[Tex/LaTex] Modifying Spirograph code to get more control

tikz-pgf

To get more control on Spirograph patterns, the Spirograph code from the answer to this question, needs to be modified.

This code is

\documentclass{beamer}
\beamertemplatenavigationsymbolsempty
\usepackage{tikz}
\usetikzlibrary{shadings}
\begin{document}
\tikzset{pics/spiro/.style={code={
\tikzset{spiro/.cd,#1}
\def\pv##1{\pgfkeysvalueof{/tikz/spiro/##1}} 
\draw[trig format=rad,pic actions] plot[variable=\t,domain=0:2*pi*\pv{nRotations}, samples=90*\pv{nRotations}+1,smooth cycle] 
(
{(\pv{R}+\pv{r})*cos(\t)+\pv{p}*cos((\pv{R}+\pv{r})*\t/\pv{r})},
{(\pv{R}+\pv{r})*sin(\t)+\pv{p}*sin((\pv{R}+\pv{r})*\t/\pv{r})}
);
}},
spiro/.cd,R/.initial=6,r/.initial=-1.5,p/.initial=1,nRotations/.initial=1}
\begin{frame}[fragile,t]
\frametitle{Spiro 1 orig}
\begin{tikzpicture}[]
 \draw
 (0,0) pic[scale=0.5, violet, line width=0.6mm, rotate=45, lower left=orange, lower right=yellow, upper left=red, upper right=magenta]{spiro}
;
\end{tikzpicture}
\end{frame}
\end{document}

Which produces

Here is what is in my mind. I need to draw the pattern in segments, and define the location of the starting point.

The distance point P has to travel, to complete a cycle (360 degrees), is expressed as 2*pi. So, to draw 0.25 of the curve, 0.5 of the curve, or the whole curve; you need to say (2*pi*0.25), (2*pi*0.5), or (2*pi*1), respectively.

It might take more than one cycle to complete the whole pattern. To draw the figure to the left, it takes 2 nRotations. The first draws the red one, the second draws the blue one, to get the figure to the right.

In the original code

t = vectorin(0, 0.05, 2*pi*nRotations)

where (0, 0.05, 2*pi) tells the programme to draw the curve in small lines, in increments of 5 percent (0.05) starting from point 0 to point 2*pi. The 0.05 increments determine the smoothness of the curve, especially for more complicated patterns.

For example using

t = vectorin(0, 0.05, 2 * pi * v_nRotations)

produces

while using

t = vectorin(0, 0.005, 2 * pi * v_nRotations)
produces a smoother curve

To draw a pattern in segments, we must control the rotation of point P

This is needed to make the one-colour left figure looks like the multiple-colour right one.

The original code

spirograph = function (R, r, p, nRotations, color)
t = vectorin(0, 0.05, 2*pi*nRotations)
spirograph(60, -15, 10,  1, green)

is modified to draw the pattern in 4 separate segments, each with a different colour. Point P also starts not from point 2*pi*0, but from point 2*pi*(0.000-.125) (to make each peak of one colour) and going in steps of 2*pi*(0.125)*2 (by multiplying in a factor) to reach point 2*pi*(0.125)*7 (to complete the 360 degrees cycle).

The idea is to begin the first segment not from position 0, but half way before it; and to end it half way after it.

Starting from location 0, produces a less attractive pattern.

The new code becomes

spirograph = function (R, r, p, start, step, stop, object, lineSize, color, scale, n, fill)

t=vectorin(start, step, stop) (which provides more control on the beginning, increment, and end of the rotation cycle than when using nRotations)

addplot(object, x*scale, y* scale, n, lineSize, color, 0, 0, fill) (x*scale, y* scale is used in other drawings to control the x and y scaling separately)

The following code: (which P starts from point 2*pi*0.0 to 2*pi*1.0) draws the pattern in Green

spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*0.00 , 0.05, 2*pi*1.00, p, 6, Green, 3.5, 0*360/4/6, "Transparent")
showplot(p, 0, 1, 1)

using the following code: which P starts from point 2*pi*(0.000-.125) to 2*pi*(0.000+.125) and then rotating the resulting peak

spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.000-.125) , 0.05, 2*pi*(0.000+.125), p, 6, Green,  3.5, 0*360/4, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.000-.125) , 0.05, 2*pi*(0.000+.125), p, 6, Blue ,  3.5, 1*360/4, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.000-.125) , 0.05, 2*pi*(0.000+.125), p, 6, Red  ,  3.5, 2*360/4, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.000-.125) , 0.05, 2*pi*(0.000+.125), p, 6, Violet, 3.5, 3*360/4, "Transparent")
showplot(p, 0, 1, 1)

or using the following code (begin from point 2*pi*(0.125)*-1 and rotated by multiplying in a factor)

spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.125)*-1 , 0.05, 2*pi*(0.125)*1, p, 6, Green , 3.5, 0*360/4, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.125)*1  , 0.05, 2*pi*(0.125)*3, p, 6, Blue  , 3.5, 0*360/4, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.125)*3  , 0.05, 2*pi*(0.125)*5, p, 6, Red   , 3.5, 0*360/4, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.125)*5  , 0.05, 2*pi*(0.125)*7, p, 6, Violet, 3.5, 0*360/4, "Transparent")
showplot(p, 0, 1, 1)

produce the left drawing in

This routine can be repeated and rotated to draw the right drawing, with filling

spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.000-.125) , 0.05, 2*pi*(0.000+.125),     p, 6, Green, 3.5, 0*360/4/6, "#5500ff00")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250*1-.125) , 0.05, 2*pi*(0.250*1+.125), p, 6, Blue,  3.5, 0*360/4/6, "#550000ff")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250*2-.125) , 0.05, 2*pi*(0.250*2+.125), p, 6, Red,   3.5, 0*360/4/6, "#55ff0000")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250*3-.125) , 0.05, 2*pi*(0.250*3+.125), p, 6, Violet,3.5, 0*360/4/6, "#55ff00ff")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.000-.125) , 0.05, 2*pi*(0.000+.125),     p, 6, Green, 3.5, 1*360/4/6, "#5500ff00")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250*1-.125) , 0.05, 2*pi*(0.250*1+.125), p, 6, Blue,  3.5, 1*360/4/6, "#550000ff")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250*2-.125) , 0.05, 2*pi*(0.250*2+.125), p, 6, Red,   3.5, 1*360/4/6, "#55ff0000")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250*3-.125) , 0.05, 2*pi*(0.250*3+.125), p, 6, Violet,3.5, 1*360/4/6, "#55ff00ff")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.000-.125) , 0.05, 2*pi*(0.000+.125),     p, 6, Green, 3.5, 2*360/4/6, "#5500ff00")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250*1-.125) , 0.05, 2*pi*(0.250*1+.125), p, 6, Blue,  3.5, 2*360/4/6, "#550000ff")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250*2-.125) , 0.05, 2*pi*(0.250*2+.125), p, 6, Red,   3.5, 2*360/4/6, "#55ff0000")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250*3-.125) , 0.05, 2*pi*(0.250*3+.125), p, 6, Violet,3.5, 2*360/4/6, "#55ff00ff")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.000-.125) , 0.05, 2*pi*(0.000+.125),     p, 6, Green, 3.5, 3*360/4/6, "#5500ff00")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250*1-.125) , 0.05, 2*pi*(0.250*1+.125), p, 6, Blue,  3.5, 3*360/4/6, "#550000ff")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250*2-.125) , 0.05, 2*pi*(0.250*2+.125), p, 6, Red,   3.5, 3*360/4/6, "#55ff0000")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250*3-.125) , 0.05, 2*pi*(0.250*3+.125), p, 6, Violet,3.5, 3*360/4/6, "#55ff00ff")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.000-.125) , 0.05, 2*pi*(0.000+.125),     p, 6, Green, 3.5, 4*360/4/6, "#5500ff00")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250*1-.125) , 0.05, 2*pi*(0.250*1+.125), p, 6, Blue,  3.5, 4*360/4/6, "#550000ff")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250*2-.125) , 0.05, 2*pi*(0.250*2+.125), p, 6, Red,   3.5, 4*360/4/6, "#55ff0000")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250*3-.125) , 0.05, 2*pi*(0.250*3+.125), p, 6, Violet,3.5, 4*360/4/6, "#55ff00ff")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.000-.125) , 0.05, 2*pi*(0.000+.125),     p, 6, Green, 3.5, 5*360/4/6, "#5500ff00")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250*1-.125) , 0.05, 2*pi*(0.250*1+.125), p, 6, Blue,  3.5, 5*360/4/6, "#550000ff")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250*2-.125) , 0.05, 2*pi*(0.250*2+.125), p, 6, Red,   3.5, 5*360/4/6, "#55ff0000")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250*3-.125) , 0.05, 2*pi*(0.250*3+.125), p, 6, Violet,3.5, 5*360/4/6, "#55ff00ff")
showplot(p, 0, 1, 1)

