Re: question on determining algebraic numbers for rotations in 2d with rational cosines.

[Michael Day <mike_liz.day@tiscali.co.uk>]

"American Citizen" posted a query a week or two ago, copied below.

I expect this has been obvious to most Pari users,  but in case not,  
here's my take.
It's easier to just consider rotation of a point with rational 
coordinates about the origin, [0,0]
Let the starting point be rational [x,y], ie [-1/6,-1/10] = [1/3,2/5] - 
[1/2,1/2]
and let the rotation angle be  cos^-1 rational r.   Let s be its sine,  
usually irrational,
and let r2 = r^2, s2 = s^2 = 1 - r2, since sin^2 + cos^2 = 1
Just dealing for now with x1,  the x-value of the rotated point,  we have
    x1 = x * r - y * s
and so its square is
   x1^2 = x^2 * r2 - 2*x*y*r*s + y^2 * s2
The middle term is likely to be irrational, so we remove it as follows:
   x1^2 - 2 * x1 * r * x  = - x^2 * r2  + y^2 * s2
yielding this quadratic in x1:
   x1^2 - 2 * x1 * r * x  + x^2 * r2  - y^2 * s2 = 0

Choosing the example from below for arccos 1/4,  American Citizen 
reports these 2 quartics:

   1/4 | 34*x^4 - 52*x^3 - 4*x^2 - 29*x + 23))  [1] | 68*x^4 - 28*x^3 + 
27*x^2 - 11*x + 1))  [2]

my x1-equation becomes 1440 * x1^2 + 120 * x1 - 11 = 0
(17:54) gp > polroots(1440*x^2 +120 * x - 11)
%56 = [-0.138491 + 0.E-19*I, 0.0551579 + 0.E-19*I]~
Restoring to the original origin by adding 1/2,  the second root becomes 
~ 0.555158, consistent
with American Citizen's quartic root:
(17:54) gp > polroots(34*x^4 - 52*x^3 - 4*x^2 - 29*x + 23)[1]
%57 = 0.555158 + 0.E-19*I
I don't see why he needs quartics in some cases since the above 
procedure for removing the
irrational term is generally applicable provided the coordinates and the 
cosine are rational;  I don't
think rotation about a point offset from the origin should demand 4th 
powers - but I haven't tried that
slightly more complicated problem directly!

