Recursive wave

A recursive wave is a self-similar curve in three-dimensional space that is constructed by iteratively adding a helix around the previous curve.

Recursive waves of depths 1, 2 and 3.

Construction

A recursive wave of depth ${\displaystyle n}$  can be constructed as following:

${\displaystyle \psi _{0}(x)=x(ai+bi+ck)}$ 

${\displaystyle \psi _{n}(x)=\psi _{n-1}(x)+R(A(n)g_{n}(x),\ \psi _{n-1}'(x),\ f(n)x+\alpha (n))}$ 

where

${\displaystyle g_{n}(x)=|{\vec {w}}\times |\psi _{n-1}'(x)||}$ 

and

${\displaystyle R({\vec {A}},{\vec {B}},\theta )=e^{{\vec {B}}\theta /2}{\vec {A}}e^{-{\vec {B}}\theta /2}}$ 

Clarification

Each wave at non-zero depth ${\displaystyle n}$  is described by an amplitude ${\displaystyle A(n)}$ , frequency ${\displaystyle f(n)}$  and phase offset ${\displaystyle \alpha (n)}$ .

${\displaystyle g_{n}(x)}$  represents a unit vector that is perpendicular to the previous curve at ${\displaystyle x}$ . An arbitrary vector ${\displaystyle {\vec {w}}}$  is chosen to be the fixed "rag" vector.

${\displaystyle R}$  is a function that rotates a vector ${\displaystyle {\vec {A}}}$  around an axis defined by a vector ${\displaystyle {\vec {B}}}$  by ${\displaystyle \theta }$  degrees. In this case it is expressed with quaternions.