Instead of drawing a whole pattern in segments, only a part of the pattern can be drawn, repeated, rotated, filled and scaled to produce more appealing drawings. Some experiments are needed to find out where the used parts meet each other,
to avoid their overlapping.

spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.000-.055), 0.05, 2*pi*(0.000+.055), p, 2, "#008800",   3.0, 0*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.250-.055), 0.05, 2*pi*(0.250+.055), p, 2, "#008800",   3.0, 0*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.500-.055), 0.05, 2*pi*(0.500+.055), p, 2, "#008800",   3.0, 0*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.750-.055), 0.05, 2*pi*(0.750+.055), p, 2, "#008800",   3.0, 0*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.000-.055), 0.05, 2*pi*(0.000+.055), p, 2, "#0000AA", 3.0, 1*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.250-.055), 0.05, 2*pi*(0.250+.055), p, 2, "#0000AA", 3.0, 1*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.500-.055), 0.05, 2*pi*(0.500+.055), p, 2, "#0000AA", 3.0, 1*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.750-.055), 0.05, 2*pi*(0.750+.055), p, 2, "#0000AA", 3.0, 1*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.000-.055), 0.05, 2*pi*(0.000+.055), p, 2, "#9933CC",   3.0, 2*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.250-.055), 0.05, 2*pi*(0.250+.055), p, 2, "#9933CC",   3.0, 2*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.500-.055), 0.05, 2*pi*(0.500+.055), p, 2, "#9933CC",   3.0, 2*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.750-.055), 0.05, 2*pi*(0.750+.055), p, 2, "#9933CC",   3.0, 2*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.000-.055), 0.05, 2*pi*(0.000+.055), p, 2, "#AA0000",  3.0, 3*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.250-.055), 0.05, 2*pi*(0.250+.055), p, 2, "#AA0000",  3.0, 3*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.500-.055), 0.05, 2*pi*(0.500+.055), p, 2, "#AA0000",  3.0, 3*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.750-.055), 0.05, 2*pi*(0.750+.055), p, 2, "#AA0000",  3.0, 3*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.000-.055), 0.05, 2*pi*(0.000+.055), p, 2, "#D2691E",   3.0, 4*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.250-.055), 0.05, 2*pi*(0.250+.055), p, 2, "#D2691E",   3.0, 4*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.500-.055), 0.05, 2*pi*(0.500+.055), p, 2, "#D2691E",   3.0, 4*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.750-.055), 0.05, 2*pi*(0.750+.055), p, 2, "#D2691E",   3.0, 4*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.000-.055), 0.05, 2*pi*(0.000+.055), p, 2, "#E9967A",   3.0, 5*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.250-.055), 0.05, 2*pi*(0.250+.055), p, 2, "#E9967A",   3.0, 5*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.500-.055), 0.05, 2*pi*(0.500+.055), p, 2, "#E9967A",   3.0, 5*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.750-.055), 0.05, 2*pi*(0.750+.055), p, 2, "#E9967A",   3.0, 5*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.000-.055), 0.05, 2*pi*(0.000+.055), p, 2, "#DAA520",   3.0, 6*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.250-.055), 0.05, 2*pi*(0.250+.055), p, 2, "#DAA520",   3.0, 6*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.500-.055), 0.05, 2*pi*(0.500+.055), p, 2, "#DAA520",   3.0, 6*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.750-.055), 0.05, 2*pi*(0.750+.055), p, 2, "#DAA520",   3.0, 6*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.000-.055), 0.05, 2*pi*(0.000+.055), p, 2, "#FFDD00", 3.0, 7*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.250-.055), 0.05, 2*pi*(0.250+.055), p, 2, "#FFDD00", 3.0, 7*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.500-.055), 0.05, 2*pi*(0.500+.055), p, 2, "#FFDD00", 3.0, 7*(360/4/8)+0, "Transparent")
spirograph(60*1.5 , -15*1.5 , 12*1.5 , 2*pi*(0.750-.055), 0.05, 2*pi*(0.750+.055), p, 2, "#FFDD00", 3.0, 7*(360/4/8)+0, "Transparent")
"showplot(p, 0, 1, 1)"
 --------------------------------------------------------------------------------------------------------- 
"p = createplot()"
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.000-.055), 0.05, 2*pi*(0.000+.055), p, 2, "#008800",   3.0, 0*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250-.055), 0.05, 2*pi*(0.250+.055), p, 2, "#008800",   3.0, 0*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.500-.055), 0.05, 2*pi*(0.500+.055), p, 2, "#008800",   3.0, 0*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.750-.055), 0.05, 2*pi*(0.750+.055), p, 2, "#008800",   3.0, 0*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.000-.055), 0.05, 2*pi*(0.000+.055), p, 2, "#0000AA",   3.0, 1*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250-.055), 0.05, 2*pi*(0.250+.055), p, 2, "#0000AA",   3.0, 1*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.500-.055), 0.05, 2*pi*(0.500+.055), p, 2, "#0000AA",   3.0, 1*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.750-.055), 0.05, 2*pi*(0.750+.055), p, 2, "#0000AA",   3.0, 1*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.000-.055), 0.05, 2*pi*(0.000+.055), p, 2, "#9933CC",   3.0, 2*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250-.055), 0.05, 2*pi*(0.250+.055), p, 2, "#9933CC",   3.0, 2*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.500-.055), 0.05, 2*pi*(0.500+.055), p, 2, "#9933CC",   3.0, 2*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.750-.055), 0.05, 2*pi*(0.750+.055), p, 2, "#9933CC",   3.0, 2*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.000-.055), 0.05, 2*pi*(0.000+.055), p, 2, "#AA0000",   3.0, 3*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250-.055), 0.05, 2*pi*(0.250+.055), p, 2, "#AA0000",   3.0, 3*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.500-.055), 0.05, 2*pi*(0.500+.055), p, 2, "#AA0000",   3.0, 3*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.750-.055), 0.05, 2*pi*(0.750+.055), p, 2, "#AA0000",   3.0, 3*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.000-.055), 0.05, 2*pi*(0.000+.055), p, 2, "#D2691E",   3.0, 4*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250-.055), 0.05, 2*pi*(0.250+.055), p, 2, "#D2691E",   3.0, 4*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.500-.055), 0.05, 2*pi*(0.500+.055), p, 2, "#D2691E",   3.0, 4*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.750-.055), 0.05, 2*pi*(0.750+.055), p, 2, "#D2691E",   3.0, 4*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.000-.055), 0.05, 2*pi*(0.000+.055), p, 2, "#E9967A",   3.0, 5*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250-.055), 0.05, 2*pi*(0.250+.055), p, 2, "#E9967A",   3.0, 5*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.500-.055), 0.05, 2*pi*(0.500+.055), p, 2, "#E9967A",   3.0, 5*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.750-.055), 0.05, 2*pi*(0.750+.055), p, 2, "#E9967A",   3.0, 5*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.000-.055), 0.05, 2*pi*(0.000+.055), p, 2, "#DAA520",   3.0, 6*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250-.055), 0.05, 2*pi*(0.250+.055), p, 2, "#DAA520",   3.0, 6*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.500-.055), 0.05, 2*pi*(0.500+.055), p, 2, "#DAA520",   3.0, 6*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.750-.055), 0.05, 2*pi*(0.750+.055), p, 2, "#DAA520",   3.0, 6*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.000-.055), 0.05, 2*pi*(0.000+.055), p, 2, "#FFDD00",   3.0, 7*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.250-.055), 0.05, 2*pi*(0.250+.055), p, 2, "#FFDD00",   3.0, 7*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.500-.055), 0.05, 2*pi*(0.500+.055), p, 2, "#FFDD00",   3.0, 7*(360/4/8)+0, "Transparent")
spirograph(60*1.25 , -15*1.25 , 12*1.25 , 2*pi*(0.750-.055), 0.05, 2*pi*(0.750+.055), p, 2, "#FFDD00",   3.0, 7*(360/4/8)+0, "Transparent")
"showplot(p, 0, 1, 1)"
---------------------------------------------------------------------------------------------------------
and so on

