Longituden

Longituden is a hypothetical, dimensionless quantity used in geometry and pattern analysis to express the length of a path relative to a characteristic length L0. It is defined for a continuous curve r(t) on an interval [a,b] as Lambda = (1/L0) ∫_a^b |dr/dt| dt, where L is the arc length. In discrete data, Lambda ≈ (1/L0) ∑_i |p_{i+1}-p_i|. The choice of L0 sets the scale; common choices include the span of a coordinate axis or a user-specified reference length.

Because L0 scales with the object's size, longituden is invariant under similarity transformations, making it useful for comparing shapes across sizes. It is a normalized measure of a curve's extent and is not a physical length by itself.

Applications include comparing trajectory lengths in different coordinate systems, pattern recognition, robotics path planning, and morphometric analyses in biology and geology. A straight line of actual length L, with L0 = L, yields longituden 1. A circle with circumference C and L0 = C also yields longituden 1. If the arc length L grows while L0 is fixed, longituden grows proportionally, capturing relative extent rather than absolute size.

Origin and usage notes: longituden is a constructed term used to discuss scale-invariant length concepts; it is not a standard mathematical quantity but serves as a convenient tool in theoretical discussions and computational analyses.