arrows arranging on a ring

Question

I have been involved in a challenge for some time: A ring with a number of arrows on its perimeter. The number of arrows depends on the angle $\theta$. For example, for $\theta=2\pi/36$, the number of arrows is 36. Moreover, the angle between any neighbors is exactly $\theta$. I'm very weak at using graphs and do not know what the starting point is. Any idea can be welcome...

Based on the @flinty post and @cvgmt I Updated the image. Then I can say that I wish to have cvgmt's image as the below one. Please look the new desired blue arrows. I wish to have blue arrows that drawn by me with paint.

Answer 1

n = 36;
len = .25;
points2d = N[CirclePoints[n]];
points3d = Append[#, 0] & /@ points2d;
tangents3d = Append[#, 0] & /@ (Cross /@ points2d);

arrows = MapThread[Arrow[{#1 - len #2, #1 + len #2}] &,
 {points3d, tangents3d}];

(* spin the arrows as we go around the circle *)
arrowsRotated = MapIndexed[
   With[{i = First[#2]},
     With[{p = points3d[[i]]},
      Rotate[#1, 2 Pi (i - 1)/n, p, p]]] &
   , arrows];

Graphics3D[{{Black, Thick, 
   ResourceFunction["Circle3D"][{0, 0, 0}, {1, 1}, Pi/2, 0]},
  Red, arrowsRotated}, Boxed -> False]

Answer 2

Reply to the comment

It means that we need to set f[s]:=-s ?

R = 8;
r[s_] := R {Cos[s], Sin[s], 0};
n[s_] := -r[s] // Normalize;
t[s_] := r'[s] // Normalize;
b[s_] := Cross[t[s], n[s]] // Normalize;
circle3 = 
  ParametricPlot3D[r[s], {s, 0, 2 π}, 
   PlotStyle -> {Opacity[.5], Green}];
d = 1;
f[s_] := -s;
Show[{circle3, 
  Table[Graphics3D[{Arrowheads[.01], Arrow[{r[s], r[s] + d*t[s]}], 
     Arrow[{r[s], r[s] + d*b[s]}], Red, 
     Arrow[{r[s] - d*(Cos[θ]*t[s] + Sin[θ]*b[s]), 
        r[s] + d*(Cos[θ]*t[s] + 
            Sin[θ]*b[s])} /. θ -> f[s]]}], {s, 0, 
    2 π, 2 π/36}]}, ViewProjection -> "Orthographic", 
 Boxed -> False, Axes -> False, PlotRange -> R + 1, 
 ImageSize -> Large, ViewPoint -> {0.60, -2.90, 1.62}]

Updated

Here we use Frenet-Serret frame $\{r(s): t(s),n(s),b(s)\}$ and set a rotation speed function f[s] so we can adjust such function to get another result.

R = 8;
r[s_] := R {Cos[s], Sin[s], 0};
n[s_] := -r[s] // Normalize;
t[s_] := r'[s] // Normalize;
b[s_] := Cross[t[s], n[s]] // Normalize;
circle3 = 
  ParametricPlot3D[r[s], {s, 0, 2 π}, PlotStyle -> Green];
d = 2;
f[s_] := s + π/2;
Show[{circle3, 
  Table[Graphics3D[{Arrowheads[.02], Red, 
     Arrow[{r[s] - d*(Cos[θ]*t[s] + Sin[θ]*b[s]), 
        r[s] + d*(Cos[θ]*t[s] + 
            Sin[θ]*b[s])} /. θ -> f[s]]}], {s, 0, 
    2 π, 2 π/36}]}, ViewProjection -> "Orthographic", 
 Boxed -> False, Axes -> True, PlotRange -> R + 1]

Original

We can rotate a line around the z-axis {0,0,1} with angle λ (2 π)/n where 0<λ<1 ( such line through the point {1,0,0} with direction dir)

n = 36;
λ = 0.8;
sol = NMaximize[{0, 
    VectorAngle[
      RotationTransform[((2 π)/n)*1, {0, 0, 1}][{x, y, z}], 
      RotationTransform[((2 π)/n)*2, {0, 0, 1}][{x, y, 
        z}]] == λ (2 π)/n, x^2 + y^2 + z^2 == 1, z > 0, 
    y > 0}, {x, y, z}];
dir = {x, y, z} /. sol[[2]];
circle3 = 
  ParametricPlot3D[{Cos[t], Sin[t], 0}, {t, 0, 2 π}, 
   PlotStyle -> Green];
Show[Graphics3D[{Arrowheads[0.02], 
   Table[GeometricTransformation[{{Red, PointSize[Large], 
       Point[{1, 0, 0}]}, 
      Arrow[{{1, 0, 0} - .3 dir, {1, 0, 0} + .3 dir}]}, 
     RotationTransform[((2 π)/n)*k, {0, 0, 1}]], {k, n}]}, 
  BoxRatios -> Automatic], circle3, Boxed -> False, Axes -> False, 
 ViewPoint -> {-1.39, -2.93, 0.97}]

Answer 3

Graphics3D[ Table[{Hue[t/(2 [Pi])], Arrow [{{Cos[t], Sin[t], 0}, {(Cos[t] - Sin[t])/ Sqrt[1 + Cos[t]^2], (Sin[t] + Cos[t])/Sqrt[1 + Cos[t]^2], Cos[t]/Sqrt[1 + Cos[t]^2]}}]}, {t, 0, 2 [Pi], [Pi]/16}], Boxed -> False]