Another method is to draw the needed parts in colour, and the unneeded parts in white or transparent.

spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.250-.045) , 0.05, 2*pi*(.250+.045), p, 2, Green,  3.0, 0*360/4/8,"#FF7CFC00")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.250+.045) , 0.05, 2*pi*(.500-.045), p, 2, Transparent,  3.0, 0*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.500-.045) , 0.05, 2*pi*(.500+.045), p, 2, Blue,   3.0, 0*360/4/8,"#AA0000FF")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.500+.045) , 0.05, 2*pi*(.750-.045), p, 2, Transparent,  3.0, 0*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.750-.045) , 0.05, 2*pi*(.750+.045), p, 2, Red,    3.0, 0*360/4/8,"#AAFF0000")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.750+.045) , 0.05, 2*pi*(1.000-.045), p, 2, Transparent, 3.0, 0*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(1.000-.045) , 0.05, 2*pi*(1.000+.045), p, 2, Violet, 3.0, 0*360/4/8,"#AAFF8C00")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(1.000+.045) , 0.05, 2*pi*(1.250-.045), p, 2, Transparent, 3.0, 0*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.250-.045) , 0.05, 2*pi*(.250+.045), p, 2, Green,  3.0, 1*360/4/8,"#FF7CFC00")
pirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.250+.045) , 0.05, 2*pi*(.500-.045), p, 2, Transparent,  3.0, 1*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.500-.045) , 0.05, 2*pi*(.500+.045), p, 2, Blue,   3.0, 1*360/4/8,"#AA0000FF")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.500+.045) , 0.05, 2*pi*(.750-.045), p, 2, Transparent,  3.0, 1*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.750-.045) , 0.05, 2*pi*(.750+.045), p, 2, Red,    3.0, 1*360/4/8,"#AAFF0000")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.750+.045) , 0.05, 2*pi*(1.000-.045), p, 2, Transparent, 3.0, 1*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(1.000-.045) , 0.05, 2*pi*(1.000+.045), p, 2, Violet, 3.0, 1*360/4/8,"#AAFF8C00")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(1.000+.045) , 0.05, 2*pi*(1.250-.045), p, 2, Transparent, 3.0, 1*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.250-.045) , 0.05, 2*pi*(.250+.045), p, 2, Green,  3.0, 2*360/4/8,"#FF7CFC00")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.250+.045) , 0.05, 2*pi*(.500-.045), p, 2, Transparent,  3.0, 2*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.500-.045) , 0.05, 2*pi*(.500+.045), p, 2, Blue,   3.0, 2*360/4/8,"#AA0000FF")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.500+.045) , 0.05, 2*pi*(.750-.045), p, 2, Transparent,  3.0, 2*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.750-.045) , 0.05, 2*pi*(.750+.045), p, 2, Red,    3.0, 2*360/4/8,"#AAFF0000")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.750+.045) , 0.05, 2*pi*(1.000-.045), p, 2, Transparent, 3.0, 2*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(1.000-.045) , 0.05, 2*pi*(1.000+.045), p, 2, Violet, 3.0, 2*360/4/8,"#AAFF8C00")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(1.000+.045) , 0.05, 2*pi*(1.250-.045), p, 2, Transparent, 3.0, 2*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.250-.045) , 0.05, 2*pi*(.250+.045), p, 2, Green,  3.0, 3*360/4/8,"#FF7CFC00")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.250+.045) , 0.05, 2*pi*(.500-.045), p, 2, Transparent,  3.0, 3*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.500-.045) , 0.05, 2*pi*(.500+.045), p, 2, Blue,   3.0, 3*360/4/8,"#AA0000FF")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.500+.045) , 0.05, 2*pi*(.750-.045), p, 2, Transparent,  3.0, 3*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.750-.045) , 0.05, 2*pi*(.750+.045), p, 2, Red,    3.0, 3*360/4/8,"#AAFF0000")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.750+.045) , 0.05, 2*pi*(1.000-.045), p, 2, Transparent, 3.0, 3*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(1.000-.045) , 0.05, 2*pi*(1.000+.045), p, 2, Violet, 3.0, 3*360/4/8,"#AAFF8C00")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(1.000+.045) , 0.05, 2*pi*(1.250-.045), p, 2, Transparent, 3.0, 3*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.250-.045) , 0.05, 2*pi*(.250+.045), p, 2, Green,  3.0, 4*360/4/8,"#FF7CFC00")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.250+.045) , 0.05, 2*pi*(.500-.045), p, 2, Transparent,  3.0, 4*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.500-.045) , 0.05, 2*pi*(.500+.045), p, 2, Blue,   3.0, 4*360/4/8,"#AA0000FF")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.500+.045) , 0.05, 2*pi*(.750-.045), p, 2, Transparent,  3.0, 4*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.750-.045) , 0.05, 2*pi*(.750+.045), p, 2, Red,    3.0, 4*360/4/8,"#AAFF0000")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.750+.045) , 0.05, 2*pi*(1.000-.045), p, 2, Transparent, 3.0, 4*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(1.000-.045) , 0.05, 2*pi*(1.000+.045), p, 2, Violet, 3.0, 4*360/4/8,"#AAFF8C00")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(1.000+.045) , 0.05, 2*pi*(1.250-.045), p, 2, Transparent, 3.0, 4*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.250-.045) , 0.05, 2*pi*(.250+.045), p, 2, Green,  3.0, 5*360/4/8,"#FF7CFC00")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.250+.045) , 0.05, 2*pi*(.500-.045), p, 2, Transparent,  3.0, 5*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.500-.045) , 0.05, 2*pi*(.500+.045), p, 2, Blue,   3.0, 5*360/4/8,"#AA0000FF")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.500+.045) , 0.05, 2*pi*(.750-.045), p, 2, Transparent,  3.0, 5*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.750-.045) , 0.05, 2*pi*(.750+.045), p, 2, Red,    3.0, 5*360/4/8,"#AAFF0000")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.750+.045) , 0.05, 2*pi*(1.000-.045), p, 2, Transparent, 3.0, 5*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(1.000-.045) , 0.05, 2*pi*(1.000+.045), p, 2, Violet, 3.0, 5*360/4/8,"#AAFF8C00")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(1.000+.045) , 0.05, 2*pi*(1.250-.045), p, 2, Transparent, 3.0, 5*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.250-.045) , 0.05, 2*pi*(.250+.045), p, 2, Green,  3.0, 6*360/4/8,"#FF7CFC00")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.250+.045) , 0.05, 2*pi*(.500-.045), p, 2, Transparent,  3.0, 6*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.500-.045) , 0.05, 2*pi*(.500+.045), p, 2, Blue,   3.0, 6*360/4/8,"#AA0000FF")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.500+.045) , 0.05, 2*pi*(.750-.045), p, 2, Transparent,  3.0, 6*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.750-.045) , 0.05, 2*pi*(.750+.045), p, 2, Red,    3.0, 6*360/4/8,"#AAFF0000")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.750+.045) , 0.05, 2*pi*(1.000-.045), p, 2, Transparent, 3.0, 6*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(1.000-.045) , 0.05, 2*pi*(1.000+.045), p, 2, Violet, 3.0, 6*360/4/8,"#AAFF8C00")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(1.000+.045) , 0.05, 2*pi*(1.250-.045), p, 2, Transparent, 3.0, 6*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.250-.045) , 0.05, 2*pi*(.250+.045), p, 2, Green,  3.0, 7*360/4/8,"#FF7CFC00")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.250+.045) , 0.05, 2*pi*(.500-.045), p, 2, Transparent,  3.0, 7*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.500-.045) , 0.05, 2*pi*(.500+.045), p, 2, Blue,   3.0, 7*360/4/8,"#AA0000FF")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.500+.045) , 0.05, 2*pi*(.750-.045), p, 2, Transparent,  3.0, 7*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.750-.045) , 0.05, 2*pi*(.750+.045), p, 2, Red,    3.0, 7*360/4/8,"#AAFF0000")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(.750+.045) , 0.05, 2*pi*(1.000-.045), p, 2, Transparent, 3.0, 7*360/4/8,"Transparent")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(1.000-.045) , 0.05, 2*pi*(1.000+.045), p, 2, Violet, 3.0, 7*360/4/8,"#AAFF8C00")
spirograph(60*1.5 , -15*1.5 , 10*1.5 , 2*pi*(1.000+.045) , 0.05, 2*pi*(1.250-.045), p, 2, Transparent, 3.0, 7*360/4/8,"Transparent")
showplot(p, 0, 1, 1)

