Deriving coordinates for a toroidal helix (helical toroid)

Visualization in 3D of Dynamics of Toroidal Helical Coils (Part #4)

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Inspired by the above, guidance for the single coiled format was (finally) located in a response by Zev Chonoles (Do these equations create a helix wrapped into a torus?, Mathematics Stack Exchange, 8 March 2013.

Although trivial from a mathematical perspective, a concern (as indicated above) was how to get from a formula to a 3D rendering through appropriate software, using a set of coordinates which are aesthetically agreeable -- however this might be understood in this context. Some of the experiments undertaken to converge on the requisite parameters for the formula are presented below.

The key parameters are for a helix of n winds wrapped around a torus of major radius R and minor radius r -- with t based on the number of sets of xyz coordinates. The quest was for an optimal balance between these four parameters -- once the implications of varying one with respect to the other were (finally) understood. The process was constrained by only modest competence in spreadsheet usage, the geometry and the 3D editing software. The formula for the coordinates in 3D are as follows:

Visualization of a toroidal helix (using 201 rows of xyz coordinates, for 9 winds; R=6, r=3)
	Column A (A1=0)	Column B (x coordinate)	Column C (y coordinate)	Column D (z coordinate)
Formula		x=(R+rcos(nt))cos(t)	y=(R+rcos(nt))sin(t)	z= r(sin(nt))
Spreadsheet (namely 2π/201= 0.062832)	A2=A1+0.062832	B1=(6+COS(A2*9))*(COS(A2))	C1=(6+COS(A2*9))*(SIN(A2))	D1=3*SIN(A2*9)
or 2π/100=0.0628318? (for 72)	A2=A1+0.0628318	B1=(6+3*COS(A2*72))*(COS(A2))	C1=(6+3*COS(A2*72))*(SIN(A2))	D1=3*SIN(A2*72)

Experimental "extremes" in the quest for desirable parameters

The decision finally taken was the following, in the light of the focus on 9 and 5 in this exercise:

• given the quest for an aesthetic solution, it appeared appropriate to make use of the golden ratio (symbolized by Φ) in seeking an appropriate relationship between R (the major radius) and r (the minor radius) of the torus. This suggested that R/r should be 1.61803. With R (arbitrarily) specified as 6, this gave r=6/1.61803 = 3.708.
  • given the quest for a sinusoidal nonagon, alternatives examined were r = 3.708/3 = 1.2361, or r = 3.708/9 = 0.412.
  • the last was chosen. Use of Φ (1.61803) on R=6 then r=6/1.61803 = 3.708 or 3.708/3= 1.23607 or 3.708/9=0.412
• the number of windings was defined by the experiment as 9, consistent with the focus on nonagons
• the number of sets of xyz coordinates appropriate to visualization of a coil was initially set arbitrarily at 201 (following the Bederov model). The more such sets, the smoother the visual rendering. Further reflection suggested that the number of segments in a given coil could be related to the 9-fold emphasis. Consideration was given to having 9x9 (namely 3⁴), 3x27 (3⁵), and 2x81 (2x9²). The last was finally chosen, namely 162, since it gave 18 segments in each sinusoidal loop (namely 2x9 in each half loop). The number of sets should be a multiple of the number of windings; a multiple of 4 gives a 4-sided form to each winding, 5 a pentagonal, etc -- the higher the multiple the smoother -- a point of consideration where the number of windings is much higher -- then possibly useful to reduce the number of digits after the decimal in each coordinate

The choices above enabled 162 sets of xyz coordinates to be constructed through spreadsheet formulae as follows, where the number of sets of coordinates is related to 2π (by 2π/162 =0.038785. [This was incorrectly stated as 0.07757 in an earlier version, which gave rise to a helix of linear elements, possibly useful for some purposes]. The formula for each column were then distributed down to A162

Column A (A1=0)	Column B (x coordinate)	Column C (y coordinate)	Column D (z coordinate)
	(R+rcos(nt))cos(t)	(R+cos(nt))sin(t)	r*sin(nt)
A1+0.038785	(R+r*COS(A1*9))*(COS(A1))	(R+r*COS(A1*9))*(SIN(A1))	r*SIN(A1*9)
4	(6+3.708*COS(A1*4))*(COS(A1))	(6+3.708*COS(A1*4))*(SIN(A1))	3.708*SIN(A1*4)
5 0.1005	(6+3.708*COS(A1*5))*(COS(A1))	(6+3.708*COS(A1*5))*(SIN(A1))	3.708*SIN(A1*5)
6 0.2856	(6+3.708*COS(A1*6))*(COS(A1))	(6+3.708*COS(A1*6))*(SIN(A1))	3.708*SIN(A1*6)
A1+0.017453 (72 for 360)	(6+3.708*COS(A1*72))*(COS(A1))	(6+3.708*COS(A1*72))*(SIN(A1))	3.708*SIN(A1*72)
A1+0.010908 (72 for 576)	(6+0.75*COS(A1*72))*(COS(A1))	(6+0.75*COS(A1*72))*(SIN(A1))	0.75*SIN(A1*72)

The further decision required was the diameter of the sinusoidal winding around an invisible torus (namely one which would not be visually rendered). This was achieved within the extrusion option of the X3D-Edit application. The choices there were:

• cross-section of the sinusoidal winding: given its relatively small diameter this could be defined by a polygon of a limited number of sides. 5 was chosen
• scale of cross section: this was scaled down to 30%
• following the previous experiment, 5 such sinusoidal coils were extruded and coloured distinctively

Convergence on desired configuration (with interactive variants)
3 toroidal coils (X3D or VRML)	Single toroidal coil (X3D or VRML)	5 toroidal coils (X3D or VRML)

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