B^2).

Standardlinjer are convenient in solving systems of linear equations, determining line intersections, and performing geometric computations such as clipping, reflection, and distance analysis. They also facilitate the description of half-planes via constraints of the form Ax + By ≤ C. While other representations exist (slope-intercept, point-normal), the standard form is a versatile default in many algorithms and proofs.