Best Answer

It is very easy to add some features like a domain and a dx to the older version of the spiro pic. I just focus on two of your screen shots for illustration, but think you can do all of them with the adjusted syntax. Please let me know if I am missing something.

\documentclass[tikz,border=3mm]{standalone}
\tikzset{pics/spiro2/.style={code={
\tikzset{spiro2/.cd,#1}
\def\pv##1{\pgfkeysvalueof{/tikz/spiro2/##1}} 
\pgfmathparse{(int(1/\pv{dx}+1)}
\tikzset{spiro2/samples=\pgfmathresult}
\draw[trig format=rad,pic actions] 
 plot[variable=\t,domain=\pv{xmin}-0.002:\pv{xmax}+0.002,
    samples=\pv{samples}] 
    ({(\pv{R}+\pv{r})*cos(\t)+\pv{p}*cos((\pv{R}+\pv{r})*\t/\pv{r})},
     {(\pv{R}+\pv{r})*sin(\t)+\pv{p}*sin((\pv{R}+\pv{r})*\t/\pv{r})});
    }},
spiro2/.cd,R/.initial=6,r/.initial=-1.5,p/.initial=1,
dx/.initial=0.005,samples/.initial=21,domain/.code args={#1:#2}{%
\pgfmathparse{#1}\tikzset{spiro2/xmin/.expanded=\pgfmathresult}
\pgfmathparse{#2}\tikzset{spiro2/xmax/.expanded=\pgfmathresult}},
xmin/.initial=0,xmax/.initial=2*pi}
\begin{document}
\begin{tikzpicture}[]
 \draw 
   (0,0) foreach \X [count=\Y starting from 0] in {blue,red,purple,orange}
   {pic[scale=0.5,draw=\X,ultra
   thick]{spiro2={domain={-pi/4+(\Y-1)*pi/2}:{-pi/4+\Y*pi/2}}}};
 \draw(0,-7) foreach \X [count=\Y starting from 0] in {blue,red,purple,orange}
   {foreach \Z in {0,...,5}
   {pic[scale=0.5,draw=\X,ultra thick,fill=\X,fill
   opacity=0.2,rotate=\Z*15]{spiro2={domain={-pi/4+(\Y-1)*pi/2}:{-pi/4+\Y*pi/2}}}}};  
\end{tikzpicture}
\end{document}

Please note that the older version used smooth cycle so one could get good results with a comparatively small number of samples. Here, on the other hand, I followed your instructions to connect the plot points by straight lines, so one needs more samples and longer time to compile. Please let me know if one should go back to the smooth case to increase the speed. Doing that yields

\documentclass[tikz,border=3mm]{standalone}
\tikzset{pics/spiro2/.style={code={
\tikzset{spiro2/.cd,#1}
\def\pv##1{\pgfkeysvalueof{/tikz/spiro2/##1}} 
\pgfmathparse{(int(1/\pv{dx}+1)}
\tikzset{spiro2/samples=\pgfmathresult}
\draw[trig format=rad,pic actions] 
 plot[variable=\t,domain=\pv{xmin}-0.002:\pv{xmax}+0.002,
    samples=\pv{samples},smooth] 
    ({(\pv{R}+\pv{r})*cos(\t)+\pv{p}*cos((\pv{R}+\pv{r})*\t/\pv{r})},
     {(\pv{R}+\pv{r})*sin(\t)+\pv{p}*sin((\pv{R}+\pv{r})*\t/\pv{r})});
    }},
spiro2/.cd,R/.initial=6,r/.initial=-1.5,p/.initial=1,
dx/.initial=0.05,samples/.initial=21,domain/.code args={#1:#2}{%
\pgfmathparse{#1}\tikzset{spiro2/xmin/.expanded=\pgfmathresult}
\pgfmathparse{#2}\tikzset{spiro2/xmax/.expanded=\pgfmathresult}},
xmin/.initial=0,xmax/.initial=2*pi}
\begin{document}
\begin{tikzpicture}
 \draw 
   (0,0) foreach \X [count=\Y starting from 0] in {blue,red,purple,orange}
   {pic[scale=0.5,draw=\X,ultra
   thick]{spiro2={domain={-pi/4+(\Y-1)*pi/2}:{-pi/4+\Y*pi/2}}}};
 \draw(0,-7) foreach \X [count=\Y starting from 0] in {blue,red,purple,orange}
   {foreach \Z in {0,...,5}
   {pic[scale=0.5,draw=\X,ultra thick,fill=\X,fill
   opacity=0.2,rotate=\Z*15]{spiro2={domain={-pi/4+(\Y-1)*pi/2}:{-pi/4+\Y*pi/2}}}}};  
\end{tikzpicture}
\end{document}

It appears natural to me to combine this with path fading.

