retificável

Retificável is an adjective used in Portuguese with several related meanings, most prominently in mathematics where it describes a curve whose length is finite. A curve, typically denoted gamma, is said to be rectifiable if its arc length L is finite. Formally, for a curve gamma: [a,b] → R^n, L is defined as the supremum of the sums ∑ ||gamma(ti) − gamma(ti−1)|| taken over all finite partitions a = t0 < t1 < ... < tm = b. If L < ∞, the curve is rectifiable. For absolutely continuous curves, the length can be computed as L = ∫_a^b ||gamma′(t)|| dt.

Examples and properties: Straight line segments and polygonal chains are rectifiable, as they have finite length. A circle is rectifiable, with length equal to its circumference. In contrast, some fractal curves, such as the Koch snowflake boundary, are non-rectifiable because their length is infinite despite other properties like finite area.

General framework: Rectifiability is tied to the concept of length and total variation. A curve is rectifiable if and only if its total variation is finite. In metric spaces, rectifiable curves can be studied via their length, and many rectifiable curves admit a parameterization by arc length, which yields a Lipschitz mapping.

Other uses: In everyday Portuguese, retificável can also mean something that can be corrected or rectified, such as a statement, document, or action that is susceptible to correction. This general sense is distinct from the geometric sense but shares the same root idea of making something right or exact.