nearperpendicular

Nearperpendicular is a term used in geometry and related fields to describe two lines, vectors, or planes that are oriented close to a right angle. It denotes an approximate perpendicular relationship defined by a specified tolerance around 90 degrees.

Mathematically, the angle theta between two nonzero objects is used to quantify nearperpendicularity. For two vectors a and b, cos(theta) = (a · b) / (|a||b|). Nearperpendicular means theta is within a chosen tolerance δ of 90 degrees, i.e., |theta − 90°| < δ, or equivalently |cos(theta)| is small, often below a chosen threshold ε. In practice, this is implemented with normalized dot products or angle checks, depending on the application.

For lines in a plane, nearperpendicular means their direction vectors form an angle close to 90 degrees. In three dimensions, the angle between two lines is typically taken as the angle between their direction vectors; this applies whether the lines are intersecting or skew. The magnitude of the cross product, |a × b| = |a||b|sin(theta), also reflects nearperpendicularity when |sin(theta)| is near 1.

Applications of nearperpendicular criteria appear in computer graphics, computational geometry, structural engineering, and robotics, where precise right-angle relationships may be impractical due to measurement error or numerical tolerances. Choosing an appropriate tolerance depends on scale, measurement accuracy, and the consequences of deviation from exact orthogonality. Related terms include approximately perpendicular and approximate orthogonality.