\documentclass[border=3mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{shadings,fadings}
\tikzset{pics/spiro2/.style={code={
\tikzset{spiro2/.cd,#1}
\def\pv##1{\pgfkeysvalueof{/tikz/spiro2/##1}} 
\pgfmathparse{(int(1/\pv{dx}+1)}
\tikzset{spiro2/samples=\pgfmathresult}
\draw[trig format=rad,pic actions] 
 plot[variable=\t,domain=\pv{xmin}-0.002:\pv{xmax}+0.002,
    samples=\pv{samples},smooth] 
    ({(\pv{R}+\pv{r})*cos(\t)+\pv{p}*cos((\pv{R}+\pv{r})*\t/\pv{r})},
     {(\pv{R}+\pv{r})*sin(\t)+\pv{p}*sin((\pv{R}+\pv{r})*\t/\pv{r})});
    }},
spiro2/.cd,R/.initial=6,r/.initial=-1.5,p/.initial=1,
dx/.initial=0.05,samples/.initial=21,domain/.code args={#1:#2}{%
\pgfmathparse{#1}\tikzset{spiro2/xmin/.expanded=\pgfmathresult}
\pgfmathparse{#2}\tikzset{spiro2/xmax/.expanded=\pgfmathresult}},
xmin/.initial=0,xmax/.initial=2*pi}
\begin{document}
\begin{tikzfadingfrompicture}[name=spiro]
\draw foreach \Y in {0,...,3}
   {foreach \Z in {0,...,5}
   {pic[scale=0.5,fill=transparent!60,
   draw=transparent!20,
   rotate=\Z*15]{spiro2={domain={-pi/4+(\Y-1)*pi/2}:{-pi/2+pi/9+\Y*pi/2}}}}};  
\end{tikzfadingfrompicture}
\begin{tikzpicture}
 \shade[shading=color wheel,
  path fading=spiro,fit fading=false] (0,0) circle [radius=4cm];
\end{tikzpicture}
\end{document}

ADDENDUM: These are the requested graphs. I added some explanations to the code. Explaining something well requires the knowledge where the other user involved is struggling. I do not have this knowledge. If you ask specific questions, I will try to answer them.

\documentclass[tikz,border=3mm]{standalone}
% The following just sets up a plot where you can contol the parameters via pgf
% keys. The central object is the plot.
\tikzset{pics/spiro2/.style={code={
\tikzset{spiro2/.cd,#1}
\def\pv##1{\pgfkeysvalueof{/tikz/spiro2/##1}} 
\pgfmathparse{(int(1/\pv{dx}+1)}
\tikzset{spiro2/samples=\pgfmathresult}
\draw[trig format=rad,pic actions] 
 plot[variable=\t,domain=\pv{xmin}-0.002:\pv{xmax}+0.002,
    samples=\pv{samples},smooth] 
    ({(\pv{R}+\pv{r})*cos(\t)+\pv{p}*cos((\pv{R}+\pv{r})*\t/\pv{r})},
     {(\pv{R}+\pv{r})*sin(\t)+\pv{p}*sin((\pv{R}+\pv{r})*\t/\pv{r})});
    }},
spiro2/.cd,R/.initial=6,r/.initial=-1.5,p/.initial=1,
dx/.initial=0.08,samples/.initial=21,domain/.code args={#1:#2}{%
\pgfmathparse{#1}\tikzset{spiro2/xmin/.expanded=\pgfmathresult}
\pgfmathparse{#2}\tikzset{spiro2/xmax/.expanded=\pgfmathresult}},
xmin/.initial=0,xmax/.initial=2*pi}
\begin{document}
\begin{tikzpicture}
 \path (0,0)  
  pic[scale=0.5,draw=yellow,ultra   thick]{spiro2={dx=0.03}}
   foreach \X [count=\Y starting from 0] in {blue,red,purple,orange}
   {pic[scale=0.5,draw=\X,ultra
   thick]{spiro2={domain={-pi/12+\Y*pi/2}:{pi/12+\Y*pi/2}}}};
 % This is a loop orgy. We loop over scale factors, overall rotations and colors.   
 \path[line cap=round] (7,0)  
   foreach \ScaleN 
    [evaluate=\ScaleN as \Scale using {pow(0.85,\ScaleN)/0.8}] % compute sale factor
    in {1,...,5} %loop over scale
     {foreach \Z in {0,...,3} %loop over 4 overall rotations
       {foreach \X [count=\Y starting from 0] in 
         {yellow,orange,red,blue,purple,cyan,magenta,green!70!black} % colors
       {pic[scale=0.5,draw=\X,rotate=\Y*90/8+\Z*90,
       scale=\Scale,line width=\Scale*2pt]
       {spiro2={domain={-pi/11.4}:{pi/11.4}}}}}};
\end{tikzpicture}
\end{document}

updated

Simply to remove some completely unneeded hack of \pgfmathfloatvalueof which I had been using for the xfp and xint answers. I had been working starting with some file from initial answer now at https://tex.stackexchange.com/a/425332/4686 and only today do I read again that answer and see that later on I had removed the hack. I did not realize my starting point was only initial answer not the final version.

So now I use directly \pgfmathfloatvalueof which is not f-expandable (or was not in April 2018) but it does not need to be, pure expandability suffices for xfp or xintexpr expressions. For shortening and aesthetics I do \def\FVof(#1){\pgfmathfloatvalueof{#1}} simply to have only parentheses within the declare function expression, no braces.

You can

[x] Think a bit and use a better mathematical formula for the given range

[x] or Use xfp which achieves 16 decimal digits of precision. For this I used the technique I had developed at https://tex.stackexchange.com/a/425332/4686, but here (as I am a good boy) to plug-in xfp's \fpeval rather than \xintthefloatexpr.

[x] or Bug the xint author to implement mathematical functions, or use the available interface to program them yourselves via series in suitable ranges.

I illustrate first alternative here: (see second and third, which were added later, below)

\documentclass[a4paper]{article}

\usepackage{tikz}
\usepackage[margin=0.3in]{geometry}
\usepackage{pgfplots}
\pgfplotsset{width=12cm,compat=1.16}

\usetikzlibrary{math}

\newcommand\alphazero{23.44} % Earth axial tilt
\newcommand\latitude{52} % latitude in degrees

\begin{document}

\begin{tikzpicture}[
    % declare function={
    %     sunrise(\d,\x) = {acos( sqrt( cos(\d)^2 - (sin(\alphazero)*cos(\x))^2 ) / cos(\d) )};
    % }
    declare function={
        sunrise(\d,\x) = {asin( sin(\alphazero)*cos(\x) / cos(\d) )};
    }
]

\begin{axis}[
    axis lines=left,
    align=center,
    grid=both,
    minor y tick num=4,
    title={\Large Sunrise},
    xlabel={Day of year angle $(x^{\circ})$},
    ylabel={Sunrise position $(\theta^{\circ})$},
]


% % Plot (1)
% % Using table calculated from Perl
% \addplot [
%     green
% ] table {perltable.dat};
% \addlegendentry{(1) Table calculated from Perl};


% Plot (2) 
% Using pgfplots `\addplot expression'
\addplot expression [
    blue,
    domain=82:90,
    only marks,
    mark size=1pt,
    samples=9,
    variable=x,
]
{sunrise(\latitude, x)};
\addlegendentry{(2) Using pgfplots `\textbackslash addplot expression'};


% Plot (3)
% Plot the Tikz Math library results
\tikzmath {
    \a = sunrise(\latitude, 86);
    \b = sunrise(\latitude, 87);
    \c = sunrise(\latitude, 88);
    \d = sunrise(\latitude, 89);
    \e = sunrise(\latitude, 90);
}

\path (axis cs:82.7,0.3) node[draw,fill=white,inner sep=3pt,anchor=south west,align=left] {
    \em Tikz Math library \\
    \em results :- \\
    $\theta(86)$ = \a \\
    $\theta(87)$ = \b \\
    $\theta(88)$ = \c \\
    $\theta(89)$ = \d \\
    $\theta(90)$ = \e
};

\addplot [
    red,
    only marks,
    mark size=1pt
] coordinates {
    (86,\a) (87,\b) (88,\c) (89,\d) (90,\e)
};
\addlegendentry{(3) Tikz Math library results};

\end{axis}

\end{tikzpicture}

\end{document}

Produces:

For comparison I used Maple with 20 digits precision and the original (numerically not good) mathematical formula:

> sunrise(52.,86.);                                                            
                                                            2
180.00000000000000000 arccos((cos(0.28888888888888888889 Pi)

                                     2                               2 1/2
     - sin(0.13022222222222222222 Pi)  cos(0.47777777777777777778 Pi) )   /

    cos(0.28888888888888888889 Pi))/Pi

> evalf(%);
                             2.5832387643466301795

> evalf(sunrise(52.,87.));
                             1.9378307812905049847

> evalf(sunrise(52.,88.));
                             1.2920783807501711501

> evalf(sunrise(52.,89.));
                            0.64609652908098415201

One can see that with the better formula TikZ achieves about 3 or 4 digits accuracy which is enough for plotting.

