A new way to visualise spread polynomials

First five spread polynomials (in the interval [0,1]

First five spread polynomials (in the interval [0,1])

SPREAD polynomials arise as solutions to the triple spread formula (or TSF)

(sa + sb + sc)2 – 2 (sa2 + sb2 + sc2) – 4 sa sb sc = 0.

By fixing two of the spreads, parametrized in the variable s, the formula reduces to a quadratic equation in the third spread. Being quadratic there are two such solutions in the spread variable corresponding, geometrically, to cases where the angle measured by the first spread is oriented positively or negatively with respect to the second. For the simplest case, using sa = sb = s, the TSF becomes:

(2s + Sc(s))2 – 2 (2s2 + Sc(s)2) – 4s2 Sc(s) = 0

   =>  Sc(s) (Sc(s) – 4s(1 – s)) = 0,

giving Sc(s) = 0 or 4s(1 – s).

S(s)  = 0 is indeed the spread when a positive angle corresponding to s combines with a negative angle of equal measure, while 4s(1 – s) (the ‘double angle formula’) is the spread produced when equal, like oriented angles are combined.

Using sa = s,  sb = 4s(1 – s), the TSF then becomes:

(s +  4s(1 – s) + Sc(s))2 – 2 (s2 + 16s2(1 – s)2 + Sc(s)2) – 64s2(1 – s) Sc(s) = 0

  =>  (Sc(s) – s)(Sc(s) – s(3 – 4s)2) = 0,

giving Sc(s) = s or s(3 – 4s)2.

The TSF also gives the ‘solution-set’ of every spread triple where three relations, rather than a single spread relation, are considered at once.

If we take the spread s plotted as  (s, 0, 0)  and its double angled spread relation  4s(1 – s)  plotted as  (s, 4s(1 – s), 0)  and allow s to run through all values in the interval  [0,1]  the familiar Logistic Curve is drawn in the unit square in the first quadrant of the x-y plane.

The pair of spread polynomials (first and second) then combine in the solution of the TSF to form an appropriate triple:

1st, 2nd and 3rd { {s, 4s(1 – s)} , s(3 – 4s)2}  or  1st, 2nd and 1st { {s, 4s(1 – s)} , s}

each entry a polynomial in s which, plotting all three entries at once, describe co-ordinates:

(s, 4s(1 – s), s(3 – 4s)2)  or  (s, 4s(1 – s), s)

lying on separate curves in three dimensions. The ambient space is referred to as a phase-space (sometimes, in 2-d, as phase portrait) when the variables plotted represent the effects of changes occurring in some actual space (rotations of angles in 2-d planar space, for example). In the phase-space either solution forms the locus of a curve whose projection in x-y and x-z planes are respectively the spread polynomials

(s, Sn(s), 0) and (s, 0, Sn-1(s))

and whose projection in the y-z plane is then an implicit curve of the form

(0, Sn(s), Sn-1(s)).

First and second spread triples - in x0y, x0z and y0z projection

First and second spread triple functions – in x0y, x0z and y0z projection

A recursive formula for spread polynomials (obtainable from the TSF) gives, for each new polynomial:

Sn+1(s) = 2(1 – 2s)Sn(s) – Sn-1(s) + 2s

This provides a means of successively generating the spread triples that make up these separate spread triple ‘functions’ S(s,n):

(s, s, S2(s)).. (s, S2(s), S3(s)).. (s, S3(s), S4(s)).. (s, S4(s), S5(s)).. (s, S5(s), S6(s))

These functions can either be varied in the spread variable or the indexing parameter ‘n’. When the latter is carried out the successive values

(s, S2(s)).. (S2(s), S3(s)).. (S3(s), S4(s))…

will all lie on an ellipse in the plane x = s. Developing these ellipses along the x direction traces out the surface resembling an inflated regular tetrahedron and called the Ellipson.

Developing the Ellipson from recurrence relation:

Developing the Ellipson from recurrence relation for spread polynomials

Alternatively, by varying the spread parameter, fixing on particular spread triples and indexing on these, the same surface is developed .

Four Spread Tiples in phase space indicating the form of the Ellipson

First four Spread Tiples in phase-space indicating the form of the Ellipson

Four spread triples with inscribed regular tetrahedron and xoz projection for comparison

Same four Spread Triples with inscribed regular tetrahedron and x0z projection

An Animated Ellipson:

Ellipson 3-D100ms

The Ellipson, f(a,b,c) = (a + b +c)² - 2(a² + b² + c²) - 4abc = 0

Ellipson 2.gif


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