Sphere via rotations of longitude line

'πœ—deg'=slider([0,360,24])

'πœ‘deg'=slider([0,180,12])

'πœ—'='πœ—deg'*degree,'πœ‘'='πœ‘deg'*degree

'lineR'=slider([0,0.1,20])

'𝐎'=vector(0,0,0)

function('𝕉'_'xy','πœƒ')=matrix(3,3,cos('πœƒ'),-sin('πœƒ'),0,sin('πœƒ'),cos('πœƒ'),0,0,0,1),function('𝕉'_'yz','πœƒ')=matrix(3,3,1,0,0,0,cos('πœƒ'),-sin('πœƒ'),0,sin('πœƒ'),cos('πœƒ')),function('𝕉'_'zx','πœƒ')=matrix(3,3,cos('πœƒ'),0,-sin('πœƒ'),0,1,0,sin('πœƒ'),0,cos('πœƒ'))

The combined rotation transformation:

function('𝕋','πœ—','πœ‘')=matrix(3,3,cos('πœ—')*cos('πœ‘'),-sin('πœ—'),-(cos('πœ—')*sin('πœ‘')),sin('πœ—')*cos('πœ‘'),cos('πœ—'),-(sin('πœ—')*sin('πœ‘')),sin('πœ‘'),0,cos('πœ‘'))

function('Cube',u,v,k,l)=branch(if(vector(u,v,k),l=1),if(vector(k,u,v),l=2),if(vector(v,k,u),l=3))

function('Sphere',r,'πœ—','πœ‘')=function('𝕉'_'xy','πœ—')*function('𝕉'_'zx','πœ‘')*r*hat(x)

––––––––––––––– Plots –––––––––––––
Longitudes and latitudes:

function('Sphere',1,k-(n*degree),2*pi*t),'radius'='lineR',in(k,set(0*degree,15*degree,ldots,165*degree)),vector(h,s,v)=vector(k/pi,0.5,1)

function('Sphere',1,pi*t-(n*degree),k),'radius'='lineR',in(k,set(0*degree,15*degree,ldots,345*degree))

The surface of the sphere:

function('Sphere',1,2*pi*u-(n*degree),-(90*degree)+pi*v),leq(2*pi*u,'πœ—'),leq(pi*v,'πœ‘')

What is the effect of the combined rotations, before we multiply by unit vector [1 0 0]?
It puts the cube on the sphere at the correct location and with the correct orientation!
The azimuth (longitude) and latitude at which to slide the cube around the globe:

'𝛼deg'=slider([0,360,12])

'πœ†deg'=slider([0,360,24])

'cubeS'=slider([0,1,50])

'𝛼'='𝛼deg'*degree,'πœ†'='πœ†deg'*degree

Axes on the unit cube, to show its orientation:

function('𝕋','𝛼'-(n*degree),'πœ†')*[hat(x)+'cubeS'*function('Cube',0,0,0,1)],function('𝕋','𝛼'-(n*degree),'πœ†')*[hat(x)+'cubeS'*function('Cube',1,0,0,1)],'radius'='lineR'

function('𝕋','𝛼'-(n*degree),'πœ†')*[hat(x)+'cubeS'*function('Cube',0,0,0,1)],function('𝕋','𝛼'-(n*degree),'πœ†')*[hat(x)+'cubeS'*function('Cube',0,1,0,1)],'radius'='lineR'

function('𝕋','𝛼'-(n*degree),'πœ†')*[hat(x)+'cubeS'*function('Cube',0,0,0,1)],function('𝕋','𝛼'-(n*degree),'πœ†')*[hat(x)+'cubeS'*function('Cube',0,0,1,1)],'radius'='lineR'

function('𝕋','𝛼'-(n*degree),'πœ†')*[hat(x)+'cubeS'*function('Cube',u,v,k,l)],in(k,set(0,1)),in(l,set(1,ldots,3))

Graph of the formula

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