Here is Maple with the better formula (and Digits set to 20):

> sunrise2:=(d,x)-> arcsin(sin(conv*23.44)*cos(conv*x)/cos(conv*d))/conv;
                                        sin(conv 23.44) cos(conv x)
                                 arcsin(---------------------------)
                                                cos(conv d)
           sunrise2 := (d, x) -> -----------------------------------
                                                conv

> evalf(sunrise2(52.,86.));                                              
                             2.5832387643466301725

> evalf(sunrise2(52.,87.));
                             1.9378307812905049928

> evalf(sunrise2(52.,88.));
                             1.2920783807501711638

> evalf(sunrise2(52.,89.));
                            0.64609652908098417757

With xfp used within declare function.

\documentclass[a4paper]{article}
\usepackage{tikz}
\usepackage[margin=0.3in]{geometry}
\usepackage{pgfplots}
\pgfplotsset{width=12cm,compat=1.16}

\usetikzlibrary{math}

\newcommand\alphazero{23.44} % Earth axial tilt
\newcommand\latitude{52} % latitude in degrees

\usepackage{xintkernel}
\usepackage{xfp}

\newcommand\FVof{}
\def\FVof(#1){\pgfmathfloatvalueof{#1}}

\begin{document}

\begin{tikzpicture}[
    declare function={
        sunrise(\d,\x) = {\fpeval{acosd( sqrt( cosd(\FVof(\d))^2 
            - (sind(\alphazero)*cosd(\FVof(\x)))^2 ) / cosd(\FVof(\d)) )}};
    }
  ]

\begin{axis}[
    axis lines=left,
    align=center,
    grid=both,
    minor y tick num=4,
    title={\Large Sunrise},
    xlabel={Day of year angle $(x^{\circ})$},
    ylabel={Sunrise position $(\theta^{\circ})$},
]


% Plot (2) 
% Using pgfplots `\addplot expression' + xfp
\addplot expression [
    blue,
    domain=82:90,
    only marks,
    mark size=1pt,
    samples=90,
    variable=x,
]
{sunrise(\latitude, x)};
\addlegendentry{Using pgfplots `\textbackslash addplot expression' and xfp and
a hack to use the latter};

% Plot (3)
% Plot the Tikz Math library results
\tikzmath {
    \a = sunrise(\latitude, 86);
    \b = sunrise(\latitude, 87);
    \c = sunrise(\latitude, 88);
    \d = sunrise(\latitude, 89);
    \e = sunrise(\latitude, 90);
}

\path (axis cs:82.7,0.3) node[draw,fill=white,inner sep=3pt,anchor=south west,align=left] {
    \em xfp results : \\
    $\theta(86)$ = \a \\
    $\theta(87)$ = \b \\
    $\theta(88)$ = \c \\
    $\theta(89)$ = \d \\
    $\theta(90)$ = \e
};

\addplot [
    red,
    only marks,
    mark size=1pt
] coordinates {
    (86,\a) (87,\b) (88,\c) (89,\d) (90,\e)
};
\addlegendentry{(3) a selection of xfp results};

\end{axis}

\end{tikzpicture}

\end{document}

With xint, but we needed to code sin, cos using high level interface. The most delicate is probably asind which I bothered defining only for usage with a small argument as enough here. I use for this illustration the "better" formula, because that was already quite an hurdle to implement by hand at high level the needed functions...

Compare the results with those of Maple (which was set at 20 digits of precision, whereas here we target only 16 digits) given above. Not bad...

I removed this code to make room for newer one, below.

I have now redone acosd/asind for xint differently, using Newton iteration rather than a series. This allows to cover the whole range.

I tested both with the "BAD" formula with acosd and the "GOOD" formula with asind. On the left xfp, on the right xint.

See code comments, for doing the plots the quantities depending only on latitude are pre-computed, but the full formula (compilation is not much longer) is also given, commented out.

\documentclass[tikz, border=12pt]{standalone}
\usepackage[T1]{fontenc}
\usepackage{tikz}
%\usepackage[margin=0.3in]{geometry}
\usepackage{pgfplots}
\pgfplotsset{width=12cm,compat=1.16}

\usetikzlibrary{math}

\newcommand\alphazero{23.44} % Earth axial tilt
\newcommand\latitude{52} % latitude in degrees

\usepackage{xfp}
\usepackage{xintexpr}

\newcommand\FVof{}
\def\FVof(#1){\pgfmathfloatvalueof{#1}}

\xintverbosetrue

% WARNING WARNING WARNING WARNING
% I HAVE TESTED ***** NOTHING ***** OF THE FOLLOWING
% EXCEPT THAT IT WORKS FOR THE CURRENT APPLICATION.
% THERE MAY BE TRIVIAL ERRORS

% \xintdeffloatvar Pi      := 3.141592653589793;

\xintdeffloatvar Piover2 := 1.570796326794897;

% \xintdeffloatvar Piover4 := 0.7853981633974483;

\xintdeffloatvar Piover180 := 0.01745329251994330;

\xintdeffloatvar invPiover180 := 57.2957795130823;

% pre-compute 1/n! for n = 1, 2, ..., 20
\xintdeffloatvar invfact\xintListWithSep{, invfact}{\xintSeq{1}{20}}
                 := seq(1/i!, i = 1..20);

% should I rather use successive divisions by (2n+1)(2n), or rather
% multiplication by their precomputed inverses, in a modified Horner scheme ?
\xintdeffloatfunc sinaux(X) := 1 - X(invfact3 - X(invfact5 - X(invfact7
                                 - X(invfact9 - X(invfact11 - X(invfact13
                                 - X(invfact15 - X * invfact17)))))));

\xintdeffloatfunc cosaux(X) := 1 - X(invfact2 - X(invfact4 - X(invfact6
                                 - X(invfact8 - X(invfact10 - X(invfact12
                                 - X(invfact14 - X(invfact16 - X * invfact18))))))));

% use this only between -pi/4 and pi/4
\xintdeffloatfunc sinAA(x) := x * sinaux(sqr(x));

% use this only between -pi/4 and pi/4
\xintdeffloatfunc cosAA(x) := cosaux(sqr(x));

% Use 1 - 2 sin(x/2)^2 formula for better cos(x) in -pi/4 < x < pi/4 ?
% And use 2 sin(x/2) cos(x/2) for sine ?

