Sinuoidika

Sinuoidika is a term used in some mathematical and applied contexts to describe a family of curves and signals that generalize sinusoidal behavior by allowing the amplitude and instantaneous frequency to vary with time. The name combines the sine root with the -oid suffix and the plural -ika, reflecting its resemblance to sine-like forms while acknowledging variation.

In two dimensions a sinuoidika curve can be described parametrically by x(t) = A(t) cos(θ(t)) and y(t) = A(t) sin(θ(t)), where A(t) is an amplitude envelope and θ(t) is a phase function with derivative θ′(t) = k(t), the instantaneous angular frequency. The amplitude envelope A(t) creates modulation, while k(t) introduces frequency modulation. A common way to express θ(t) is θ(t) = θ0 + ∫0^t k(τ) dτ.

A simple example is A(t) = 1 + 0.5 sin(0.1 t) and k(t) = 2π f0 + α t, which yields a locally sinusoidal path with gradually increasing frequency.

In one dimension, sinuoidika also refer to non-stationary sinusoidal signals often modeled as A(t) sin(∫0^t k(τ) dτ + φ0). In practice, the concept is used in signal processing, music synthesis, and computer graphics to describe and analyze signals whose amplitude and frequency evolve over time.

Notes: There is no single universally accepted definition, and the term may be employed differently in various sources. See also sinusoid, chirp, amplitude modulation, frequency modulation, envelope.