Recursive wave: Difference between revisions

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{{Multiple issues|{{Orphan|date=September 2020}}{{Unreferenced|date=November 2020}}

== Introduction ==

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[[File:Recwave1.gif|thumbnail|ARecursive recursive wavewaves of depthdepths 1, 2 and  3.]]

A '''recursive wave''' is a self-similiarsimilar curve in [[three -dimensional space.]] Itthat is constructed by iteratively adding a [[helix]] around the previous curve.

[[File:Recwave1.gif|thumbnail|A recursive wave of depth 1, 2 and 3.]]

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

== Construction ==

A recursive wave of depth <math>n</math> can be constructed as following:

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{{Equation box 1 |indent =: |title= |equation = <math>\psi_0(x) = x(ai + bi + ck)</math> |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}}

{{Equation box 1 |indent =: |title= |equation = <math>\psi_{n}(x) = \psi_{n-1}(x) + R(A(n)g_n(x),\ \psi_{n-1}'(x),\ f(n)x + \alpha(n))</math> |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}}

where

Where

{{Equation box 1 |indent =: |title= |equation = <math>g_n(x) = |\vec{w} \times |\psi_{n-1}'(x)||</math> |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}}

and

And

{{Equation box 1 |indent =: |title= |equation = <math>R(\vec{A}, \vec{B}, \theta) = e^{\vec{B}\theta /2}\vec{A}e^{-\vec{B}\theta /2}</math> |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}}

=== Clarification ===

Each wave at non-zero depth <math>n</math> is described by an [[amplitude]] <math>A(n)</math>, [[frequency]] <math>f(n)</math> and [[phase offset]] <math>\alpha(n)</math>.

Each wave at non-zero depth <math>ng_n(x)</math> represents a unit [[Vector space|vector]] that is describedperpendicular byto anthe amplitudeprevious <math>A(n)</math>,curve frequencyat <math>f(n)x</math>. andAn phasearbitrary offsetvector <math>\alpha(n)vec{w}</math> is chosen to be the fixed "rag" vector.

<math>g_n(x)R</math> representsis ana unit vectorfunction that isrotates perpendiculara to the previous curve atvector <math>x\vec{A}</math>. Anaround arbitraryan axis defined by a vector <math>\vec{wB}</math> isby chosen<math>\theta</math> todegrees. beIn thethis fixedcase "rag"it vector.is expressed with [[Quaternion|quaternions]].

==See also==

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

*[[Solenoid (DNA)]]

[[Category:Curves]]

== Additional Notes ==

The related animation was recorded from an interactive WebGL demo.

Contact me at [e-mail] if you're interested, or need help recreating something similiar.