Thanks,
Mike


On 30/07/2025 03:01, American Citizen wrote:
> I forgot to mention the all important root, it is given as [1] or [2], 
> etc. as the nth root of the polroots() command
>
> They are all given on the row for the particular rational cosine.
>
> On 7/29/25 18:59, American Citizen wrote:
> > Suppose we place a point on the x-y plane in the Cartesian 2d system. 
> > We then rotate this point counterclockwise in such a way that the 
> > rotation angle cosine is rational. (the sine may or may not be 
> > rational).
> >
> > My question is specifically, how do we predict these 4 degree 
> > equations and their coefficients, depending upon the rational cosines?
> >
> > Original Point [1/3, 2/5]
> > Center = [1/2, 1/2]
> > acos(angle)
> > point is rotated ccw on the x-y plane.
> > ----------------------------------------------------------------
> > acos        x-coordinate                      y-coordinate
> > ----------------------------------------------------------------
> > 1/2 | 1800*x^2 - 1500*x + 299 [2] | 600*x^2 - 540*x + 109   [1]
> > 1/3 | 72*x^4 - 6*x^3 + 33*x^2 - 44*x + 9))   [2] | 32*x^4 + 53*x^3 - 
> > 56*x^2 - 21*x + 10)) [3]
> > 2/3 | 405*x^2 - 315*x + 59 [2] | 2025*x^2 - 1755*x + 349 [1]
> > 1/4 | 34*x^4 - 52*x^3 - 4*x^2 - 29*x + 23))  [1] | 68*x^4 - 28*x^3 + 
> > 27*x^2 - 11*x + 1))  [2]
> > 3/4 | 800*x^2 - 600*x + 109   [2] | 7200*x^2 - 6120*x + 1213  [1]
> > 1/5 | 31*x^4 - 54*x^3 - 37*x^2 - 33*x + 37)) [1] | 46*x^4 + 29*x^3 + 
> > 4*x^2 - 53*x + 15))  [2]
> > 2/5 | 61*x^4 - 46*x^3 - 4*x^2 + 25*x - 10))  [2] | 31*x^4 - 27*x^3 + 
> > 42*x^2 - 8*x - 1))   [2]
> > 3/5 | [12/25, 23/75] | 25*x - 12 | 75*x - 23
> > 4/5 | [32/75, 8/25] | 75*x - 32 | 25*x - 8
> > 1/6 | 15*x^4 + 49*x^3 + 7*x^2 - 21*x - 1))   [4] | 3*x^4 - 28*x^3 + 
> > 51*x^2 - 48*x + 11))  [1]
> > 5/6 | 16200*x^2 - 11700*x + 2063 [2] | 648*x^2 - 540*x + 107  [1]
> > 1/7 | 48*x^4 + 9*x^3 - 63*x^2 + 5*x + 11))   [3] | 8*x^4 + 38*x^3 + 
> > 84*x^2 + 25*x - 18))  [2]
> > 2/7 | 29*x^4 + 21*x^3 - 8*x^2 - 14*x + 4))   [2] | 48*x^4 + 24*x^3 - 
> > 29*x^2 + 63*x - 18)) [2]
> > 3/7 | 245*x^2 - 210*x + 43 [2] | 11025*x^2 - 10080*x + 2054 [1]
> > 4/7 | 11025*x^2 - 8925*x + 1732  [2] | 3675*x^2 - 3255*x + 652 [1]
> > 5/7 | 11025*x^2 - 8400*x + 1546  [2] | 147*x^2 - 126*x + 25 [1]
> > 6/7 | 1225*x^2 - 875*x + 153  [2] | 11025*x^2 - 9135*x + 1811 [1]
> > 1/8 | 12*x^4 + 9*x^3 + 76*x^2 - 32*x - 10))  [2] | 21*x^4 + 42*x^3 - 
> > 4*x^2 + 49*x - 17))  [2]
> > 3/8 | 640*x^2 - 560*x + 117   [2] | 28800*x^2 - 26640*x + 5473 [1]
> > 5/8 | 28800*x^2 - 22800*x + 4337 [2] | 384*x^2 - 336*x + 67 [1]
> > 7/8 | 55*x^4 + 30*x^3 + 19*x^2 + 46*x - 25 [2] | 25*x^4 - 71*x^3 + 
> > 28*x - 7 [2]
> > 1/9 | 12*x^4 - 9*x^3 - 28*x^2 + 29*x - 7))   [3] | 38*x^4 + 7*x^3 + 
> > 16*x^2 - x - 2))   [2]
> > 2/9 | 51*x^4 - 35*x^3 + 10*x^2 - 25*x + 12)) [1] | 54*x^4 + 23*x^3 - 
> > 9*x^2 - 17*x + 5))   [1]
> > 4/9 | 3645*x^2 - 3105*x + 632 [2] | 18225*x^2 - 16605*x + 3376 [1]
> > 5/9 | 18225*x^2 - 14850*x + 2899 [2] | 729*x^2 - 648*x + 130  [1]
> > 7/9 | 18225*x^2 - 13500*x + 2428 [2] | 18225*x^2 - 15390*x + 3049 [1]
> > 8/9 | 18225*x^2 - 12825*x + 2218 [2] | 18225*x^2 - 14985*x + 2974 [1]
> > 1/10 | 25*x^4 + 24*x^3 - 10*x^2 - 21*x + 8)) [2] | 49*x^4 + 10*x^3 - 
> > 20*x^2 - 24*x + 9))  [1]
> > 3/10 | 5000*x^2 - 4500*x + 967   [2] | 45000*x^2 - 42300*x + 8803 [1]
> > 7/10 | 45000*x^2 - 34500*x + 6383   [2] | 15000*x^2 - 12900*x + 2561 [1]
> > 9/10 | 5000*x^2 - 3500*x + 603   [2] | 45000*x^2 - 36900*x + 7327 [1]
> >
> > - Randall
> >
> > P.S. This will apply in 3d space, when we rotate on a plane.

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