envelopedetectoria

Envelopedetectoria is a neologism used in geometry and dynamical systems to describe the envelope of a family of trajectories, i.e., the boundary curve that is tangent to members of the family as a parameter varies. It refers to the locus of points that are tangent to at least one trajectory, and thus marks the limit of the region traced by a moving point under changing conditions or rules of motion.

In mathematical terms, suppose a family of trajectories is described implicitly by F(x,y,s)=0 for a parameter s. The envelopedetectoria is the set of points (x,y) for which there exists s with F(x,y,s)=0 and ∂F/∂s=0. A simple example is the family of lines y = m x + b(m). Here F(x,y,m) = y - m x - b(m). The envelope follows from F=0 and ∂F/∂m = -x - b'(m) = 0, giving x = -b'(m) and y = -m b'(m) + b(m).

Applications include robotics and motion planning, where it characterizes the boundary of the reachable region; optics and mechanics, where envelopes describe caustics and boundary light paths; and computer graphics, where it helps with collision approximation and shading.

The term blends envelope (enveloping boundary) with trayectoria (trajectory) in Spanish and is used mainly in theoretical discussions as a conceptual tool rather than a standard classical term.

See also: envelope (mathematics), trajectory, caustic, family of curves.