involutes

Involutes are curves that can be defined as the locus traced by the end of a taut string as it is unwound from another curve. This construction, often called the involute of the base curve, is a standard object in differential geometry and has practical applications in gear design and computer-aided geometry.

Mathematically, let γ(s) be a smooth plane curve parameterized by arc length s, with unit tangent T(s) = γ'(s). The involute I(s) of γ is I(s) = γ(s) − s T(s), obtained by unwinding a string of length s from γ. The string is tangent to γ at γ(s), and the segment joining γ(s) to I(s) lies along the tangent line. The evolute of the involute is the original base curve γ.

A canonical example is the circle. For a circle of radius a, γ(t) = (a cos t, a sin t). The unit tangent is T(t) = (−sin t, cos t), so the involute is I(t) = γ(t) − t T(t) = (a cos t + t sin t, a sin t − t cos t). This involute starts at (a, 0) when t = 0 and expands outward as t increases; it is a classical curve often cited in geometry and kinematics.

Extensions and applications: the concept generalizes to space curves and can be used in the design of involute gears, where the tooth profile is based on the involute to ensure a constant velocity ratio during meshing. Involutes also relate to the geometry of evolutes and have a role in certain computer-aided design and manufacturing contexts.