% WARNING WARNING WARNING WARNING
% I HAVE TESTED ***** NOTHING ***** OF THE FOLLOWING
% EXCEPT THAT IT WORKS FOR THE CURRENT APPLICATION.
% THERE MAY BE TRIVIAL ERRORS

\xintdeffloatfunc sind_0(x) := sinAA(x * Piover180);
\xintdeffloatfunc sind_1(x) := cosAA((x - 90) * Piover180);
\xintdeffloatfunc sind_2(x) := sind_1(x);
\xintdeffloatfunc sind_3(x) := - sinAA((x - 180) * Piover180);
\xintdeffloatfunc sind_4(x) := sind_3(x);
\xintdeffloatfunc sind_5(x) := - cosAA((x - 270) * Piover180);
\xintdeffloatfunc sind_6(x) := sind_5(x);
\xintdeffloatfunc sind_7(x) := sinAA((x - 360) * Piover180);

\xintdeffloatfunc cosd_0(x) := cosAA(x * Piover180);
\xintdeffloatfunc cosd_1(x) := -sinAA((x - 90) * Piover180);
\xintdeffloatfunc cosd_2(x) := cosd_1(x);
\xintdeffloatfunc cosd_3(x) := -cosAA((x - 180) * Piover180);
\xintdeffloatfunc cosd_4(x) := cosd_3(x);
\xintdeffloatfunc cosd_5(x) := sinAA((x - 270) * Piover180);
\xintdeffloatfunc cosd_6(x) := cosd_5(x);
\xintdeffloatfunc cosd_7(x) := cosAA((x - 360) * Piover180);

% Not trying to define it as a genuine function due to original
% dispatch, perhaps I can succeed with other syntax, of course
% goal is to evaluate only right function.

\xintNewFunction{sind_}[1]{sind_\xinttheiiexpr num(#1)//45\relax(#1)}

\xintdeffloatfunc sind(x) := sind_(x/:360);

\xintNewFunction{cosd_}[1]{cosd_\xinttheiiexpr num(#1)//45\relax(#1)}

\xintdeffloatfunc cosd(x) := cosd_(x/:360);


% WARNING WARNING WARNING WARNING
% I HAVE TESTED ***** NOTHING ***** OF THE FOLLOWING
% EXCEPT THAT IT WORKS FOR THE CURRENT APPLICATION.
% THERE MAY BE TRIVIAL ERRORS

% Compute asin(x) via Newton method
% We do the computation via a fixed point termination,
% I sense this is not very good away (should use sinaux ?)
% but will stick with that for now.
% And of course it should be written at lower level for efficiency.
\xintNewFunction{asinAA}[1]{%
   % we assume 0<= t=#1<= 0.72 and want x with sin(x) - t = 0
   % start with x=t, and iterate x <- x + (t - sin(x))/cos(x)
   % deltas will always be positive (if t non-zero)
   iter(#1;%
        subs((D<=1e-16)? % Am I sure no neverending loop due to rounding ?
                         % I should do with increased precision 
                         % and round at end
             {break(@+D)}{@+D}, 
              D=(#1 - sinAA(@))/cosAA(@)),
        i=1++ % dummy iteration index, not used but needed by iter()
   )
}

% non-negative argument only
\xintdeffloatfunc asinA(t) := 
    if(t<0.72, asinAA(t), Piover2 - asinAA(sqrt(1-sqr(t))));
\xintdeffloatfunc asindA(t) := 
    if(t<0.72, asinAA(t)*invPiover180, 90 - asinAA(sqrt(1-sqr(t)))*invPiover180);

\xintdeffloatfunc asin(t) := sgn(t) * asinA(abs(t));
\xintdeffloatfunc asind(t) := sgn(t) * asindA(abs(t));

% acos(x), acosd(x) from asin() function

\xintdeffloatfunc acos(t) := Piover2 - asin(t);
\xintdeffloatfunc acosd(t):= 90 - asind(t);


% alphazero = 23.44
% latitude ("d") = 52

% The real 2-variable xint function with no pre-computation
% "BAD" formula deliberately used to stress-test math engines
% NOTICE THAT sind(alphazero) IS COMPUTED HERE AT TIME OF THIS DEFINITION
\xintdeffloatfunc sunriseBAD(d, x) :=
    acosd( sqrt( sqr(cosd(d)) - sqr(sind(\alphazero)*cosd(x))) / cosd(d) );

% "good" formula:
\xintdeffloatfunc sunrise(d,x) := asind(sind(\alphazero) * cosd(x) / cosd(d));

% THE "CHEATING" WAY (always advisable for plotting)
% AND PRECOMPUTE SOME COEFFICIENTS
% the axis of earth is less of a variable than latitude ;-)
\xintdeffloatvar alphazero := \alphazero;%
\xintdeffloatvar sindalphazero := sind(\alphazero);% sind(23.44)
\xintdeffloatvar cosdlatitude  := cosd(\latitude);
\xintdeffloatvar cosdlatitude2 := cosdlatitude**2;

\xintdeffloatfunc sunrisecheatBAD(d, x) :=
    acosd( sqrt( cosdlatitude2 - sqr(sindalphazero*cosd(x))) / cosdlatitude );

\xintdeffloatfunc sunrisecheat(d, x) :=
    asind(sindalphazero * cosd(x) / cosdlatitude);

% WE CAN ALSO CHEAT WITH XFP

\edef\cosdlatitudesquared{\fpeval{cosd(\latitude)^2}}
\edef\cosdlatitude       {\fpeval{cosd(\latitude)}}
\edef\sindalphazero      {\fpeval{sind(\alphazero)}}
\begin{document}

\begin{tikzpicture}[
   declare function={
% in real life plotting use the cheating variants:
     sunrisexint(\d,\x) = 
        {\xintfloateval{sunrisecheatBAD(\FVof(\d),\FVof(\x))}};
     sunrisexfp(\d,\x) = % cheating variant for xfp too
        {\fpeval{
      acosd( sqrt( \cosdlatitudesquared 
                   - (\sindalphazero*cosd(\FVof(\x)))^2 
                 ) / \cosdlatitude )}};
% functions doing it again and again : 
    %  sunrisexint(\d,\x) = 
    %     {\xintfloateval{sunrise(\FVof(\d),\FVof(\x))}};
    %  sunrisexfp(\d,\x) =
    %     {\fpeval{acosd( sqrt( cosd(\FVof(\d))^2
    %                      - (sind(\alphazero)*cosd(\FVof(\x)))^2 )
    %                / cosd(\FVof(\d)))}};
    }
  ]

\begin{axis}[
    axis lines=left,
    align=center,
    grid=both,
    minor y tick num=4,
    title={\Large Sunrise using «BAD» formula},
    xlabel={Day of year angle $(x^{\circ})$},
    ylabel={Sunrise position $(\theta^{\circ})$},
]


% Plot (1) 
% Using pgfplots `\addplot expression' + xfp
\addplot expression [
    green,
    domain=-90:0,
    only marks,
    mark size=1.5pt,
    samples=31,
    variable=x,
]
{sunrisexfp(\latitude, x)};
%\addlegendentry{Using pgfplots `\textbackslash addplot expression' with the
%  «acosd» formula via xfp};

% Plot (2) 
% Using pgfplots `\addplot expression' + xint
\addplot expression [
    blue,
    domain=0:90,
    only marks,
    mark size=1.5pt,
    samples=31,
    variable=x,
]
{sunrisexint(\latitude, x)};
%\addlegendentry{Using pgfplots `\textbackslash addplot expression' with the
%  «acosd» formula via xint};

% Plot (3)
% Plot the Tikz Math library results
\tikzmath {
    \W = sunrisexint(\latitude, 0);
    \Wa = sunrisexint(\latitude, 10);
    \Wb = sunrisexint(\latitude, 20);
    \Wc = sunrisexint(\latitude, 30);
    \Wd = sunrisexint(\latitude, 40);
    \Z = sunrisexint(\latitude, 50);
    \A = sunrisexint(\latitude, 81);
    \B = sunrisexint(\latitude, 82);
    \C = sunrisexint(\latitude, 83);
    \D = sunrisexint(\latitude, 84);
    \E = sunrisexint(\latitude, 85);
    \a = sunrisexint(\latitude, 86);
    \b = sunrisexint(\latitude, 87);
    \c = sunrisexint(\latitude, 88);
    \d = sunrisexint(\latitude, 89);
    \e = sunrisexint(\latitude, 90);
%
    \Wxfp = sunrisexfp(\latitude, 0);
    \Waxfp = sunrisexfp(\latitude, 10);
    \Wbxfp = sunrisexfp(\latitude, 20);
    \Wcxfp = sunrisexfp(\latitude, 30);
    \Wdxfp = sunrisexfp(\latitude, 40);
    \Zxfp = sunrisexfp(\latitude, 50);
    \Axfp = sunrisexfp(\latitude, 81);
    \Bxfp = sunrisexfp(\latitude, 82);
    \Cxfp = sunrisexfp(\latitude, 83);
    \Dxfp = sunrisexfp(\latitude, 84);
    \Exfp = sunrisexfp(\latitude, 85);
    \axfp = sunrisexfp(\latitude, 86);
    \bxfp = sunrisexfp(\latitude, 87);
    \cxfp = sunrisexfp(\latitude, 88);
    \dxfp = sunrisexfp(\latitude, 89);
    \exfp = sunrisexfp(\latitude, 90);
}

\path (axis cs:-75,0.3) node[draw,fill=white,inner sep=3pt,anchor=south
west,align=left] {
  \begin{tabular}{rcc}
    &xfp&xint\\
    $\theta(0)$ &\Wxfp&\W \\
    $\theta(10)$ &\Waxfp&\Wa \\
    $\theta(20)$ &\Wbxfp&\Wb \\
    $\theta(30)$ &\Wcxfp&\Wc \\
    $\theta(40)$ &\Wdxfp&\Wd \\
    $\theta(50)$&\Zxfp&\Z \\
    $\theta(81)$&\Axfp&\A \\
    $\theta(82)$&\Bxfp&\B \\
    $\theta(83)$&\Cxfp&\C \\
    $\theta(84)$&\Dxfp&\D \\
    $\theta(85)$&\Exfp&\E \\
    $\theta(86)$&\axfp&\a \\
    $\theta(87)$&\bxfp&\b \\
    $\theta(88)$&\cxfp&\c \\
    $\theta(89)$&\dxfp&\d \\
    $\theta(90)$&\exfp&\e
  \end{tabular}
};

\end{axis}

\end{tikzpicture}

% Recall this:
% \xintdeffloatfunc sunrisecheat(d, x):=
%        asind(sindalphazero * cosd(x) / cosdlatitude);

\begin{tikzpicture}[
   declare function={
% in real life plotting use the cheating variants:
     sunrisexint(\d,\x) = 
        {\xintfloateval{sunrisecheat(\FVof(\d),\FVof(\x))}};
     sunrisexfp(\d,\x) = % cheating variant for xfp too
        {\fpeval{asind(\sindalphazero*cosd(\FVof(\x))/\cosdlatitude)}};
    }
  ]

\begin{axis}[
    axis lines=left,
    align=center,
    grid=both,
    minor y tick num=4,
    title={\Large Sunrise using «GOOD» formula},
    xlabel={Day of year angle $(x^{\circ})$},
    ylabel={Sunrise position $(\theta^{\circ})$},
]


% Plot (1) 
% Using pgfplots `\addplot expression' + xfp
\addplot expression [
    green,
    domain=-90:0,
    only marks,
    mark size=1.5pt,
    samples=31,
    variable=x,
]
{sunrisexfp(\latitude, x)};
%\addlegendentry{Using pgfplots `\textbackslash addplot expression' with the
%  «acosd» formula via xfp};

% Plot (2) 
% Using pgfplots `\addplot expression' + xint
\addplot expression [
    blue,
    domain=0:90,
    only marks,
    mark size=1.5pt,
    samples=31,
    variable=x,
]
{sunrisexint(\latitude, x)};
%\addlegendentry{Using pgfplots `\textbackslash addplot expression' with the
%  «acosd» formula via xint};

% Plot (3)
% Plot the Tikz Math library results
\tikzmath {
    \W = sunrisexint(\latitude, 0);
    \Wa = sunrisexint(\latitude, 10);
    \Wb = sunrisexint(\latitude, 20);
    \Wc = sunrisexint(\latitude, 30);
    \Wd = sunrisexint(\latitude, 40);
    \Z = sunrisexint(\latitude, 50);
    \A = sunrisexint(\latitude, 81);
    \B = sunrisexint(\latitude, 82);
    \C = sunrisexint(\latitude, 83);
    \D = sunrisexint(\latitude, 84);
    \E = sunrisexint(\latitude, 85);
    \a = sunrisexint(\latitude, 86);
    \b = sunrisexint(\latitude, 87);
    \c = sunrisexint(\latitude, 88);
    \d = sunrisexint(\latitude, 89);
    \e = sunrisexint(\latitude, 90);
%
    \Wxfp = sunrisexfp(\latitude, 0);
    \Waxfp = sunrisexfp(\latitude, 10);
    \Wbxfp = sunrisexfp(\latitude, 20);
    \Wcxfp = sunrisexfp(\latitude, 30);
    \Wdxfp = sunrisexfp(\latitude, 40);
    \Zxfp = sunrisexfp(\latitude, 50);
    \Axfp = sunrisexfp(\latitude, 81);
    \Bxfp = sunrisexfp(\latitude, 82);
    \Cxfp = sunrisexfp(\latitude, 83);
    \Dxfp = sunrisexfp(\latitude, 84);
    \Exfp = sunrisexfp(\latitude, 85);
    \axfp = sunrisexfp(\latitude, 86);
    \bxfp = sunrisexfp(\latitude, 87);
    \cxfp = sunrisexfp(\latitude, 88);
    \dxfp = sunrisexfp(\latitude, 89);
    \exfp = sunrisexfp(\latitude, 90);
}

\path (axis cs:-75,0.3) node[draw,fill=white,inner sep=3pt,anchor=south
west,align=left] {
  \begin{tabular}{rcc}
    &xfp&xint\\
    $\theta(0)$ &\Wxfp&\W \\
    $\theta(10)$ &\Waxfp&\Wa \\
    $\theta(20)$ &\Wbxfp&\Wb \\
    $\theta(30)$ &\Wcxfp&\Wc \\
    $\theta(40)$ &\Wdxfp&\Wd \\
    $\theta(50)$&\Zxfp&\Z \\
    $\theta(81)$&\Axfp&\A \\
    $\theta(82)$&\Bxfp&\B \\
    $\theta(83)$&\Cxfp&\C \\
    $\theta(84)$&\Dxfp&\D \\
    $\theta(85)$&\Exfp&\E \\
    $\theta(86)$&\axfp&\a \\
    $\theta(87)$&\bxfp&\b \\
    $\theta(88)$&\cxfp&\c \\
    $\theta(89)$&\dxfp&\d \\
    $\theta(90)$&\exfp&\e
  \end{tabular}
};

\end{axis}

\end{tikzpicture}

\end{document}

Reminder of Maple results with "bad" formula:

> evalf(sunrise(52.,86.));                                              
                             2.5832387643466301795

> evalf(sunrise(52.,87.));
                             1.9378307812905049847

> evalf(sunrise(52.,88.));
                             1.2920783807501711501

> evalf(sunrise(52.,89.));
                            0.64609652908098415201

Reminder of Maple results with "good" formula:

> evalf(sunrise2(52.,86.));                                              
                             2.5832387643466301725

> evalf(sunrise2(52.,87.));
                             1.9378307812905049928

> evalf(sunrise2(52.,88.));
                             1.2920783807501711638

> evalf(sunrise2(52.,89.));
                            0.64609652908098417757

Regarding xint results, in the medium range, recall that they are not using well-thought out numerical algorithms but hand-written high level routines. In particular Newton method for asind should be done with more digits precision than the target 16 precision.

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